lincresunit3lem1 - Mathbox for Alexander van der Vekens

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Theorem lincresunit3lem1 42268

Description: Lemma 1 for lincresunit3 42270. (Contributed by AV, 17-May-2019.)

**Hypotheses**
Ref	Expression
lincresunit.b	⊢ 𝐵 = (Base‘𝑀)
lincresunit.r	⊢ 𝑅 = (Scalar‘𝑀)
lincresunit.e	⊢ 𝐸 = (Base‘𝑅)
lincresunit.u	⊢ 𝑈 = (Unit‘𝑅)
lincresunit.0	⊢ 0 = (0_g‘𝑅)
lincresunit.z	⊢ 𝑍 = (0_g‘𝑀)
lincresunit.n	⊢ 𝑁 = (inv_g‘𝑅)
lincresunit.i	⊢ 𝐼 = (inv_r‘𝑅)
lincresunit.t	⊢ · = (._r‘𝑅)
lincresunit.g	⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠)))

**Assertion**
Ref	Expression
lincresunit3lem1	⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) → ((𝑁‘(𝐹‘𝑋))( ·_𝑠 ‘𝑀)((𝐺‘𝑧)( ·_𝑠 ‘𝑀)𝑧)) = ((𝐹‘𝑧)( ·_𝑠 ‘𝑀)𝑧))

Distinct variable groups: 𝐵,𝑠 𝐸,𝑠 𝐹,𝑠 𝑀,𝑠 𝑆,𝑠 𝑋,𝑠 𝑈,𝑠 𝐼,𝑠 𝑁,𝑠 · ,𝑠 𝑧,𝑠 Allowed substitution hints: 𝐵(𝑧) 𝑅(𝑧,𝑠) 𝑆(𝑧) · (𝑧) 𝑈(𝑧) 𝐸(𝑧) 𝐹(𝑧) 𝐺(𝑧,𝑠) 𝐼(𝑧) 𝑀(𝑧) 𝑁(𝑧) 𝑋(𝑧) 0 (𝑧,𝑠) 𝑍(𝑧,𝑠)

**Proof of Theorem lincresunit3lem1**
Step	Hyp	Ref	Expression
1		lincresunit.g	. . . . . 6 ⊢ 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠)))
2	1	a1i 11	. . . . 5 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) → 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠))))
3		fveq2 6191	. . . . . . 7 ⊢ (𝑠 = 𝑧 → (𝐹‘𝑠) = (𝐹‘𝑧))
4	3	oveq2d 6666	. . . . . 6 ⊢ (𝑠 = 𝑧 → ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠)) = ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑧)))
5	4	adantl 482	. . . . 5 ⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) ∧ 𝑠 = 𝑧) → ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑠)) = ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑧)))
6		simpr3 1069	. . . . 5 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) → 𝑧 ∈ (𝑆 ∖ {𝑋}))
7		ovexd 6680	. . . . 5 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) → ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑧)) ∈ V)
8	2, 5, 6, 7	fvmptd 6288	. . . 4 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) → (𝐺‘𝑧) = ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑧)))
9	8	oveq1d 6665	. . 3 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) → ((𝐺‘𝑧)( ·_𝑠 ‘𝑀)𝑧) = (((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑧))( ·_𝑠 ‘𝑀)𝑧))
10	9	oveq2d 6666	. 2 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) → ((𝑁‘(𝐹‘𝑋))( ·_𝑠 ‘𝑀)((𝐺‘𝑧)( ·_𝑠 ‘𝑀)𝑧)) = ((𝑁‘(𝐹‘𝑋))( ·_𝑠 ‘𝑀)(((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑧))( ·_𝑠 ‘𝑀)𝑧)))
11		simp2 1062	. . . 4 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑀 ∈ LMod)
12	11	adantr 481	. . 3 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) → 𝑀 ∈ LMod)
13		lincresunit.r	. . . . . 6 ⊢ 𝑅 = (Scalar‘𝑀)
14	13	lmodfgrp 18872	. . . . 5 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ Grp)
15	14	3ad2ant2 1083	. . . 4 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑅 ∈ Grp)
16		lincresunit.e	. . . . . 6 ⊢ 𝐸 = (Base‘𝑅)
17		lincresunit.u	. . . . . 6 ⊢ 𝑈 = (Unit‘𝑅)
18	16, 17	unitcl 18659	. . . . 5 ⊢ ((𝐹‘𝑋) ∈ 𝑈 → (𝐹‘𝑋) ∈ 𝐸)
19	18	3ad2ant2 1083	. . . 4 ⊢ ((𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋})) → (𝐹‘𝑋) ∈ 𝐸)
20		lincresunit.n	. . . . 5 ⊢ 𝑁 = (inv_g‘𝑅)
21	16, 20	grpinvcl 17467	. . . 4 ⊢ ((𝑅 ∈ Grp ∧ (𝐹‘𝑋) ∈ 𝐸) → (𝑁‘(𝐹‘𝑋)) ∈ 𝐸)
22	15, 19, 21	syl2an 494	. . 3 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) → (𝑁‘(𝐹‘𝑋)) ∈ 𝐸)
23		3simpa 1058	. . . . 5 ⊢ ((𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋})) → (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈))
24	23	anim2i 593	. . . 4 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) → ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)))
25		eldifi 3732	. . . . . 6 ⊢ (𝑧 ∈ (𝑆 ∖ {𝑋}) → 𝑧 ∈ 𝑆)
26	25	3ad2ant3 1084	. . . . 5 ⊢ ((𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋})) → 𝑧 ∈ 𝑆)
27	26	adantl 482	. . . 4 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) → 𝑧 ∈ 𝑆)
28		lincresunit.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑀)
29		lincresunit.0	. . . . 5 ⊢ 0 = (0_g‘𝑅)
30		lincresunit.z	. . . . 5 ⊢ 𝑍 = (0_g‘𝑀)
31		lincresunit.i	. . . . 5 ⊢ 𝐼 = (inv_r‘𝑅)
32		lincresunit.t	. . . . 5 ⊢ · = (._r‘𝑅)
33	28, 13, 16, 17, 29, 30, 20, 31, 32, 1	lincresunitlem2 42265	. . . 4 ⊢ ((((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈)) ∧ 𝑧 ∈ 𝑆) → ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑧)) ∈ 𝐸)
34	24, 27, 33	syl2anc 693	. . 3 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) → ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑧)) ∈ 𝐸)
35		elpwi 4168	. . . . . . . . 9 ⊢ (𝑆 ∈ 𝒫 𝐵 → 𝑆 ⊆ 𝐵)
36	35	sseld 3602	. . . . . . . 8 ⊢ (𝑆 ∈ 𝒫 𝐵 → (𝑧 ∈ 𝑆 → 𝑧 ∈ 𝐵))
37	25, 36	syl5com 31	. . . . . . 7 ⊢ (𝑧 ∈ (𝑆 ∖ {𝑋}) → (𝑆 ∈ 𝒫 𝐵 → 𝑧 ∈ 𝐵))
38	37	3ad2ant3 1084	. . . . . 6 ⊢ ((𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋})) → (𝑆 ∈ 𝒫 𝐵 → 𝑧 ∈ 𝐵))
39	38	com12 32	. . . . 5 ⊢ (𝑆 ∈ 𝒫 𝐵 → ((𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋})) → 𝑧 ∈ 𝐵))
40	39	3ad2ant1 1082	. . . 4 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋})) → 𝑧 ∈ 𝐵))
41	40	imp 445	. . 3 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) → 𝑧 ∈ 𝐵)
42		eqid 2622	. . . . 5 ⊢ ( ·_𝑠 ‘𝑀) = ( ·_𝑠 ‘𝑀)
43	28, 13, 42, 16, 32	lmodvsass 18888	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ ((𝑁‘(𝐹‘𝑋)) ∈ 𝐸 ∧ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑧)) ∈ 𝐸 ∧ 𝑧 ∈ 𝐵)) → (((𝑁‘(𝐹‘𝑋)) · ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑧)))( ·_𝑠 ‘𝑀)𝑧) = ((𝑁‘(𝐹‘𝑋))( ·_𝑠 ‘𝑀)(((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑧))( ·_𝑠 ‘𝑀)𝑧)))
44	43	eqcomd 2628	. . 3 ⊢ ((𝑀 ∈ LMod ∧ ((𝑁‘(𝐹‘𝑋)) ∈ 𝐸 ∧ ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑧)) ∈ 𝐸 ∧ 𝑧 ∈ 𝐵)) → ((𝑁‘(𝐹‘𝑋))( ·_𝑠 ‘𝑀)(((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑧))( ·_𝑠 ‘𝑀)𝑧)) = (((𝑁‘(𝐹‘𝑋)) · ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑧)))( ·_𝑠 ‘𝑀)𝑧))
45	12, 22, 34, 41, 44	syl13anc 1328	. 2 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) → ((𝑁‘(𝐹‘𝑋))( ·_𝑠 ‘𝑀)(((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑧))( ·_𝑠 ‘𝑀)𝑧)) = (((𝑁‘(𝐹‘𝑋)) · ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑧)))( ·_𝑠 ‘𝑀)𝑧))
46	13	lmodring 18871	. . . . . 6 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ Ring)
47	46	3ad2ant2 1083	. . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) → 𝑅 ∈ Ring)
48	47	adantr 481	. . . 4 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) → 𝑅 ∈ Ring)
49		elmapi 7879	. . . . . . 7 ⊢ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) → 𝐹:𝑆⟶𝐸)
50		ffvelrn 6357	. . . . . . 7 ⊢ ((𝐹:𝑆⟶𝐸 ∧ 𝑧 ∈ 𝑆) → (𝐹‘𝑧) ∈ 𝐸)
51	49, 25, 50	syl2an 494	. . . . . 6 ⊢ ((𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ 𝑧 ∈ (𝑆 ∖ {𝑋})) → (𝐹‘𝑧) ∈ 𝐸)
52	51	3adant2 1080	. . . . 5 ⊢ ((𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋})) → (𝐹‘𝑧) ∈ 𝐸)
53	52	adantl 482	. . . 4 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) → (𝐹‘𝑧) ∈ 𝐸)
54		simp2 1062	. . . . 5 ⊢ ((𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋})) → (𝐹‘𝑋) ∈ 𝑈)
55	54	adantl 482	. . . 4 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) → (𝐹‘𝑋) ∈ 𝑈)
56	16, 17, 20, 31, 32	invginvrid 42148	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐹‘𝑧) ∈ 𝐸 ∧ (𝐹‘𝑋) ∈ 𝑈) → ((𝑁‘(𝐹‘𝑋)) · ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑧))) = (𝐹‘𝑧))
57	48, 53, 55, 56	syl3anc 1326	. . 3 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) → ((𝑁‘(𝐹‘𝑋)) · ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑧))) = (𝐹‘𝑧))
58	57	oveq1d 6665	. 2 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) → (((𝑁‘(𝐹‘𝑋)) · ((𝐼‘(𝑁‘(𝐹‘𝑋))) · (𝐹‘𝑧)))( ·_𝑠 ‘𝑀)𝑧) = ((𝐹‘𝑧)( ·_𝑠 ‘𝑀)𝑧))
59	10, 45, 58	3eqtrd 2660	1 ⊢ (((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod ∧ 𝑋 ∈ 𝑆) ∧ (𝐹 ∈ (𝐸 ↑_𝑚 𝑆) ∧ (𝐹‘𝑋) ∈ 𝑈 ∧ 𝑧 ∈ (𝑆 ∖ {𝑋}))) → ((𝑁‘(𝐹‘𝑋))( ·_𝑠 ‘𝑀)((𝐺‘𝑧)( ·_𝑠 ‘𝑀)𝑧)) = ((𝐹‘𝑧)( ·_𝑠 ‘𝑀)𝑧))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∖ cdif 3571 𝒫 cpw 4158 {csn 4177 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑_𝑚 cmap 7857 Basecbs 15857 ._rcmulr 15942 Scalarcsca 15944 ·_𝑠 cvsca 15945 0_gc0g 16100 Grpcgrp 17422 inv_gcminusg 17423 Ringcrg 18547 Unitcui 18639 inv_rcinvr 18671 LModclmod 18863

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-lmod 18865

This theorem is referenced by: lincresunit3lem2 42269

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