Theorem islmhm 21217

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 21212	. . 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 7659	. 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LMod ∧ 𝑇 ∈ LMod))
3		oveq12 7426	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 GrpHom 𝑡) = (𝑆 GrpHom 𝑇))
4		fvexd 6897	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (Scalar‘𝑠) ∈ V)
5		simplr 781	. . . . . . . . . . 11 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑡 = 𝑇)
6	5	fveq2d 6886	. . . . . . . . . 10 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Scalar‘𝑡) = (Scalar‘𝑇))
7		islmhm.l	. . . . . . . . . 10 ⊢ 𝐿 = (Scalar‘𝑇)
8	6, 7	eqtr4di 2815	. . . . . . . . 9 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Scalar‘𝑡) = 𝐿)
9		simpr 490	. . . . . . . . . . 11 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑤 = (Scalar‘𝑠))
10		simpll 779	. . . . . . . . . . . 12 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑠 = 𝑆)
11	10	fveq2d 6886	. . . . . . . . . . 11 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Scalar‘𝑠) = (Scalar‘𝑆))
12	9, 11	eqtrd 2797	. . . . . . . . . 10 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑤 = (Scalar‘𝑆))
13		islmhm.k	. . . . . . . . . 10 ⊢ 𝐾 = (Scalar‘𝑆)
14	12, 13	eqtr4di 2815	. . . . . . . . 9 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → 𝑤 = 𝐾)
15	8, 14	eqeq12d 2778	. . . . . . . 8 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ((Scalar‘𝑡) = 𝑤 ↔ 𝐿 = 𝐾))
16	14	fveq2d 6886	. . . . . . . . . 10 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑤) = (Base‘𝐾))
17		islmhm.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐾)
18	16, 17	eqtr4di 2815	. . . . . . . . 9 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑤) = 𝐵)
19	10	fveq2d 6886	. . . . . . . . . . 11 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑠) = (Base‘𝑆))
20		islmhm.e	. . . . . . . . . . 11 ⊢ 𝐸 = (Base‘𝑆)
21	19, 20	eqtr4di 2815	. . . . . . . . . 10 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (Base‘𝑠) = 𝐸)
22	10	fveq2d 6886	. . . . . . . . . . . . . 14 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ( ·𝑠 ‘𝑠) = ( ·𝑠 ‘𝑆))
23		islmhm.m	. . . . . . . . . . . . . 14 ⊢ · = ( ·𝑠 ‘𝑆)
24	22, 23	eqtr4di 2815	. . . . . . . . . . . . 13 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ( ·𝑠 ‘𝑠) = · )
25	24	oveqd 7434	. . . . . . . . . . . 12 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (𝑥( ·𝑠 ‘𝑠)𝑦) = (𝑥 · 𝑦))
26	25	fveq2d 6886	. . . . . . . . . . 11 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (𝑓‘(𝑥( ·𝑠 ‘𝑠)𝑦)) = (𝑓‘(𝑥 · 𝑦)))
27	5	fveq2d 6886	. . . . . . . . . . . . 13 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ( ·𝑠 ‘𝑡) = ( ·𝑠 ‘𝑇))
28		islmhm.n	. . . . . . . . . . . . 13 ⊢ × = ( ·𝑠 ‘𝑇)
29	27, 28	eqtr4di 2815	. . . . . . . . . . . 12 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → ( ·𝑠 ‘𝑡) = × )
30	29	oveqd 7434	. . . . . . . . . . 11 ⊢ (((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) ∧ 𝑤 = (Scalar‘𝑠)) → (𝑥( ·𝑠 ‘𝑡)(𝑓‘𝑦)) = (𝑥 × (𝑓‘𝑦)))
31	26, 30	eqeq12d 2778	. . . . . . . . . 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 3336	. . . . . . . . 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 3336	. . . . . . . 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 644	. . . . . . 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 3785	. . . . . 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 3432	. . . . 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 7450	. . . . . 6 ⊢ (𝑆 GrpHom 𝑇) ∈ V
38	37	rabex 5307	. . . . 5 ⊢ {𝑓 ∈ (𝑆 GrpHom 𝑇) ∣ (𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)))} ∈ V
39	36, 1, 38	ovmpoa 7572	. . . 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 2848	. . 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 6881	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 · 𝑦)) = (𝐹‘(𝑥 · 𝑦)))
42		fveq1 6881	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
43	42	oveq2d 7433	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑥 × (𝑓‘𝑦)) = (𝑥 × (𝐹‘𝑦)))
44	41, 43	eqeq12d 2778	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥 · 𝑦)) = (𝑥 × (𝑓‘𝑦)) ↔ (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))
45	44	2ralbidv 3228	. . . . . 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 642	. . . . 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 3648	. . . 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 1111	. . . 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 281	. . 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 290	. 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 834	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)))))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078  {crab 3414  Vcvv 3453  [wsbc 3742  ‘cfv 6537  (class class class)co 7417  Basecbs 17307  Scalarcsca 17351   ·_𝑠 cvsca 17352   GrpHom cghm 19346  LModclmod 21050   LMHom clmhm 21209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-lmhm 21212

This theorem is used by:  islmhm3  21218  lmhmlem  21219  lmhmlin  21225  islmhmd  21229  reslmhm  21242  lmhmpropd  21263  evls1maplmhm  22608  lactlmhm  34152