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Theorem lmhmpropd 18840
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 18695 . . . . 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 18695 . . . . 5 (𝜑 → (𝐾 ∈ LMod ↔ 𝑀 ∈ LMod))
178, 16anbi12d 742 . . . 4 (𝜑 → ((𝐽 ∈ LMod ∧ 𝐾 ∈ LMod) ↔ (𝐿 ∈ LMod ∧ 𝑀 ∈ LMod)))
187oveqrspc2v 6550 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝑃𝑤𝐵)) → (𝑧( ·𝑠𝐽)𝑤) = (𝑧( ·𝑠𝐿)𝑤))
1918adantlr 746 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (𝑧( ·𝑠𝐽)𝑤) = (𝑧( ·𝑠𝐿)𝑤))
2019fveq2d 6092 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑓‘(𝑧( ·𝑠𝐿)𝑤)))
21 simpll 785 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝜑)
22 simprl 789 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝑧𝑃)
23 simplrr 796 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝐺 = 𝐹)
2423fveq2d 6092 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (Base‘𝐺) = (Base‘𝐹))
2524, 14, 63eqtr4g 2668 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝑄 = 𝑃)
2622, 25eleqtrrd 2690 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝑧𝑄)
27 simplrl 795 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝑓 ∈ (𝐽 GrpHom 𝐾))
28 eqid 2609 . . . . . . . . . . . . . 14 (Base‘𝐽) = (Base‘𝐽)
29 eqid 2609 . . . . . . . . . . . . . 14 (Base‘𝐾) = (Base‘𝐾)
3028, 29ghmf 17433 . . . . . . . . . . . . 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 791 . . . . . . . . . . . . 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 2689 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝑤 ∈ (Base‘𝐽))
3531, 34ffvelrnd 6253 . . . . . . . . . . 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 2690 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (𝑓𝑤) ∈ 𝐶)
3815oveqrspc2v 6550 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝑄 ∧ (𝑓𝑤) ∈ 𝐶)) → (𝑧( ·𝑠𝐾)(𝑓𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤)))
3921, 26, 37, 38syl12anc 1315 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (𝑧( ·𝑠𝐾)(𝑓𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤)))
4020, 39eqeq12d 2624 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → ((𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤)) ↔ (𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤))))
41402ralbidva 2970 . . . . . . 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 670 . . . . . 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 1032 . . . . . 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 1032 . . . . . 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 301 . . . . 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 2624 . . . . . 6 (𝜑 → (𝐺 = 𝐹 ↔ (Scalar‘𝐾) = (Scalar‘𝐽)))
474fveq2d 6092 . . . . . . . 8 (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐽)))
486, 47syl5eq 2655 . . . . . . 7 (𝜑𝑃 = (Base‘(Scalar‘𝐽)))
491raleqdv 3120 . . . . . . 7 (𝜑 → (∀𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤)) ↔ ∀𝑤 ∈ (Base‘𝐽)(𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤))))
5048, 49raleqbidv 3128 . . . . . 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 1393 . . . . 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 17467 . . . . . . 7 (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀))
5352eleq2d 2672 . . . . . 6 (𝜑 → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐿 GrpHom 𝑀)))
5413, 5eqeq12d 2624 . . . . . 6 (𝜑 → (𝐺 = 𝐹 ↔ (Scalar‘𝑀) = (Scalar‘𝐿)))
555fveq2d 6092 . . . . . . . 8 (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐿)))
566, 55syl5eq 2655 . . . . . . 7 (𝜑𝑃 = (Base‘(Scalar‘𝐿)))
572raleqdv 3120 . . . . . . 7 (𝜑 → (∀𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤)) ↔ ∀𝑤 ∈ (Base‘𝐿)(𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤))))
5856, 57raleqbidv 3128 . . . . . 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 1390 . . . . 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 296 . . . 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 742 . . 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 2609 . . . 4 (Scalar‘𝐽) = (Scalar‘𝐽)
63 eqid 2609 . . . 4 (Scalar‘𝐾) = (Scalar‘𝐾)
64 eqid 2609 . . . 4 (Base‘(Scalar‘𝐽)) = (Base‘(Scalar‘𝐽))
65 eqid 2609 . . . 4 ( ·𝑠𝐽) = ( ·𝑠𝐽)
66 eqid 2609 . . . 4 ( ·𝑠𝐾) = ( ·𝑠𝐾)
6762, 63, 64, 28, 65, 66islmhm 18794 . . 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 2609 . . . 4 (Scalar‘𝐿) = (Scalar‘𝐿)
69 eqid 2609 . . . 4 (Scalar‘𝑀) = (Scalar‘𝑀)
70 eqid 2609 . . . 4 (Base‘(Scalar‘𝐿)) = (Base‘(Scalar‘𝐿))
71 eqid 2609 . . . 4 (Base‘𝐿) = (Base‘𝐿)
72 eqid 2609 . . . 4 ( ·𝑠𝐿) = ( ·𝑠𝐿)
73 eqid 2609 . . . 4 ( ·𝑠𝑀) = ( ·𝑠𝑀)
7468, 69, 70, 71, 72, 73islmhm 18794 . . 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 301 . 2 (𝜑 → (𝑓 ∈ (𝐽 LMHom 𝐾) ↔ 𝑓 ∈ (𝐿 LMHom 𝑀)))
7675eqrdv 2607 1 (𝜑 → (𝐽 LMHom 𝐾) = (𝐿 LMHom 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  wf 5786  cfv 5790  (class class class)co 6527  Basecbs 15641  +gcplusg 15714  Scalarcsca 15717   ·𝑠 cvsca 15718   GrpHom cghm 17426  LModclmod 18632   LMHom clmhm 18786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-ndx 15644  df-slot 15645  df-base 15646  df-sets 15647  df-plusg 15727  df-0g 15871  df-mgm 17011  df-sgrp 17053  df-mnd 17064  df-mhm 17104  df-grp 17194  df-ghm 17427  df-mgp 18259  df-ur 18271  df-ring 18318  df-lmod 18634  df-lmhm 18789
This theorem is referenced by:  phlpropd  19764
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