MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lmhmpropd Structured version   Visualization version   GIF version

Theorem lmhmpropd 19540
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 19392 . . . . 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 19392 . . . . 5 (𝜑 → (𝐾 ∈ LMod ↔ 𝑀 ∈ LMod))
178, 16anbi12d 630 . . . 4 (𝜑 → ((𝐽 ∈ LMod ∧ 𝐾 ∈ LMod) ↔ (𝐿 ∈ LMod ∧ 𝑀 ∈ LMod)))
187oveqrspc2v 7048 . . . . . . . . . . 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 6547 . . . . . . . . 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 6547 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (Base‘𝐺) = (Base‘𝐹))
2524, 14, 63eqtr4g 2856 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝑄 = 𝑃)
2622, 25eleqtrrd 2886 . . . . . . . . . 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 2795 . . . . . . . . . . . . . 14 (Base‘𝐽) = (Base‘𝐽)
29 eqid 2795 . . . . . . . . . . . . . 14 (Base‘𝐾) = (Base‘𝐾)
3028, 29ghmf 18108 . . . . . . . . . . . . 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 2885 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → 𝑤 ∈ (Base‘𝐽))
3531, 34ffvelrnd 6722 . . . . . . . . . . 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 2886 . . . . . . . . . 10 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (𝑓𝑤) ∈ 𝐶)
3815oveqrspc2v 7048 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝑄 ∧ (𝑓𝑤) ∈ 𝐶)) → (𝑧( ·𝑠𝐾)(𝑓𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤)))
3921, 26, 37, 38syl12anc 833 . . . . . . . . 9 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → (𝑧( ·𝑠𝐾)(𝑓𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤)))
4020, 39eqeq12d 2810 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧𝑃𝑤𝐵)) → ((𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤)) ↔ (𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤))))
41402ralbidva 3165 . . . . . . 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 1082 . . . . . 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 1082 . . . . . 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 2810 . . . . . 6 (𝜑 → (𝐺 = 𝐹 ↔ (Scalar‘𝐾) = (Scalar‘𝐽)))
474fveq2d 6547 . . . . . . . 8 (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐽)))
486, 47syl5eq 2843 . . . . . . 7 (𝜑𝑃 = (Base‘(Scalar‘𝐽)))
491raleqdv 3375 . . . . . . 7 (𝜑 → (∀𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤)) ↔ ∀𝑤 ∈ (Base‘𝐽)(𝑓‘(𝑧( ·𝑠𝐽)𝑤)) = (𝑧( ·𝑠𝐾)(𝑓𝑤))))
5048, 49raleqbidv 3361 . . . . . 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 1431 . . . . 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 18142 . . . . . . 7 (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀))
5352eleq2d 2868 . . . . . 6 (𝜑 → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐿 GrpHom 𝑀)))
5413, 5eqeq12d 2810 . . . . . 6 (𝜑 → (𝐺 = 𝐹 ↔ (Scalar‘𝑀) = (Scalar‘𝐿)))
555fveq2d 6547 . . . . . . . 8 (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐿)))
566, 55syl5eq 2843 . . . . . . 7 (𝜑𝑃 = (Base‘(Scalar‘𝐿)))
572raleqdv 3375 . . . . . . 7 (𝜑 → (∀𝑤𝐵 (𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤)) ↔ ∀𝑤 ∈ (Base‘𝐿)(𝑓‘(𝑧( ·𝑠𝐿)𝑤)) = (𝑧( ·𝑠𝑀)(𝑓𝑤))))
5856, 57raleqbidv 3361 . . . . . 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 1428 . . . . 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 2795 . . . 4 (Scalar‘𝐽) = (Scalar‘𝐽)
63 eqid 2795 . . . 4 (Scalar‘𝐾) = (Scalar‘𝐾)
64 eqid 2795 . . . 4 (Base‘(Scalar‘𝐽)) = (Base‘(Scalar‘𝐽))
65 eqid 2795 . . . 4 ( ·𝑠𝐽) = ( ·𝑠𝐽)
66 eqid 2795 . . . 4 ( ·𝑠𝐾) = ( ·𝑠𝐾)
6762, 63, 64, 28, 65, 66islmhm 19494 . . 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 2795 . . . 4 (Scalar‘𝐿) = (Scalar‘𝐿)
69 eqid 2795 . . . 4 (Scalar‘𝑀) = (Scalar‘𝑀)
70 eqid 2795 . . . 4 (Base‘(Scalar‘𝐿)) = (Base‘(Scalar‘𝐿))
71 eqid 2795 . . . 4 (Base‘𝐿) = (Base‘𝐿)
72 eqid 2795 . . . 4 ( ·𝑠𝐿) = ( ·𝑠𝐿)
73 eqid 2795 . . . 4 ( ·𝑠𝑀) = ( ·𝑠𝑀)
7468, 69, 70, 71, 72, 73islmhm 19494 . . 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 2793 1 (𝜑 → (𝐽 LMHom 𝐾) = (𝐿 LMHom 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1080   = wceq 1522  wcel 2081  wral 3105  wf 6226  cfv 6230  (class class class)co 7021  Basecbs 16317  +gcplusg 16399  Scalarcsca 16402   ·𝑠 cvsca 16403   GrpHom cghm 18101  LModclmod 19329   LMHom clmhm 19486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324  ax-cnex 10444  ax-resscn 10445  ax-1cn 10446  ax-icn 10447  ax-addcl 10448  ax-addrcl 10449  ax-mulcl 10450  ax-mulrcl 10451  ax-mulcom 10452  ax-addass 10453  ax-mulass 10454  ax-distr 10455  ax-i2m1 10456  ax-1ne0 10457  ax-1rid 10458  ax-rnegex 10459  ax-rrecex 10460  ax-cnre 10461  ax-pre-lttri 10462  ax-pre-lttrn 10463  ax-pre-ltadd 10464  ax-pre-mulgt0 10465
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-pred 6028  df-ord 6074  df-on 6075  df-lim 6076  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-riota 6982  df-ov 7024  df-oprab 7025  df-mpo 7026  df-om 7442  df-wrecs 7803  df-recs 7865  df-rdg 7903  df-er 8144  df-map 8263  df-en 8363  df-dom 8364  df-sdom 8365  df-pnf 10528  df-mnf 10529  df-xr 10530  df-ltxr 10531  df-le 10532  df-sub 10724  df-neg 10725  df-nn 11492  df-2 11553  df-ndx 16320  df-slot 16321  df-base 16323  df-sets 16324  df-plusg 16412  df-0g 16549  df-mgm 17686  df-sgrp 17728  df-mnd 17739  df-mhm 17779  df-grp 17869  df-ghm 18102  df-mgp 18935  df-ur 18947  df-ring 18994  df-lmod 19331  df-lmhm 19489
This theorem is referenced by:  phlpropd  20486
  Copyright terms: Public domain W3C validator