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Theorem lmhmpropd 21090
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 20940 . . . . 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 20940 . . . . 5 (𝜑 → (𝐾 ∈ LMod ↔ 𝑀 ∈ LMod))
178, 16anbi12d 632 . . . 4 (𝜑 → ((𝐽 ∈ LMod ∧ 𝐾 ∈ LMod) ↔ (𝐿 ∈ LMod ∧ 𝑀 ∈ LMod)))
187oveqrspc2v 7458 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝑃𝑤𝐵)) → (𝑧( ·𝑠𝐽)𝑤) = (𝑧( ·𝑠𝐿)𝑤))
1918adantlr 715 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (𝑧( ·𝑠𝐽)𝑤) = (𝑧( ·𝑠𝐿)𝑤))
2019fveq2d 6911 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑓‘(𝑧( ·𝑠𝐿)𝑤)))
21 simpll 767 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝜑)
22 simprl 771 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝑧𝑃)
23 simplrr 778 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝐺 = 𝐹)
2423fveq2d 6911 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (Base‘𝐺) = (Base‘𝐹))
2524, 14, 63eqtr4g 2800 . . . . . . . . . . 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 777 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝑓 ∈ (𝐽 GrpHom 𝐾))
28 eqid 2735 . . . . . . . . . . . . . 14 (Base‘𝐽) = (Base‘𝐽)
29 eqid 2735 . . . . . . . . . . . . . 14 (Base‘𝐾) = (Base‘𝐾)
3028, 29ghmf 19251 . . . . . . . . . . . . 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 773 . . . . . . . . . . . . 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, 34ffvelcdmd 7105 . . . . . . . . . . 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 7458 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝑄 ∧ (𝑓𝑤) ∈ 𝐶)) → (𝑧( ·𝑠𝐾)(𝑓𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤)))
3921, 26, 37, 38syl12anc 837 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (𝑧( ·𝑠𝐾)(𝑓𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤)))
4020, 39eqeq12d 2751 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → ((𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤)) ↔ (𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤))))
41402ralbidva 3217 . . . . . . 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 1088 . . . . . 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 1088 . . . . . 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 314 . . . . 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 2751 . . . . . 6 (𝜑 → (𝐺 = 𝐹 ↔ (Scalar‘𝐾) = (Scalar‘𝐽)))
474fveq2d 6911 . . . . . . . 8 (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐽)))
486, 47eqtrid 2787 . . . . . . 7 (𝜑𝑃 = (Base‘(Scalar‘𝐽)))
491raleqdv 3324 . . . . . . 7 (𝜑 → (∀𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤)) ↔ ∀𝑤 ∈ (Base‘𝐽)(𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤))))
5048, 49raleqbidv 3344 . . . . . 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 1438 . . . . 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 19287 . . . . . . 7 (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀))
5352eleq2d 2825 . . . . . 6 (𝜑 → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐿 GrpHom 𝑀)))
5413, 5eqeq12d 2751 . . . . . 6 (𝜑 → (𝐺 = 𝐹 ↔ (Scalar‘𝑀) = (Scalar‘𝐿)))
555fveq2d 6911 . . . . . . . 8 (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐿)))
566, 55eqtrid 2787 . . . . . . 7 (𝜑𝑃 = (Base‘(Scalar‘𝐿)))
572raleqdv 3324 . . . . . . 7 (𝜑 → (∀𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤)) ↔ ∀𝑤 ∈ (Base‘𝐿)(𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤))))
5856, 57raleqbidv 3344 . . . . . 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 1435 . . . . 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 309 . . . 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 632 . . 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 2735 . . . 4 (Scalar‘𝐽) = (Scalar‘𝐽)
63 eqid 2735 . . . 4 (Scalar‘𝐾) = (Scalar‘𝐾)
64 eqid 2735 . . . 4 (Base‘(Scalar‘𝐽)) = (Base‘(Scalar‘𝐽))
65 eqid 2735 . . . 4 ( ·𝑠𝐽) = ( ·𝑠𝐽)
66 eqid 2735 . . . 4 ( ·𝑠𝐾) = ( ·𝑠𝐾)
6762, 63, 64, 28, 65, 66islmhm 21044 . . 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 2735 . . . 4 (Scalar‘𝐿) = (Scalar‘𝐿)
69 eqid 2735 . . . 4 (Scalar‘𝑀) = (Scalar‘𝑀)
70 eqid 2735 . . . 4 (Base‘(Scalar‘𝐿)) = (Base‘(Scalar‘𝐿))
71 eqid 2735 . . . 4 (Base‘𝐿) = (Base‘𝐿)
72 eqid 2735 . . . 4 ( ·𝑠𝐿) = ( ·𝑠𝐿)
73 eqid 2735 . . . 4 ( ·𝑠𝑀) = ( ·𝑠𝑀)
7468, 69, 70, 71, 72, 73islmhm 21044 . . 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 314 . 2 (𝜑 → (𝑓 ∈ (𝐽 LMHom 𝐾) ↔ 𝑓 ∈ (𝐿 LMHom 𝑀)))
7675eqrdv 2733 1 (𝜑 → (𝐽 LMHom 𝐾) = (𝐿 LMHom 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Scalarcsca 17301   ·𝑠 cvsca 17302   GrpHom cghm 19243  LModclmod 20875   LMHom clmhm 21036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-plusg 17311  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-grp 18967  df-ghm 19244  df-mgp 20153  df-ur 20200  df-ring 20253  df-lmod 20877  df-lmhm 21039
This theorem is referenced by:  phlpropd  21691
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