Theorem islmhm 21024

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 = (Scalar‘𝑆)
islmhm.l	⊢ 𝐿 = (Scalar‘𝑇)
islmhm.b	⊢ 𝐵 = (Base‘𝐾)
islmhm.e	⊢ 𝐸 = (Base‘𝑆)
islmhm.m	⊢ · = ( ·𝑠 ‘𝑆)
islmhm.n	⊢ × = ( ·𝑠 ‘𝑇)

Assertion

Ref	Expression
islmhm	⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))))

Distinct variable groups:   𝑥,𝐵   𝑦,𝐸   𝑥,𝑦,𝑆   𝑥,𝐹,𝑦   𝑥,𝑇,𝑦

Allowed substitution hints:   𝐵(𝑦)   · (𝑥,𝑦)   × (𝑥,𝑦)   𝐸(𝑥)   𝐾(𝑥,𝑦)   𝐿(𝑥,𝑦)

Proof of Theorem islmhm

Dummy variables 𝑓 𝑠 𝑡 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-lmhm 21019	. . 3 ⊢ LMHom = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑓 ∈ (𝑠 GrpHom 𝑡) ∣ [(Scalar‘𝑠) / 𝑤]((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠 ‘𝑠)𝑦)) = (𝑥( ·𝑠 ‘𝑡)(𝑓‘𝑦)))})
2	1	elmpocl 7604	. 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LMod ∧ 𝑇 ∈ LMod))
3		oveq12 7372	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 GrpHom 𝑡) = (𝑆 GrpHom 𝑇))
4		fvexd 6849	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (Scalar‘𝑠) ∈ V)
5		simplr 774	. . . . . . . . . . 11 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑡 = 𝑇)
6	5	fveq2d 6838	. . . . . . . . . 10 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Scalar‘𝑡) = (Scalar‘𝑇))
7		islmhm.l	. . . . . . . . . 10 ⊢ 𝐿 = (Scalar‘𝑇)
8	6, 7	eqtr4di 2793	. . . . . . . . 9 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Scalar‘𝑡) = 𝐿)
9		simpr 485	. . . . . . . . . . 11 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑤 = (Scalar‘𝑠))
10		simpll 772	. . . . . . . . . . . 12 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑠 = 𝑆)
11	10	fveq2d 6838	. . . . . . . . . . 11 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Scalar‘𝑠) = (Scalar‘𝑆))
12	9, 11	eqtrd 2775	. . . . . . . . . 10 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑤 = (Scalar‘𝑆))
13		islmhm.k	. . . . . . . . . 10 ⊢ 𝐾 = (Scalar‘𝑆)
14	12, 13	eqtr4di 2793	. . . . . . . . 9 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑤 = 𝐾)
15	8, 14	eqeq12d 2756	. . . . . . . 8 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ((Scalar‘𝑡) = 𝑤 ↔ 𝐿 = 𝐾))
16	14	fveq2d 6838	. . . . . . . . . 10 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑤) = (Base‘𝐾))
17		islmhm.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐾)
18	16, 17	eqtr4di 2793	. . . . . . . . 9 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑤) = 𝐵)
19	10	fveq2d 6838	. . . . . . . . . . 11 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑠) = (Base‘𝑆))
20		islmhm.e	. . . . . . . . . . 11 ⊢ 𝐸 = (Base‘𝑆)
21	19, 20	eqtr4di 2793	. . . . . . . . . 10 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑠) = 𝐸)
22	10	fveq2d 6838	. . . . . . . . . . . . . 14 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ( ·𝑠 ‘𝑠) = ( ·𝑠 ‘𝑆))
23		islmhm.m	. . . . . . . . . . . . . 14 ⊢ · = ( ·𝑠 ‘𝑆)
24	22, 23	eqtr4di 2793	. . . . . . . . . . . . 13 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ( ·𝑠 ‘𝑠) = · )
25	24	oveqd 7380	. . . . . . . . . . . 12 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (𝑥( ·𝑠 ‘𝑠)𝑦) = (𝑥 · 𝑦))
26	25	fveq2d 6838	. . . . . . . . . . 11 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (𝑓‘(𝑥( ·𝑠 ‘𝑠)𝑦)) = (𝑓‘(𝑥 · 𝑦)))
27	5	fveq2d 6838	. . . . . . . . . . . . 13 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ( ·𝑠 ‘𝑡) = ( ·𝑠 ‘𝑇))
28		islmhm.n	. . . . . . . . . . . . 13 ⊢ × = ( ·𝑠 ‘𝑇)
29	27, 28	eqtr4di 2793	. . . . . . . . . . . 12 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ( ·𝑠 ‘𝑡) = × )
30	29	oveqd 7380	. . . . . . . . . . 11 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (𝑥( ·𝑠 ‘𝑡)(𝑓‘𝑦)) = (𝑥 × (𝑓‘𝑦)))
31	26, 30	eqeq12d 2756	. . . . . . . . . 10 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ((𝑓‘(𝑥( ·𝑠 ‘𝑠)𝑦)) = (𝑥( ·𝑠 ‘𝑡)(𝑓‘𝑦)) ↔ (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦))))
32	21, 31	raleqbidv 3314	. . . . . . . . 9 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠 ‘𝑠)𝑦)) = (𝑥( ·𝑠 ‘𝑡)(𝑓‘𝑦)) ↔ ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦))))
33	18, 32	raleqbidv 3314	. . . . . . . 8 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠 ‘𝑠)𝑦)) = (𝑥( ·𝑠 ‘𝑡)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦))))
34	15, 33	anbi12d 638	. . . . . . 7 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠 ‘𝑠)𝑦)) = (𝑥( ·𝑠 ‘𝑡)(𝑓‘𝑦))) ↔ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)))))
35	4, 34	sbcied 3773	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ([(Scalar‘𝑠) / 𝑤]((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠 ‘𝑠)𝑦)) = (𝑥( ·𝑠 ‘𝑡)(𝑓‘𝑦))) ↔ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)))))
36	3, 35	rabeqbidv 3410	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → {𝑓 ∈ (𝑠 GrpHom 𝑡) ∣ [(Scalar‘𝑠) / 𝑤]((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠 ‘𝑠)𝑦)) = (𝑥( ·𝑠 ‘𝑡)(𝑓‘𝑦)))} = {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)))})
37		ovex 7396	. . . . . 6 ⊢ (𝑆 GrpHom 𝑇) ∈ V
38	37	rabex 5274	. . . . 5 ⊢ {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)))} ∈ V
39	36, 1, 38	ovmpoa 7518	. . . 4 ⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝑆 LMHom 𝑇) = {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)))})
40	39	eleq2d 2826	. . 3 ⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ 𝐹 ∈ {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)))}))
41		fveq1 6833	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 · 𝑦)) = (𝐹‘(𝑥 · 𝑦)))
42		fveq1 6833	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
43	42	oveq2d 7379	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑥 × (𝑓‘𝑦)) = (𝑥 × (𝐹‘𝑦)))
44	41, 43	eqeq12d 2756	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)) ↔ (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))
45	44	2ralbidv 3204	. . . . . 6 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))
46	45	anbi2d 636	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦))) ↔ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))))
47	46	elrab 3636	. . . 4 ⊢ (𝐹 ∈ {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)))} ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))))
48		3anass 1100	. . . 4 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))))
49	47, 48	bitr4i 279	. . 3 ⊢ (𝐹 ∈ {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)))} ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))
50	40, 49	bitrdi 288	. 2 ⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))))
51	2, 50	biadanii 827	1 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦)))))

Syntax hints:   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054  {crab 3392  Vcvv 3432  [wsbc 3730  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  Scalarcsca 17221   ·_𝑠 cvsca 17222   GrpHom cghm 19185  LModclmod 20857   LMHom clmhm 21016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-lmhm 21019

This theorem is referenced by:  islmhm3  21025  lmhmlem  21026  lmhmlin  21032  islmhmd  21036  reslmhm  21049  lmhmpropd  21070  evls1maplmhm  22370  lactlmhm  33825