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Theorem lmhmpropd 19776
Description: Module homomorphism depends only on the module attributes of structures. (Contributed by Mario Carneiro, 8-Oct-2015.)
Hypotheses
Ref Expression
lmhmpropd.a (𝜑𝐵 = (Base‘𝐽))
lmhmpropd.b (𝜑𝐶 = (Base‘𝐾))
lmhmpropd.c (𝜑𝐵 = (Base‘𝐿))
lmhmpropd.d (𝜑𝐶 = (Base‘𝑀))
lmhmpropd.1 (𝜑𝐹 = (Scalar‘𝐽))
lmhmpropd.2 (𝜑𝐺 = (Scalar‘𝐾))
lmhmpropd.3 (𝜑𝐹 = (Scalar‘𝐿))
lmhmpropd.4 (𝜑𝐺 = (Scalar‘𝑀))
lmhmpropd.p 𝑃 = (Base‘𝐹)
lmhmpropd.q 𝑄 = (Base‘𝐺)
lmhmpropd.e ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐽)𝑦) = (𝑥(+g𝐿)𝑦))
lmhmpropd.f ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦))
lmhmpropd.g ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐽)𝑦) = (𝑥( ·𝑠𝐿)𝑦))
lmhmpropd.h ((𝜑 ∧ (𝑥𝑄𝑦𝐶)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝑀)𝑦))
Assertion
Ref Expression
lmhmpropd (𝜑 → (𝐽 LMHom 𝐾) = (𝐿 LMHom 𝑀))
Distinct variable groups:   𝑥,𝑦,𝐶   𝑥,𝐽,𝑦   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝑥,𝑀,𝑦   𝑥,𝑃,𝑦   𝜑,𝑥,𝑦   𝑥,𝐵,𝑦   𝑥,𝑄,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem lmhmpropd
Dummy variables 𝑧 𝑤 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmhmpropd.a . . . . . 6 (𝜑𝐵 = (Base‘𝐽))
2 lmhmpropd.c . . . . . 6 (𝜑𝐵 = (Base‘𝐿))
3 lmhmpropd.e . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐽)𝑦) = (𝑥(+g𝐿)𝑦))
4 lmhmpropd.1 . . . . . 6 (𝜑𝐹 = (Scalar‘𝐽))
5 lmhmpropd.3 . . . . . 6 (𝜑𝐹 = (Scalar‘𝐿))
6 lmhmpropd.p . . . . . 6 𝑃 = (Base‘𝐹)
7 lmhmpropd.g . . . . . 6 ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐽)𝑦) = (𝑥( ·𝑠𝐿)𝑦))
81, 2, 3, 4, 5, 6, 7lmodpropd 19628 . . . . 5 (𝜑 → (𝐽 ∈ LMod ↔ 𝐿 ∈ LMod))
9 lmhmpropd.b . . . . . 6 (𝜑𝐶 = (Base‘𝐾))
10 lmhmpropd.d . . . . . 6 (𝜑𝐶 = (Base‘𝑀))
11 lmhmpropd.f . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦))
12 lmhmpropd.2 . . . . . 6 (𝜑𝐺 = (Scalar‘𝐾))
13 lmhmpropd.4 . . . . . 6 (𝜑𝐺 = (Scalar‘𝑀))
14 lmhmpropd.q . . . . . 6 𝑄 = (Base‘𝐺)
15 lmhmpropd.h . . . . . 6 ((𝜑 ∧ (𝑥𝑄𝑦𝐶)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝑀)𝑦))
169, 10, 11, 12, 13, 14, 15lmodpropd 19628 . . . . 5 (𝜑 → (𝐾 ∈ LMod ↔ 𝑀 ∈ LMod))
178, 16anbi12d 630 . . . 4 (𝜑 → ((𝐽 ∈ LMod ∧ 𝐾 ∈ LMod) ↔ (𝐿 ∈ LMod ∧ 𝑀 ∈ LMod)))
187oveqrspc2v 7172 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝑃𝑤𝐵)) → (𝑧( ·𝑠𝐽)𝑤) = (𝑧( ·𝑠𝐿)𝑤))
1918adantlr 711 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (𝑧( ·𝑠𝐽)𝑤) = (𝑧( ·𝑠𝐿)𝑤))
2019fveq2d 6668 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑓‘(𝑧( ·𝑠𝐿)𝑤)))
21 simpll 763 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝜑)
22 simprl 767 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝑧𝑃)
23 simplrr 774 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝐺 = 𝐹)
2423fveq2d 6668 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (Base‘𝐺) = (Base‘𝐹))
2524, 14, 63eqtr4g 2881 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝑄 = 𝑃)
2622, 25eleqtrrd 2916 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝑧𝑄)
27 simplrl 773 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝑓 ∈ (𝐽 GrpHom 𝐾))
28 eqid 2821 . . . . . . . . . . . . . 14 (Base‘𝐽) = (Base‘𝐽)
29 eqid 2821 . . . . . . . . . . . . . 14 (Base‘𝐾) = (Base‘𝐾)
3028, 29ghmf 18302 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐽 GrpHom 𝐾) → 𝑓:(Base‘𝐽)⟶(Base‘𝐾))
3127, 30syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝑓:(Base‘𝐽)⟶(Base‘𝐾))
32 simprr 769 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝑤𝐵)
3321, 1syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝐵 = (Base‘𝐽))
3432, 33eleqtrd 2915 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝑤 ∈ (Base‘𝐽))
3531, 34ffvelrnd 6845 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (𝑓𝑤) ∈ (Base‘𝐾))
3621, 9syl 17 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝐶 = (Base‘𝐾))
3735, 36eleqtrrd 2916 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (𝑓𝑤) ∈ 𝐶)
3815oveqrspc2v 7172 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝑄 ∧ (𝑓𝑤) ∈ 𝐶)) → (𝑧( ·𝑠𝐾)(𝑓𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤)))
3921, 26, 37, 38syl12anc 832 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (𝑧( ·𝑠𝐾)(𝑓𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤)))
4020, 39eqeq12d 2837 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → ((𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤)) ↔ (𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤))))
41402ralbidva 3198 . . . . . . 7 ((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) → (∀𝑧𝑃𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤)) ↔ ∀𝑧𝑃𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤))))
4241pm5.32da 579 . . . . . 6 (𝜑 → (((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹) ∧ ∀𝑧𝑃𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤))) ↔ ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹) ∧ ∀𝑧𝑃𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤)))))
43 df-3an 1081 . . . . . 6 ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹 ∧ ∀𝑧𝑃𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤))) ↔ ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹) ∧ ∀𝑧𝑃𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤))))
44 df-3an 1081 . . . . . 6 ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹 ∧ ∀𝑧𝑃𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤))) ↔ ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹) ∧ ∀𝑧𝑃𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤))))
4542, 43, 443bitr4g 315 . . . . 5 (𝜑 → ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹 ∧ ∀𝑧𝑃𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤))) ↔ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹 ∧ ∀𝑧𝑃𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤)))))
4612, 4eqeq12d 2837 . . . . . 6 (𝜑 → (𝐺 = 𝐹 ↔ (Scalar‘𝐾) = (Scalar‘𝐽)))
474fveq2d 6668 . . . . . . . 8 (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐽)))
486, 47syl5eq 2868 . . . . . . 7 (𝜑𝑃 = (Base‘(Scalar‘𝐽)))
491raleqdv 3416 . . . . . . 7 (𝜑 → (∀𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤)) ↔ ∀𝑤 ∈ (Base‘𝐽)(𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤))))
5048, 49raleqbidv 3402 . . . . . 6 (𝜑 → (∀𝑧𝑃𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤)) ↔ ∀𝑧 ∈ (Base‘(Scalar‘𝐽))∀𝑤 ∈ (Base‘𝐽)(𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤))))
5146, 503anbi23d 1430 . . . . 5 (𝜑 → ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹 ∧ ∀𝑧𝑃𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤))) ↔ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ (Scalar‘𝐾) = (Scalar‘𝐽) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐽))∀𝑤 ∈ (Base‘𝐽)(𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤)))))
521, 9, 2, 10, 3, 11ghmpropd 18336 . . . . . . 7 (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀))
5352eleq2d 2898 . . . . . 6 (𝜑 → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐿 GrpHom 𝑀)))
5413, 5eqeq12d 2837 . . . . . 6 (𝜑 → (𝐺 = 𝐹 ↔ (Scalar‘𝑀) = (Scalar‘𝐿)))
555fveq2d 6668 . . . . . . . 8 (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐿)))
566, 55syl5eq 2868 . . . . . . 7 (𝜑𝑃 = (Base‘(Scalar‘𝐿)))
572raleqdv 3416 . . . . . . 7 (𝜑 → (∀𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤)) ↔ ∀𝑤 ∈ (Base‘𝐿)(𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤))))
5856, 57raleqbidv 3402 . . . . . 6 (𝜑 → (∀𝑧𝑃𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤)) ↔ ∀𝑧 ∈ (Base‘(Scalar‘𝐿))∀𝑤 ∈ (Base‘𝐿)(𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤))))
5953, 54, 583anbi123d 1427 . . . . 5 (𝜑 → ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹 ∧ ∀𝑧𝑃𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤))) ↔ (𝑓 ∈ (𝐿 GrpHom 𝑀) ∧ (Scalar‘𝑀) = (Scalar‘𝐿) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐿))∀𝑤 ∈ (Base‘𝐿)(𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤)))))
6045, 51, 593bitr3d 310 . . . 4 (𝜑 → ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ (Scalar‘𝐾) = (Scalar‘𝐽) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐽))∀𝑤 ∈ (Base‘𝐽)(𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤))) ↔ (𝑓 ∈ (𝐿 GrpHom 𝑀) ∧ (Scalar‘𝑀) = (Scalar‘𝐿) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐿))∀𝑤 ∈ (Base‘𝐿)(𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤)))))
6117, 60anbi12d 630 . . 3 (𝜑 → (((𝐽 ∈ LMod ∧ 𝐾 ∈ LMod) ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ (Scalar‘𝐾) = (Scalar‘𝐽) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐽))∀𝑤 ∈ (Base‘𝐽)(𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤)))) ↔ ((𝐿 ∈ LMod ∧ 𝑀 ∈ LMod) ∧ (𝑓 ∈ (𝐿 GrpHom 𝑀) ∧ (Scalar‘𝑀) = (Scalar‘𝐿) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐿))∀𝑤 ∈ (Base‘𝐿)(𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤))))))
62 eqid 2821 . . . 4 (Scalar‘𝐽) = (Scalar‘𝐽)
63 eqid 2821 . . . 4 (Scalar‘𝐾) = (Scalar‘𝐾)
64 eqid 2821 . . . 4 (Base‘(Scalar‘𝐽)) = (Base‘(Scalar‘𝐽))
65 eqid 2821 . . . 4 ( ·𝑠𝐽) = ( ·𝑠𝐽)
66 eqid 2821 . . . 4 ( ·𝑠𝐾) = ( ·𝑠𝐾)
6762, 63, 64, 28, 65, 66islmhm 19730 . . 3 (𝑓 ∈ (𝐽 LMHom 𝐾) ↔ ((𝐽 ∈ LMod ∧ 𝐾 ∈ LMod) ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ (Scalar‘𝐾) = (Scalar‘𝐽) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐽))∀𝑤 ∈ (Base‘𝐽)(𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤)))))
68 eqid 2821 . . . 4 (Scalar‘𝐿) = (Scalar‘𝐿)
69 eqid 2821 . . . 4 (Scalar‘𝑀) = (Scalar‘𝑀)
70 eqid 2821 . . . 4 (Base‘(Scalar‘𝐿)) = (Base‘(Scalar‘𝐿))
71 eqid 2821 . . . 4 (Base‘𝐿) = (Base‘𝐿)
72 eqid 2821 . . . 4 ( ·𝑠𝐿) = ( ·𝑠𝐿)
73 eqid 2821 . . . 4 ( ·𝑠𝑀) = ( ·𝑠𝑀)
7468, 69, 70, 71, 72, 73islmhm 19730 . . 3 (𝑓 ∈ (𝐿 LMHom 𝑀) ↔ ((𝐿 ∈ LMod ∧ 𝑀 ∈ LMod) ∧ (𝑓 ∈ (𝐿 GrpHom 𝑀) ∧ (Scalar‘𝑀) = (Scalar‘𝐿) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝐿))∀𝑤 ∈ (Base‘𝐿)(𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤)))))
7561, 67, 743bitr4g 315 . 2 (𝜑 → (𝑓 ∈ (𝐽 LMHom 𝐾) ↔ 𝑓 ∈ (𝐿 LMHom 𝑀)))
7675eqrdv 2819 1 (𝜑 → (𝐽 LMHom 𝐾) = (𝐿 LMHom 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3138  wf 6345  cfv 6349  (class class class)co 7145  Basecbs 16473  +gcplusg 16555  Scalarcsca 16558   ·𝑠 cvsca 16559   GrpHom cghm 18295  LModclmod 19565   LMHom clmhm 19722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-sdom 8501  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-plusg 16568  df-0g 16705  df-mgm 17842  df-sgrp 17891  df-mnd 17902  df-mhm 17946  df-grp 18046  df-ghm 18296  df-mgp 19171  df-ur 19183  df-ring 19230  df-lmod 19567  df-lmhm 19725
This theorem is referenced by:  phlpropd  20729
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