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Theorem lmhmpropd 20549
Description: Module homomorphism depends only on the module attributes of structures. (Contributed by Mario Carneiro, 8-Oct-2015.)
Hypotheses
Ref Expression
lmhmpropd.a (πœ‘ β†’ 𝐡 = (Baseβ€˜π½))
lmhmpropd.b (πœ‘ β†’ 𝐢 = (Baseβ€˜πΎ))
lmhmpropd.c (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))
lmhmpropd.d (πœ‘ β†’ 𝐢 = (Baseβ€˜π‘€))
lmhmpropd.1 (πœ‘ β†’ 𝐹 = (Scalarβ€˜π½))
lmhmpropd.2 (πœ‘ β†’ 𝐺 = (Scalarβ€˜πΎ))
lmhmpropd.3 (πœ‘ β†’ 𝐹 = (Scalarβ€˜πΏ))
lmhmpropd.4 (πœ‘ β†’ 𝐺 = (Scalarβ€˜π‘€))
lmhmpropd.p 𝑃 = (Baseβ€˜πΉ)
lmhmpropd.q 𝑄 = (Baseβ€˜πΊ)
lmhmpropd.e ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜π½)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))
lmhmpropd.f ((πœ‘ ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜π‘€)𝑦))
lmhmpropd.g ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯( ·𝑠 β€˜π½)𝑦) = (π‘₯( ·𝑠 β€˜πΏ)𝑦))
lmhmpropd.h ((πœ‘ ∧ (π‘₯ ∈ 𝑄 ∧ 𝑦 ∈ 𝐢)) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) = (π‘₯( ·𝑠 β€˜π‘€)𝑦))
Assertion
Ref Expression
lmhmpropd (πœ‘ β†’ (𝐽 LMHom 𝐾) = (𝐿 LMHom 𝑀))
Distinct variable groups:   π‘₯,𝑦,𝐢   π‘₯,𝐽,𝑦   π‘₯,𝐾,𝑦   π‘₯,𝐿,𝑦   π‘₯,𝑀,𝑦   π‘₯,𝑃,𝑦   πœ‘,π‘₯,𝑦   π‘₯,𝐡,𝑦   π‘₯,𝑄,𝑦
Allowed substitution hints:   𝐹(π‘₯,𝑦)   𝐺(π‘₯,𝑦)

Proof of Theorem lmhmpropd
Dummy variables 𝑧 𝑀 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmhmpropd.a . . . . . 6 (πœ‘ β†’ 𝐡 = (Baseβ€˜π½))
2 lmhmpropd.c . . . . . 6 (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))
3 lmhmpropd.e . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜π½)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))
4 lmhmpropd.1 . . . . . 6 (πœ‘ β†’ 𝐹 = (Scalarβ€˜π½))
5 lmhmpropd.3 . . . . . 6 (πœ‘ β†’ 𝐹 = (Scalarβ€˜πΏ))
6 lmhmpropd.p . . . . . 6 𝑃 = (Baseβ€˜πΉ)
7 lmhmpropd.g . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯( ·𝑠 β€˜π½)𝑦) = (π‘₯( ·𝑠 β€˜πΏ)𝑦))
81, 2, 3, 4, 5, 6, 7lmodpropd 20400 . . . . 5 (πœ‘ β†’ (𝐽 ∈ LMod ↔ 𝐿 ∈ LMod))
9 lmhmpropd.b . . . . . 6 (πœ‘ β†’ 𝐢 = (Baseβ€˜πΎ))
10 lmhmpropd.d . . . . . 6 (πœ‘ β†’ 𝐢 = (Baseβ€˜π‘€))
11 lmhmpropd.f . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜π‘€)𝑦))
12 lmhmpropd.2 . . . . . 6 (πœ‘ β†’ 𝐺 = (Scalarβ€˜πΎ))
13 lmhmpropd.4 . . . . . 6 (πœ‘ β†’ 𝐺 = (Scalarβ€˜π‘€))
14 lmhmpropd.q . . . . . 6 𝑄 = (Baseβ€˜πΊ)
15 lmhmpropd.h . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝑄 ∧ 𝑦 ∈ 𝐢)) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) = (π‘₯( ·𝑠 β€˜π‘€)𝑦))
169, 10, 11, 12, 13, 14, 15lmodpropd 20400 . . . . 5 (πœ‘ β†’ (𝐾 ∈ LMod ↔ 𝑀 ∈ LMod))
178, 16anbi12d 632 . . . 4 (πœ‘ β†’ ((𝐽 ∈ LMod ∧ 𝐾 ∈ LMod) ↔ (𝐿 ∈ LMod ∧ 𝑀 ∈ LMod)))
187oveqrspc2v 7385 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑧 ∈ 𝑃 ∧ 𝑀 ∈ 𝐡)) β†’ (𝑧( ·𝑠 β€˜π½)𝑀) = (𝑧( ·𝑠 β€˜πΏ)𝑀))
1918adantlr 714 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑀 ∈ 𝐡)) β†’ (𝑧( ·𝑠 β€˜π½)𝑀) = (𝑧( ·𝑠 β€˜πΏ)𝑀))
2019fveq2d 6847 . . . . . . . . 9 (((πœ‘ ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑀 ∈ 𝐡)) β†’ (π‘“β€˜(𝑧( ·𝑠 β€˜π½)𝑀)) = (π‘“β€˜(𝑧( ·𝑠 β€˜πΏ)𝑀)))
21 simpll 766 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑀 ∈ 𝐡)) β†’ πœ‘)
22 simprl 770 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑀 ∈ 𝐡)) β†’ 𝑧 ∈ 𝑃)
23 simplrr 777 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑀 ∈ 𝐡)) β†’ 𝐺 = 𝐹)
2423fveq2d 6847 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑀 ∈ 𝐡)) β†’ (Baseβ€˜πΊ) = (Baseβ€˜πΉ))
2524, 14, 63eqtr4g 2798 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑀 ∈ 𝐡)) β†’ 𝑄 = 𝑃)
2622, 25eleqtrrd 2837 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑀 ∈ 𝐡)) β†’ 𝑧 ∈ 𝑄)
27 simplrl 776 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑀 ∈ 𝐡)) β†’ 𝑓 ∈ (𝐽 GrpHom 𝐾))
28 eqid 2733 . . . . . . . . . . . . . 14 (Baseβ€˜π½) = (Baseβ€˜π½)
29 eqid 2733 . . . . . . . . . . . . . 14 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
3028, 29ghmf 19017 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐽 GrpHom 𝐾) β†’ 𝑓:(Baseβ€˜π½)⟢(Baseβ€˜πΎ))
3127, 30syl 17 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑀 ∈ 𝐡)) β†’ 𝑓:(Baseβ€˜π½)⟢(Baseβ€˜πΎ))
32 simprr 772 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑀 ∈ 𝐡)) β†’ 𝑀 ∈ 𝐡)
3321, 1syl 17 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑀 ∈ 𝐡)) β†’ 𝐡 = (Baseβ€˜π½))
3432, 33eleqtrd 2836 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑀 ∈ 𝐡)) β†’ 𝑀 ∈ (Baseβ€˜π½))
3531, 34ffvelcdmd 7037 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑀 ∈ 𝐡)) β†’ (π‘“β€˜π‘€) ∈ (Baseβ€˜πΎ))
3621, 9syl 17 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑀 ∈ 𝐡)) β†’ 𝐢 = (Baseβ€˜πΎ))
3735, 36eleqtrrd 2837 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑀 ∈ 𝐡)) β†’ (π‘“β€˜π‘€) ∈ 𝐢)
3815oveqrspc2v 7385 . . . . . . . . . 10 ((πœ‘ ∧ (𝑧 ∈ 𝑄 ∧ (π‘“β€˜π‘€) ∈ 𝐢)) β†’ (𝑧( ·𝑠 β€˜πΎ)(π‘“β€˜π‘€)) = (𝑧( ·𝑠 β€˜π‘€)(π‘“β€˜π‘€)))
3921, 26, 37, 38syl12anc 836 . . . . . . . . 9 (((πœ‘ ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑀 ∈ 𝐡)) β†’ (𝑧( ·𝑠 β€˜πΎ)(π‘“β€˜π‘€)) = (𝑧( ·𝑠 β€˜π‘€)(π‘“β€˜π‘€)))
4020, 39eqeq12d 2749 . . . . . . . 8 (((πœ‘ ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) ∧ (𝑧 ∈ 𝑃 ∧ 𝑀 ∈ 𝐡)) β†’ ((π‘“β€˜(𝑧( ·𝑠 β€˜π½)𝑀)) = (𝑧( ·𝑠 β€˜πΎ)(π‘“β€˜π‘€)) ↔ (π‘“β€˜(𝑧( ·𝑠 β€˜πΏ)𝑀)) = (𝑧( ·𝑠 β€˜π‘€)(π‘“β€˜π‘€))))
41402ralbidva 3207 . . . . . . 7 ((πœ‘ ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹)) β†’ (βˆ€π‘§ ∈ 𝑃 βˆ€π‘€ ∈ 𝐡 (π‘“β€˜(𝑧( ·𝑠 β€˜π½)𝑀)) = (𝑧( ·𝑠 β€˜πΎ)(π‘“β€˜π‘€)) ↔ βˆ€π‘§ ∈ 𝑃 βˆ€π‘€ ∈ 𝐡 (π‘“β€˜(𝑧( ·𝑠 β€˜πΏ)𝑀)) = (𝑧( ·𝑠 β€˜π‘€)(π‘“β€˜π‘€))))
4241pm5.32da 580 . . . . . 6 (πœ‘ β†’ (((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹) ∧ βˆ€π‘§ ∈ 𝑃 βˆ€π‘€ ∈ 𝐡 (π‘“β€˜(𝑧( ·𝑠 β€˜π½)𝑀)) = (𝑧( ·𝑠 β€˜πΎ)(π‘“β€˜π‘€))) ↔ ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹) ∧ βˆ€π‘§ ∈ 𝑃 βˆ€π‘€ ∈ 𝐡 (π‘“β€˜(𝑧( ·𝑠 β€˜πΏ)𝑀)) = (𝑧( ·𝑠 β€˜π‘€)(π‘“β€˜π‘€)))))
43 df-3an 1090 . . . . . 6 ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹 ∧ βˆ€π‘§ ∈ 𝑃 βˆ€π‘€ ∈ 𝐡 (π‘“β€˜(𝑧( ·𝑠 β€˜π½)𝑀)) = (𝑧( ·𝑠 β€˜πΎ)(π‘“β€˜π‘€))) ↔ ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹) ∧ βˆ€π‘§ ∈ 𝑃 βˆ€π‘€ ∈ 𝐡 (π‘“β€˜(𝑧( ·𝑠 β€˜π½)𝑀)) = (𝑧( ·𝑠 β€˜πΎ)(π‘“β€˜π‘€))))
44 df-3an 1090 . . . . . 6 ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹 ∧ βˆ€π‘§ ∈ 𝑃 βˆ€π‘€ ∈ 𝐡 (π‘“β€˜(𝑧( ·𝑠 β€˜πΏ)𝑀)) = (𝑧( ·𝑠 β€˜π‘€)(π‘“β€˜π‘€))) ↔ ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹) ∧ βˆ€π‘§ ∈ 𝑃 βˆ€π‘€ ∈ 𝐡 (π‘“β€˜(𝑧( ·𝑠 β€˜πΏ)𝑀)) = (𝑧( ·𝑠 β€˜π‘€)(π‘“β€˜π‘€))))
4542, 43, 443bitr4g 314 . . . . 5 (πœ‘ β†’ ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹 ∧ βˆ€π‘§ ∈ 𝑃 βˆ€π‘€ ∈ 𝐡 (π‘“β€˜(𝑧( ·𝑠 β€˜π½)𝑀)) = (𝑧( ·𝑠 β€˜πΎ)(π‘“β€˜π‘€))) ↔ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹 ∧ βˆ€π‘§ ∈ 𝑃 βˆ€π‘€ ∈ 𝐡 (π‘“β€˜(𝑧( ·𝑠 β€˜πΏ)𝑀)) = (𝑧( ·𝑠 β€˜π‘€)(π‘“β€˜π‘€)))))
4612, 4eqeq12d 2749 . . . . . 6 (πœ‘ β†’ (𝐺 = 𝐹 ↔ (Scalarβ€˜πΎ) = (Scalarβ€˜π½)))
474fveq2d 6847 . . . . . . . 8 (πœ‘ β†’ (Baseβ€˜πΉ) = (Baseβ€˜(Scalarβ€˜π½)))
486, 47eqtrid 2785 . . . . . . 7 (πœ‘ β†’ 𝑃 = (Baseβ€˜(Scalarβ€˜π½)))
491raleqdv 3312 . . . . . . 7 (πœ‘ β†’ (βˆ€π‘€ ∈ 𝐡 (π‘“β€˜(𝑧( ·𝑠 β€˜π½)𝑀)) = (𝑧( ·𝑠 β€˜πΎ)(π‘“β€˜π‘€)) ↔ βˆ€π‘€ ∈ (Baseβ€˜π½)(π‘“β€˜(𝑧( ·𝑠 β€˜π½)𝑀)) = (𝑧( ·𝑠 β€˜πΎ)(π‘“β€˜π‘€))))
5048, 49raleqbidv 3318 . . . . . 6 (πœ‘ β†’ (βˆ€π‘§ ∈ 𝑃 βˆ€π‘€ ∈ 𝐡 (π‘“β€˜(𝑧( ·𝑠 β€˜π½)𝑀)) = (𝑧( ·𝑠 β€˜πΎ)(π‘“β€˜π‘€)) ↔ βˆ€π‘§ ∈ (Baseβ€˜(Scalarβ€˜π½))βˆ€π‘€ ∈ (Baseβ€˜π½)(π‘“β€˜(𝑧( ·𝑠 β€˜π½)𝑀)) = (𝑧( ·𝑠 β€˜πΎ)(π‘“β€˜π‘€))))
5146, 503anbi23d 1440 . . . . 5 (πœ‘ β†’ ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹 ∧ βˆ€π‘§ ∈ 𝑃 βˆ€π‘€ ∈ 𝐡 (π‘“β€˜(𝑧( ·𝑠 β€˜π½)𝑀)) = (𝑧( ·𝑠 β€˜πΎ)(π‘“β€˜π‘€))) ↔ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ (Scalarβ€˜πΎ) = (Scalarβ€˜π½) ∧ βˆ€π‘§ ∈ (Baseβ€˜(Scalarβ€˜π½))βˆ€π‘€ ∈ (Baseβ€˜π½)(π‘“β€˜(𝑧( ·𝑠 β€˜π½)𝑀)) = (𝑧( ·𝑠 β€˜πΎ)(π‘“β€˜π‘€)))))
521, 9, 2, 10, 3, 11ghmpropd 19051 . . . . . . 7 (πœ‘ β†’ (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀))
5352eleq2d 2820 . . . . . 6 (πœ‘ β†’ (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐿 GrpHom 𝑀)))
5413, 5eqeq12d 2749 . . . . . 6 (πœ‘ β†’ (𝐺 = 𝐹 ↔ (Scalarβ€˜π‘€) = (Scalarβ€˜πΏ)))
555fveq2d 6847 . . . . . . . 8 (πœ‘ β†’ (Baseβ€˜πΉ) = (Baseβ€˜(Scalarβ€˜πΏ)))
566, 55eqtrid 2785 . . . . . . 7 (πœ‘ β†’ 𝑃 = (Baseβ€˜(Scalarβ€˜πΏ)))
572raleqdv 3312 . . . . . . 7 (πœ‘ β†’ (βˆ€π‘€ ∈ 𝐡 (π‘“β€˜(𝑧( ·𝑠 β€˜πΏ)𝑀)) = (𝑧( ·𝑠 β€˜π‘€)(π‘“β€˜π‘€)) ↔ βˆ€π‘€ ∈ (Baseβ€˜πΏ)(π‘“β€˜(𝑧( ·𝑠 β€˜πΏ)𝑀)) = (𝑧( ·𝑠 β€˜π‘€)(π‘“β€˜π‘€))))
5856, 57raleqbidv 3318 . . . . . 6 (πœ‘ β†’ (βˆ€π‘§ ∈ 𝑃 βˆ€π‘€ ∈ 𝐡 (π‘“β€˜(𝑧( ·𝑠 β€˜πΏ)𝑀)) = (𝑧( ·𝑠 β€˜π‘€)(π‘“β€˜π‘€)) ↔ βˆ€π‘§ ∈ (Baseβ€˜(Scalarβ€˜πΏ))βˆ€π‘€ ∈ (Baseβ€˜πΏ)(π‘“β€˜(𝑧( ·𝑠 β€˜πΏ)𝑀)) = (𝑧( ·𝑠 β€˜π‘€)(π‘“β€˜π‘€))))
5953, 54, 583anbi123d 1437 . . . . 5 (πœ‘ β†’ ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ 𝐺 = 𝐹 ∧ βˆ€π‘§ ∈ 𝑃 βˆ€π‘€ ∈ 𝐡 (π‘“β€˜(𝑧( ·𝑠 β€˜πΏ)𝑀)) = (𝑧( ·𝑠 β€˜π‘€)(π‘“β€˜π‘€))) ↔ (𝑓 ∈ (𝐿 GrpHom 𝑀) ∧ (Scalarβ€˜π‘€) = (Scalarβ€˜πΏ) ∧ βˆ€π‘§ ∈ (Baseβ€˜(Scalarβ€˜πΏ))βˆ€π‘€ ∈ (Baseβ€˜πΏ)(π‘“β€˜(𝑧( ·𝑠 β€˜πΏ)𝑀)) = (𝑧( ·𝑠 β€˜π‘€)(π‘“β€˜π‘€)))))
6045, 51, 593bitr3d 309 . . . 4 (πœ‘ β†’ ((𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ (Scalarβ€˜πΎ) = (Scalarβ€˜π½) ∧ βˆ€π‘§ ∈ (Baseβ€˜(Scalarβ€˜π½))βˆ€π‘€ ∈ (Baseβ€˜π½)(π‘“β€˜(𝑧( ·𝑠 β€˜π½)𝑀)) = (𝑧( ·𝑠 β€˜πΎ)(π‘“β€˜π‘€))) ↔ (𝑓 ∈ (𝐿 GrpHom 𝑀) ∧ (Scalarβ€˜π‘€) = (Scalarβ€˜πΏ) ∧ βˆ€π‘§ ∈ (Baseβ€˜(Scalarβ€˜πΏ))βˆ€π‘€ ∈ (Baseβ€˜πΏ)(π‘“β€˜(𝑧( ·𝑠 β€˜πΏ)𝑀)) = (𝑧( ·𝑠 β€˜π‘€)(π‘“β€˜π‘€)))))
6117, 60anbi12d 632 . . 3 (πœ‘ β†’ (((𝐽 ∈ LMod ∧ 𝐾 ∈ LMod) ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ (Scalarβ€˜πΎ) = (Scalarβ€˜π½) ∧ βˆ€π‘§ ∈ (Baseβ€˜(Scalarβ€˜π½))βˆ€π‘€ ∈ (Baseβ€˜π½)(π‘“β€˜(𝑧( ·𝑠 β€˜π½)𝑀)) = (𝑧( ·𝑠 β€˜πΎ)(π‘“β€˜π‘€)))) ↔ ((𝐿 ∈ LMod ∧ 𝑀 ∈ LMod) ∧ (𝑓 ∈ (𝐿 GrpHom 𝑀) ∧ (Scalarβ€˜π‘€) = (Scalarβ€˜πΏ) ∧ βˆ€π‘§ ∈ (Baseβ€˜(Scalarβ€˜πΏ))βˆ€π‘€ ∈ (Baseβ€˜πΏ)(π‘“β€˜(𝑧( ·𝑠 β€˜πΏ)𝑀)) = (𝑧( ·𝑠 β€˜π‘€)(π‘“β€˜π‘€))))))
62 eqid 2733 . . . 4 (Scalarβ€˜π½) = (Scalarβ€˜π½)
63 eqid 2733 . . . 4 (Scalarβ€˜πΎ) = (Scalarβ€˜πΎ)
64 eqid 2733 . . . 4 (Baseβ€˜(Scalarβ€˜π½)) = (Baseβ€˜(Scalarβ€˜π½))
65 eqid 2733 . . . 4 ( ·𝑠 β€˜π½) = ( ·𝑠 β€˜π½)
66 eqid 2733 . . . 4 ( ·𝑠 β€˜πΎ) = ( ·𝑠 β€˜πΎ)
6762, 63, 64, 28, 65, 66islmhm 20503 . . 3 (𝑓 ∈ (𝐽 LMHom 𝐾) ↔ ((𝐽 ∈ LMod ∧ 𝐾 ∈ LMod) ∧ (𝑓 ∈ (𝐽 GrpHom 𝐾) ∧ (Scalarβ€˜πΎ) = (Scalarβ€˜π½) ∧ βˆ€π‘§ ∈ (Baseβ€˜(Scalarβ€˜π½))βˆ€π‘€ ∈ (Baseβ€˜π½)(π‘“β€˜(𝑧( ·𝑠 β€˜π½)𝑀)) = (𝑧( ·𝑠 β€˜πΎ)(π‘“β€˜π‘€)))))
68 eqid 2733 . . . 4 (Scalarβ€˜πΏ) = (Scalarβ€˜πΏ)
69 eqid 2733 . . . 4 (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€)
70 eqid 2733 . . . 4 (Baseβ€˜(Scalarβ€˜πΏ)) = (Baseβ€˜(Scalarβ€˜πΏ))
71 eqid 2733 . . . 4 (Baseβ€˜πΏ) = (Baseβ€˜πΏ)
72 eqid 2733 . . . 4 ( ·𝑠 β€˜πΏ) = ( ·𝑠 β€˜πΏ)
73 eqid 2733 . . . 4 ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘€)
7468, 69, 70, 71, 72, 73islmhm 20503 . . 3 (𝑓 ∈ (𝐿 LMHom 𝑀) ↔ ((𝐿 ∈ LMod ∧ 𝑀 ∈ LMod) ∧ (𝑓 ∈ (𝐿 GrpHom 𝑀) ∧ (Scalarβ€˜π‘€) = (Scalarβ€˜πΏ) ∧ βˆ€π‘§ ∈ (Baseβ€˜(Scalarβ€˜πΏ))βˆ€π‘€ ∈ (Baseβ€˜πΏ)(π‘“β€˜(𝑧( ·𝑠 β€˜πΏ)𝑀)) = (𝑧( ·𝑠 β€˜π‘€)(π‘“β€˜π‘€)))))
7561, 67, 743bitr4g 314 . 2 (πœ‘ β†’ (𝑓 ∈ (𝐽 LMHom 𝐾) ↔ 𝑓 ∈ (𝐿 LMHom 𝑀)))
7675eqrdv 2731 1 (πœ‘ β†’ (𝐽 LMHom 𝐾) = (𝐿 LMHom 𝑀))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  Scalarcsca 17141   ·𝑠 cvsca 17142   GrpHom cghm 19010  LModclmod 20336   LMHom clmhm 20495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-plusg 17151  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-mhm 18606  df-grp 18756  df-ghm 19011  df-mgp 19902  df-ur 19919  df-ring 19971  df-lmod 20338  df-lmhm 20498
This theorem is referenced by:  phlpropd  21075
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