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In this paper, we establish new generalizations and results in shift-invariant subspaces of mixed-norm Lebesgue spaces <inline-formula><math display="inline"><semantics><mrow><msub><mi>L</mi><mover accent="true"><mi>p</mi><mo stretchy="false">→</mo></mover></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. We obtain a mixed-norm Hölder inequality, a mixed-norm Minkowski inequality, a mixed-norm convolution inequality, a convolution-Hölder type inequality and a stability theorem to mixed-norm case in the setting of shift-invariant subspace of <inline-formula><math display="inline"><semantics><mrow><msub><mi>L</mi><mover accent="true"><mi>p</mi><mo stretchy="false">→</mo></mover></msub><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>d</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Our new results unify and refine the existing results in the literature.