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Theorem lmhmpropd 19275
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 19128 . . . . 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 19128 . . . . 5 (𝜑 → (𝐾 ∈ LMod ↔ 𝑀 ∈ LMod))
178, 16anbi12d 749 . . . 4 (𝜑 → ((𝐽 ∈ LMod ∧ 𝐾 ∈ LMod) ↔ (𝐿 ∈ LMod ∧ 𝑀 ∈ LMod)))
187oveqrspc2v 6836 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝑃𝑤𝐵)) → (𝑧( ·𝑠𝐽)𝑤) = (𝑧( ·𝑠𝐿)𝑤))
1918adantlr 753 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (𝑧( ·𝑠𝐽)𝑤) = (𝑧( ·𝑠𝐿)𝑤))
2019fveq2d 6356 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑓‘(𝑧( ·𝑠𝐿)𝑤)))
21 simpll 807 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝜑)
22 simprl 811 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝑧𝑃)
23 simplrr 820 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝐺 = 𝐹)
2423fveq2d 6356 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (Base‘𝐺) = (Base‘𝐹))
2524, 14, 63eqtr4g 2819 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝑄 = 𝑃)
2622, 25eleqtrrd 2842 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝑧𝑄)
27 simplrl 819 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝑓 ∈ (𝐽 GrpHom 𝐾))
28 eqid 2760 . . . . . . . . . . . . . 14 (Base‘𝐽) = (Base‘𝐽)
29 eqid 2760 . . . . . . . . . . . . . 14 (Base‘𝐾) = (Base‘𝐾)
3028, 29ghmf 17865 . . . . . . . . . . . . 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 813 . . . . . . . . . . . . 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 2841 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝑤 ∈ (Base‘𝐽))
3531, 34ffvelrnd 6523 . . . . . . . . . . 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 2842 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (𝑓𝑤) ∈ 𝐶)
3815oveqrspc2v 6836 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝑄 ∧ (𝑓𝑤) ∈ 𝐶)) → (𝑧( ·𝑠𝐾)(𝑓𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤)))
3921, 26, 37, 38syl12anc 1475 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (𝑧( ·𝑠𝐾)(𝑓𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤)))
4020, 39eqeq12d 2775 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → ((𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤)) ↔ (𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤))))
41402ralbidva 3126 . . . . . . 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 676 . . . . . 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 1074 . . . . . 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 1074 . . . . . 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 303 . . . . 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 2775 . . . . . 6 (𝜑 → (𝐺 = 𝐹 ↔ (Scalar‘𝐾) = (Scalar‘𝐽)))
474fveq2d 6356 . . . . . . . 8 (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐽)))
486, 47syl5eq 2806 . . . . . . 7 (𝜑𝑃 = (Base‘(Scalar‘𝐽)))
491raleqdv 3283 . . . . . . 7 (𝜑 → (∀𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤)) ↔ ∀𝑤 ∈ (Base‘𝐽)(𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤))))
5048, 49raleqbidv 3291 . . . . . 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 1551 . . . . 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 17899 . . . . . . 7 (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀))
5352eleq2d 2825 . . . . . 6 (𝜑 → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐿 GrpHom 𝑀)))
5413, 5eqeq12d 2775 . . . . . 6 (𝜑 → (𝐺 = 𝐹 ↔ (Scalar‘𝑀) = (Scalar‘𝐿)))
555fveq2d 6356 . . . . . . . 8 (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐿)))
566, 55syl5eq 2806 . . . . . . 7 (𝜑𝑃 = (Base‘(Scalar‘𝐿)))
572raleqdv 3283 . . . . . . 7 (𝜑 → (∀𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤)) ↔ ∀𝑤 ∈ (Base‘𝐿)(𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤))))
5856, 57raleqbidv 3291 . . . . . 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 1548 . . . . 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 298 . . . 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 749 . . 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 2760 . . . 4 (Scalar‘𝐽) = (Scalar‘𝐽)
63 eqid 2760 . . . 4 (Scalar‘𝐾) = (Scalar‘𝐾)
64 eqid 2760 . . . 4 (Base‘(Scalar‘𝐽)) = (Base‘(Scalar‘𝐽))
65 eqid 2760 . . . 4 ( ·𝑠𝐽) = ( ·𝑠𝐽)
66 eqid 2760 . . . 4 ( ·𝑠𝐾) = ( ·𝑠𝐾)
6762, 63, 64, 28, 65, 66islmhm 19229 . . 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 2760 . . . 4 (Scalar‘𝐿) = (Scalar‘𝐿)
69 eqid 2760 . . . 4 (Scalar‘𝑀) = (Scalar‘𝑀)
70 eqid 2760 . . . 4 (Base‘(Scalar‘𝐿)) = (Base‘(Scalar‘𝐿))
71 eqid 2760 . . . 4 (Base‘𝐿) = (Base‘𝐿)
72 eqid 2760 . . . 4 ( ·𝑠𝐿) = ( ·𝑠𝐿)
73 eqid 2760 . . . 4 ( ·𝑠𝑀) = ( ·𝑠𝑀)
7468, 69, 70, 71, 72, 73islmhm 19229 . . 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 303 . 2 (𝜑 → (𝑓 ∈ (𝐽 LMHom 𝐾) ↔ 𝑓 ∈ (𝐿 LMHom 𝑀)))
7675eqrdv 2758 1 (𝜑 → (𝐽 LMHom 𝐾) = (𝐿 LMHom 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wf 6045  cfv 6049  (class class class)co 6813  Basecbs 16059  +gcplusg 16143  Scalarcsca 16146   ·𝑠 cvsca 16147   GrpHom cghm 17858  LModclmod 19065   LMHom clmhm 19221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-plusg 16156  df-0g 16304  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-mhm 17536  df-grp 17626  df-ghm 17859  df-mgp 18690  df-ur 18702  df-ring 18749  df-lmod 19067  df-lmhm 19224
This theorem is referenced by:  phlpropd  20202
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