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Theorem islmhm 20503
Description: Property of being a homomorphism of left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof shortened by Mario Carneiro, 30-Apr-2015.)
Hypotheses
Ref Expression
islmhm.k 𝐾 = (Scalarβ€˜π‘†)
islmhm.l 𝐿 = (Scalarβ€˜π‘‡)
islmhm.b 𝐡 = (Baseβ€˜πΎ)
islmhm.e 𝐸 = (Baseβ€˜π‘†)
islmhm.m Β· = ( ·𝑠 β€˜π‘†)
islmhm.n Γ— = ( ·𝑠 β€˜π‘‡)
Assertion
Ref Expression
islmhm (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐸 (πΉβ€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (πΉβ€˜π‘¦)))))
Distinct variable groups:   π‘₯,𝐡   𝑦,𝐸   π‘₯,𝑦,𝑆   π‘₯,𝐹,𝑦   π‘₯,𝑇,𝑦
Allowed substitution hints:   𝐡(𝑦)   Β· (π‘₯,𝑦)   Γ— (π‘₯,𝑦)   𝐸(π‘₯)   𝐾(π‘₯,𝑦)   𝐿(π‘₯,𝑦)

Proof of Theorem islmhm
Dummy variables 𝑓 𝑠 𝑑 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-lmhm 20498 . . 3 LMHom = (𝑠 ∈ LMod, 𝑑 ∈ LMod ↦ {𝑓 ∈ (𝑠 GrpHom 𝑑) ∣ [(Scalarβ€˜π‘ ) / 𝑀]((Scalarβ€˜π‘‘) = 𝑀 ∧ βˆ€π‘₯ ∈ (Baseβ€˜π‘€)βˆ€π‘¦ ∈ (Baseβ€˜π‘ )(π‘“β€˜(π‘₯( ·𝑠 β€˜π‘ )𝑦)) = (π‘₯( ·𝑠 β€˜π‘‘)(π‘“β€˜π‘¦)))})
21elmpocl 7596 . 2 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ (𝑆 ∈ LMod ∧ 𝑇 ∈ LMod))
3 oveq12 7367 . . . . . 6 ((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) β†’ (𝑠 GrpHom 𝑑) = (𝑆 GrpHom 𝑇))
4 fvexd 6858 . . . . . . 7 ((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) β†’ (Scalarβ€˜π‘ ) ∈ V)
5 simplr 768 . . . . . . . . . . 11 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ 𝑑 = 𝑇)
65fveq2d 6847 . . . . . . . . . 10 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ (Scalarβ€˜π‘‘) = (Scalarβ€˜π‘‡))
7 islmhm.l . . . . . . . . . 10 𝐿 = (Scalarβ€˜π‘‡)
86, 7eqtr4di 2791 . . . . . . . . 9 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ (Scalarβ€˜π‘‘) = 𝐿)
9 simpr 486 . . . . . . . . . . 11 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ 𝑀 = (Scalarβ€˜π‘ ))
10 simpll 766 . . . . . . . . . . . 12 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ 𝑠 = 𝑆)
1110fveq2d 6847 . . . . . . . . . . 11 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ (Scalarβ€˜π‘ ) = (Scalarβ€˜π‘†))
129, 11eqtrd 2773 . . . . . . . . . 10 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ 𝑀 = (Scalarβ€˜π‘†))
13 islmhm.k . . . . . . . . . 10 𝐾 = (Scalarβ€˜π‘†)
1412, 13eqtr4di 2791 . . . . . . . . 9 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ 𝑀 = 𝐾)
158, 14eqeq12d 2749 . . . . . . . 8 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ ((Scalarβ€˜π‘‘) = 𝑀 ↔ 𝐿 = 𝐾))
1614fveq2d 6847 . . . . . . . . . 10 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ (Baseβ€˜π‘€) = (Baseβ€˜πΎ))
17 islmhm.b . . . . . . . . . 10 𝐡 = (Baseβ€˜πΎ)
1816, 17eqtr4di 2791 . . . . . . . . 9 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ (Baseβ€˜π‘€) = 𝐡)
1910fveq2d 6847 . . . . . . . . . . 11 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ (Baseβ€˜π‘ ) = (Baseβ€˜π‘†))
20 islmhm.e . . . . . . . . . . 11 𝐸 = (Baseβ€˜π‘†)
2119, 20eqtr4di 2791 . . . . . . . . . 10 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ (Baseβ€˜π‘ ) = 𝐸)
2210fveq2d 6847 . . . . . . . . . . . . . 14 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ ( ·𝑠 β€˜π‘ ) = ( ·𝑠 β€˜π‘†))
23 islmhm.m . . . . . . . . . . . . . 14 Β· = ( ·𝑠 β€˜π‘†)
2422, 23eqtr4di 2791 . . . . . . . . . . . . 13 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ ( ·𝑠 β€˜π‘ ) = Β· )
2524oveqd 7375 . . . . . . . . . . . 12 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ (π‘₯( ·𝑠 β€˜π‘ )𝑦) = (π‘₯ Β· 𝑦))
2625fveq2d 6847 . . . . . . . . . . 11 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ (π‘“β€˜(π‘₯( ·𝑠 β€˜π‘ )𝑦)) = (π‘“β€˜(π‘₯ Β· 𝑦)))
275fveq2d 6847 . . . . . . . . . . . . 13 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ ( ·𝑠 β€˜π‘‘) = ( ·𝑠 β€˜π‘‡))
28 islmhm.n . . . . . . . . . . . . 13 Γ— = ( ·𝑠 β€˜π‘‡)
2927, 28eqtr4di 2791 . . . . . . . . . . . 12 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ ( ·𝑠 β€˜π‘‘) = Γ— )
3029oveqd 7375 . . . . . . . . . . 11 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ (π‘₯( ·𝑠 β€˜π‘‘)(π‘“β€˜π‘¦)) = (π‘₯ Γ— (π‘“β€˜π‘¦)))
3126, 30eqeq12d 2749 . . . . . . . . . 10 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ ((π‘“β€˜(π‘₯( ·𝑠 β€˜π‘ )𝑦)) = (π‘₯( ·𝑠 β€˜π‘‘)(π‘“β€˜π‘¦)) ↔ (π‘“β€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (π‘“β€˜π‘¦))))
3221, 31raleqbidv 3318 . . . . . . . . 9 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ (βˆ€π‘¦ ∈ (Baseβ€˜π‘ )(π‘“β€˜(π‘₯( ·𝑠 β€˜π‘ )𝑦)) = (π‘₯( ·𝑠 β€˜π‘‘)(π‘“β€˜π‘¦)) ↔ βˆ€π‘¦ ∈ 𝐸 (π‘“β€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (π‘“β€˜π‘¦))))
3318, 32raleqbidv 3318 . . . . . . . 8 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ (βˆ€π‘₯ ∈ (Baseβ€˜π‘€)βˆ€π‘¦ ∈ (Baseβ€˜π‘ )(π‘“β€˜(π‘₯( ·𝑠 β€˜π‘ )𝑦)) = (π‘₯( ·𝑠 β€˜π‘‘)(π‘“β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐸 (π‘“β€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (π‘“β€˜π‘¦))))
3415, 33anbi12d 632 . . . . . . 7 (((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) ∧ 𝑀 = (Scalarβ€˜π‘ )) β†’ (((Scalarβ€˜π‘‘) = 𝑀 ∧ βˆ€π‘₯ ∈ (Baseβ€˜π‘€)βˆ€π‘¦ ∈ (Baseβ€˜π‘ )(π‘“β€˜(π‘₯( ·𝑠 β€˜π‘ )𝑦)) = (π‘₯( ·𝑠 β€˜π‘‘)(π‘“β€˜π‘¦))) ↔ (𝐿 = 𝐾 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐸 (π‘“β€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (π‘“β€˜π‘¦)))))
354, 34sbcied 3785 . . . . . 6 ((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) β†’ ([(Scalarβ€˜π‘ ) / 𝑀]((Scalarβ€˜π‘‘) = 𝑀 ∧ βˆ€π‘₯ ∈ (Baseβ€˜π‘€)βˆ€π‘¦ ∈ (Baseβ€˜π‘ )(π‘“β€˜(π‘₯( ·𝑠 β€˜π‘ )𝑦)) = (π‘₯( ·𝑠 β€˜π‘‘)(π‘“β€˜π‘¦))) ↔ (𝐿 = 𝐾 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐸 (π‘“β€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (π‘“β€˜π‘¦)))))
363, 35rabeqbidv 3423 . . . . 5 ((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) β†’ {𝑓 ∈ (𝑠 GrpHom 𝑑) ∣ [(Scalarβ€˜π‘ ) / 𝑀]((Scalarβ€˜π‘‘) = 𝑀 ∧ βˆ€π‘₯ ∈ (Baseβ€˜π‘€)βˆ€π‘¦ ∈ (Baseβ€˜π‘ )(π‘“β€˜(π‘₯( ·𝑠 β€˜π‘ )𝑦)) = (π‘₯( ·𝑠 β€˜π‘‘)(π‘“β€˜π‘¦)))} = {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐸 (π‘“β€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (π‘“β€˜π‘¦)))})
37 ovex 7391 . . . . . 6 (𝑆 GrpHom 𝑇) ∈ V
3837rabex 5290 . . . . 5 {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐸 (π‘“β€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (π‘“β€˜π‘¦)))} ∈ V
3936, 1, 38ovmpoa 7511 . . . 4 ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) β†’ (𝑆 LMHom 𝑇) = {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐸 (π‘“β€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (π‘“β€˜π‘¦)))})
4039eleq2d 2820 . . 3 ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) β†’ (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ 𝐹 ∈ {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐸 (π‘“β€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (π‘“β€˜π‘¦)))}))
41 fveq1 6842 . . . . . . . 8 (𝑓 = 𝐹 β†’ (π‘“β€˜(π‘₯ Β· 𝑦)) = (πΉβ€˜(π‘₯ Β· 𝑦)))
42 fveq1 6842 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘¦) = (πΉβ€˜π‘¦))
4342oveq2d 7374 . . . . . . . 8 (𝑓 = 𝐹 β†’ (π‘₯ Γ— (π‘“β€˜π‘¦)) = (π‘₯ Γ— (πΉβ€˜π‘¦)))
4441, 43eqeq12d 2749 . . . . . . 7 (𝑓 = 𝐹 β†’ ((π‘“β€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (π‘“β€˜π‘¦)) ↔ (πΉβ€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (πΉβ€˜π‘¦))))
45442ralbidv 3209 . . . . . 6 (𝑓 = 𝐹 β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐸 (π‘“β€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (π‘“β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐸 (πΉβ€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (πΉβ€˜π‘¦))))
4645anbi2d 630 . . . . 5 (𝑓 = 𝐹 β†’ ((𝐿 = 𝐾 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐸 (π‘“β€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (π‘“β€˜π‘¦))) ↔ (𝐿 = 𝐾 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐸 (πΉβ€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (πΉβ€˜π‘¦)))))
4746elrab 3646 . . . 4 (𝐹 ∈ {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐸 (π‘“β€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (π‘“β€˜π‘¦)))} ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝐿 = 𝐾 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐸 (πΉβ€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (πΉβ€˜π‘¦)))))
48 3anass 1096 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐸 (πΉβ€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (πΉβ€˜π‘¦))) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝐿 = 𝐾 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐸 (πΉβ€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (πΉβ€˜π‘¦)))))
4947, 48bitr4i 278 . . 3 (𝐹 ∈ {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐸 (π‘“β€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (π‘“β€˜π‘¦)))} ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐸 (πΉβ€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (πΉβ€˜π‘¦))))
5040, 49bitrdi 287 . 2 ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) β†’ (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐸 (πΉβ€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (πΉβ€˜π‘¦)))))
512, 50biadanii 821 1 (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐸 (πΉβ€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (πΉβ€˜π‘¦)))))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  {crab 3406  Vcvv 3444  [wsbc 3740  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  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-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-lmhm 20498
This theorem is referenced by:  islmhm3  20504  lmhmlem  20505  lmhmlin  20511  islmhmd  20515  reslmhm  20528  lmhmpropd  20549
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