lindslinindimp2lem4 - Mathbox for Alexander van der Vekens

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Theorem lindslinindimp2lem4 47452

Description: Lemma 4 for lindslinindsimp2 47454. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.) (Proof shortened by II, 16-Feb-2023.)

**Hypotheses**
Ref	Expression
lindslinind.r	⊢ 𝑅 = (Scalar‘𝑀)
lindslinind.b	⊢ 𝐵 = (Base‘𝑅)
lindslinind.0	⊢ 0 = (0_g‘𝑅)
lindslinind.z	⊢ 𝑍 = (0_g‘𝑀)
lindslinind.y	⊢ 𝑌 = ((inv_g‘𝑅)‘(𝑓‘𝑥))
lindslinind.g	⊢ 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))

**Assertion**
Ref	Expression
lindslinindimp2lem4	⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) ∧ (𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (𝑀 Σ_g (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·_𝑠 ‘𝑀)𝑦))) = (𝑌( ·_𝑠 ‘𝑀)𝑥))

Distinct variable groups: 𝐵,𝑓,𝑦 𝑓,𝑀,𝑦 𝑅,𝑓,𝑥 𝑆,𝑓,𝑥,𝑦 𝑦,𝑉 𝑓,𝑍,𝑦 0 ,𝑓,𝑥,𝑦 𝑦,𝐺 Allowed substitution hints: 𝐵(𝑥) 𝑅(𝑦) 𝐺(𝑥,𝑓) 𝑀(𝑥) 𝑉(𝑥,𝑓) 𝑌(𝑥,𝑦,𝑓) 𝑍(𝑥)

**Proof of Theorem lindslinindimp2lem4**
Step	Hyp	Ref	Expression
1		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → 𝑀 ∈ LMod)
2	1	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → 𝑀 ∈ LMod)
3		simprl 770	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → 𝑆 ⊆ (Base‘𝑀))
4		elpwg 4601	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
5	4	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
6	3, 5	mpbird 257	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → 𝑆 ∈ 𝒫 (Base‘𝑀))
7		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
8	7	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → 𝑥 ∈ 𝑆)
9	2, 6, 8	3jca 1126	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))
10	9	adantl 481	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))
11		simpl 482	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ))
12		lindslinind.g	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))
13	12	a1i 11	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥})))
14		eqid 2727	. . . . . . . . . . 11 ⊢ (Base‘𝑀) = (Base‘𝑀)
15		lindslinind.r	. . . . . . . . . . 11 ⊢ 𝑅 = (Scalar‘𝑀)
16		lindslinind.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅)
17		eqid 2727	. . . . . . . . . . 11 ⊢ ( ·_𝑠 ‘𝑀) = ( ·_𝑠 ‘𝑀)
18		eqid 2727	. . . . . . . . . . 11 ⊢ (+_g‘𝑀) = (+_g‘𝑀)
19		lindslinind.0	. . . . . . . . . . 11 ⊢ 0 = (0_g‘𝑅)
20	14, 15, 16, 17, 18, 19	lincdifsn 47415	. . . . . . . . . 10 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) ∧ (𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))) → (𝑓( linC ‘𝑀)𝑆) = ((𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥}))(+_g‘𝑀)((𝑓‘𝑥)( ·_𝑠 ‘𝑀)𝑥)))
21	10, 11, 13, 20	syl3anc 1369	. . . . . . . . 9 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (𝑓( linC ‘𝑀)𝑆) = ((𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥}))(+_g‘𝑀)((𝑓‘𝑥)( ·_𝑠 ‘𝑀)𝑥)))
22	21	eqeq1d 2729	. . . . . . . 8 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 ↔ ((𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥}))(+_g‘𝑀)((𝑓‘𝑥)( ·_𝑠 ‘𝑀)𝑥)) = 𝑍))
23		lmodgrp 20739	. . . . . . . . . . 11 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
24	23	adantl 481	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → 𝑀 ∈ Grp)
25	24	ad2antrl 727	. . . . . . . . 9 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → 𝑀 ∈ Grp)
26	1	ad2antrl 727	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → 𝑀 ∈ LMod)
27		elmapi 8859	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐵 ↑_m 𝑆) → 𝑓:𝑆⟶𝐵)
28		ffvelcdm 7085	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝑆⟶𝐵 ∧ 𝑥 ∈ 𝑆) → (𝑓‘𝑥) ∈ 𝐵)
29	28	expcom 413	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑆 → (𝑓:𝑆⟶𝐵 → (𝑓‘𝑥) ∈ 𝐵))
30	29	ad2antll 728	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑓:𝑆⟶𝐵 → (𝑓‘𝑥) ∈ 𝐵))
31	27, 30	syl5com 31	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝐵 ↑_m 𝑆) → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑓‘𝑥) ∈ 𝐵))
32	31	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑓‘𝑥) ∈ 𝐵))
33	32	imp 406	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (𝑓‘𝑥) ∈ 𝐵)
34		ssel2 3973	. . . . . . . . . . 11 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝑀))
35	34	ad2antll 728	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → 𝑥 ∈ (Base‘𝑀))
36	14, 15, 17, 16	lmodvscl 20750	. . . . . . . . . 10 ⊢ ((𝑀 ∈ LMod ∧ (𝑓‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ (Base‘𝑀)) → ((𝑓‘𝑥)( ·_𝑠 ‘𝑀)𝑥) ∈ (Base‘𝑀))
37	26, 33, 35, 36	syl3anc 1369	. . . . . . . . 9 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → ((𝑓‘𝑥)( ·_𝑠 ‘𝑀)𝑥) ∈ (Base‘𝑀))
38		difexg 5323	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∖ {𝑥}) ∈ V)
39	38	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∖ {𝑥}) ∈ V)
40		ssdifss 4131	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
41	40	ad2antrl 727	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
42	39, 41	jca 511	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → ((𝑆 ∖ {𝑥}) ∈ V ∧ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)))
43	42	adantl 481	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → ((𝑆 ∖ {𝑥}) ∈ V ∧ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)))
44		simprl 770	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod))
45		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ (Base‘𝑀))
46	45	ad2antll 728	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → 𝑆 ⊆ (Base‘𝑀))
47	7	ad2antll 728	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → 𝑥 ∈ 𝑆)
48		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) → 𝑓 ∈ (𝐵 ↑_m 𝑆))
49	48	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → 𝑓 ∈ (𝐵 ↑_m 𝑆))
50		lindslinind.z	. . . . . . . . . . . . 13 ⊢ 𝑍 = (0_g‘𝑀)
51		lindslinind.y	. . . . . . . . . . . . 13 ⊢ 𝑌 = ((inv_g‘𝑅)‘(𝑓‘𝑥))
52	15, 16, 19, 50, 51, 12	lindslinindimp2lem2 47450	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑_m 𝑆))) → 𝐺 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑥})))
53	44, 46, 47, 49, 52	syl13anc 1370	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → 𝐺 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑥})))
54		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))
55	54	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))
56	15, 16, 19, 50, 51, 12	lindslinindimp2lem3 47451	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) ∧ (𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 )) → 𝐺 finSupp 0 )
57	44, 55, 11, 56	syl3anc 1369	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → 𝐺 finSupp 0 )
58	53, 57	jca 511	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (𝐺 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑥})) ∧ 𝐺 finSupp 0 ))
59	14, 15, 16, 19	lincfsuppcl 47404	. . . . . . . . . 10 ⊢ ((𝑀 ∈ LMod ∧ ((𝑆 ∖ {𝑥}) ∈ V ∧ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)) ∧ (𝐺 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑥})) ∧ 𝐺 finSupp 0 )) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ∈ (Base‘𝑀))
60	26, 43, 58, 59	syl3anc 1369	. . . . . . . . 9 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ∈ (Base‘𝑀))
61		eqid 2727	. . . . . . . . . 10 ⊢ (inv_g‘𝑀) = (inv_g‘𝑀)
62	14, 18, 50, 61	grpinvid2 18940	. . . . . . . . 9 ⊢ ((𝑀 ∈ Grp ∧ ((𝑓‘𝑥)( ·_𝑠 ‘𝑀)𝑥) ∈ (Base‘𝑀) ∧ (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ∈ (Base‘𝑀)) → (((inv_g‘𝑀)‘((𝑓‘𝑥)( ·_𝑠 ‘𝑀)𝑥)) = (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ ((𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥}))(+_g‘𝑀)((𝑓‘𝑥)( ·_𝑠 ‘𝑀)𝑥)) = 𝑍))
63	25, 37, 60, 62	syl3anc 1369	. . . . . . . 8 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (((inv_g‘𝑀)‘((𝑓‘𝑥)( ·_𝑠 ‘𝑀)𝑥)) = (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ ((𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥}))(+_g‘𝑀)((𝑓‘𝑥)( ·_𝑠 ‘𝑀)𝑥)) = 𝑍))
64	22, 63	bitr4d 282	. . . . . . 7 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 ↔ ((inv_g‘𝑀)‘((𝑓‘𝑥)( ·_𝑠 ‘𝑀)𝑥)) = (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
65		eqcom 2734	. . . . . . . 8 ⊢ (((inv_g‘𝑀)‘((𝑓‘𝑥)( ·_𝑠 ‘𝑀)𝑥)) = (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) = ((inv_g‘𝑀)‘((𝑓‘𝑥)( ·_𝑠 ‘𝑀)𝑥)))
66	15	fveq2i 6894	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑅) = (Base‘(Scalar‘𝑀))
67	16, 66	eqtri 2755	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘(Scalar‘𝑀))
68	67	oveq1i 7424	. . . . . . . . . . . 12 ⊢ (𝐵 ↑_m (𝑆 ∖ {𝑥})) = ((Base‘(Scalar‘𝑀)) ↑_m (𝑆 ∖ {𝑥}))
69	53, 68	eleqtrdi 2838	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → 𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑_m (𝑆 ∖ {𝑥})))
70	39, 41	elpwd 4604	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
71	70	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
72		lincval 47400	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ LMod ∧ 𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑_m (𝑆 ∖ {𝑥})) ∧ (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀)) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) = (𝑀 Σ_g (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝐺‘𝑦)( ·_𝑠 ‘𝑀)𝑦))))
73	26, 69, 71, 72	syl3anc 1369	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) = (𝑀 Σ_g (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝐺‘𝑦)( ·_𝑠 ‘𝑀)𝑦))))
74	73	eqeq1d 2729	. . . . . . . . 9 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → ((𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) = ((inv_g‘𝑀)‘((𝑓‘𝑥)( ·_𝑠 ‘𝑀)𝑥)) ↔ (𝑀 Σ_g (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝐺‘𝑦)( ·_𝑠 ‘𝑀)𝑦))) = ((inv_g‘𝑀)‘((𝑓‘𝑥)( ·_𝑠 ‘𝑀)𝑥))))
75	12	fveq1i 6892	. . . . . . . . . . . . . . . 16 ⊢ (𝐺‘𝑦) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑦)
76	75	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) ∧ 𝑦 ∈ (𝑆 ∖ {𝑥})) → (𝐺‘𝑦) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑦))
77		fvres 6910	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (𝑆 ∖ {𝑥}) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑦) = (𝑓‘𝑦))
78	77	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) ∧ 𝑦 ∈ (𝑆 ∖ {𝑥})) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑦) = (𝑓‘𝑦))
79	76, 78	eqtrd 2767	. . . . . . . . . . . . . 14 ⊢ ((((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) ∧ 𝑦 ∈ (𝑆 ∖ {𝑥})) → (𝐺‘𝑦) = (𝑓‘𝑦))
80	79	oveq1d 7429	. . . . . . . . . . . . 13 ⊢ ((((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) ∧ 𝑦 ∈ (𝑆 ∖ {𝑥})) → ((𝐺‘𝑦)( ·_𝑠 ‘𝑀)𝑦) = ((𝑓‘𝑦)( ·_𝑠 ‘𝑀)𝑦))
81	80	mpteq2dva 5242	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝐺‘𝑦)( ·_𝑠 ‘𝑀)𝑦)) = (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·_𝑠 ‘𝑀)𝑦)))
82	81	oveq2d 7430	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (𝑀 Σ_g (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝐺‘𝑦)( ·_𝑠 ‘𝑀)𝑦))) = (𝑀 Σ_g (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·_𝑠 ‘𝑀)𝑦))))
83		eqid 2727	. . . . . . . . . . . . 13 ⊢ (inv_g‘𝑅) = (inv_g‘𝑅)
84	14, 15, 17, 61, 16, 83, 26, 35, 33	lmodvsneg 20778	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → ((inv_g‘𝑀)‘((𝑓‘𝑥)( ·_𝑠 ‘𝑀)𝑥)) = (((inv_g‘𝑅)‘(𝑓‘𝑥))( ·_𝑠 ‘𝑀)𝑥))
85	51	eqcomi 2736	. . . . . . . . . . . . . 14 ⊢ ((inv_g‘𝑅)‘(𝑓‘𝑥)) = 𝑌
86	85	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → ((inv_g‘𝑅)‘(𝑓‘𝑥)) = 𝑌)
87	86	oveq1d 7429	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (((inv_g‘𝑅)‘(𝑓‘𝑥))( ·_𝑠 ‘𝑀)𝑥) = (𝑌( ·_𝑠 ‘𝑀)𝑥))
88	84, 87	eqtrd 2767	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → ((inv_g‘𝑀)‘((𝑓‘𝑥)( ·_𝑠 ‘𝑀)𝑥)) = (𝑌( ·_𝑠 ‘𝑀)𝑥))
89	82, 88	eqeq12d 2743	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → ((𝑀 Σ_g (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝐺‘𝑦)( ·_𝑠 ‘𝑀)𝑦))) = ((inv_g‘𝑀)‘((𝑓‘𝑥)( ·_𝑠 ‘𝑀)𝑥)) ↔ (𝑀 Σ_g (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·_𝑠 ‘𝑀)𝑦))) = (𝑌( ·_𝑠 ‘𝑀)𝑥)))
90	89	biimpd 228	. . . . . . . . 9 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → ((𝑀 Σ_g (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝐺‘𝑦)( ·_𝑠 ‘𝑀)𝑦))) = ((inv_g‘𝑀)‘((𝑓‘𝑥)( ·_𝑠 ‘𝑀)𝑥)) → (𝑀 Σ_g (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·_𝑠 ‘𝑀)𝑦))) = (𝑌( ·_𝑠 ‘𝑀)𝑥)))
91	74, 90	sylbid 239	. . . . . . . 8 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → ((𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) = ((inv_g‘𝑀)‘((𝑓‘𝑥)( ·_𝑠 ‘𝑀)𝑥)) → (𝑀 Σ_g (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·_𝑠 ‘𝑀)𝑦))) = (𝑌( ·_𝑠 ‘𝑀)𝑥)))
92	65, 91	biimtrid 241	. . . . . . 7 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (((inv_g‘𝑀)‘((𝑓‘𝑥)( ·_𝑠 ‘𝑀)𝑥)) = (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) → (𝑀 Σ_g (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·_𝑠 ‘𝑀)𝑦))) = (𝑌( ·_𝑠 ‘𝑀)𝑥)))
93	64, 92	sylbid 239	. . . . . 6 ⊢ (((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → (𝑀 Σ_g (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·_𝑠 ‘𝑀)𝑦))) = (𝑌( ·_𝑠 ‘𝑀)𝑥)))
94	93	ex 412	. . . . 5 ⊢ ((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → (𝑀 Σ_g (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·_𝑠 ‘𝑀)𝑦))) = (𝑌( ·_𝑠 ‘𝑀)𝑥))))
95	94	com23 86	. . . 4 ⊢ ((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑀 Σ_g (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·_𝑠 ‘𝑀)𝑦))) = (𝑌( ·_𝑠 ‘𝑀)𝑥))))
96	95	3impia 1115	. . 3 ⊢ ((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑀 Σ_g (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·_𝑠 ‘𝑀)𝑦))) = (𝑌( ·_𝑠 ‘𝑀)𝑥)))
97	96	com12 32	. 2 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → ((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → (𝑀 Σ_g (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·_𝑠 ‘𝑀)𝑦))) = (𝑌( ·_𝑠 ‘𝑀)𝑥)))
98	97	3impia 1115	1 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) ∧ (𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (𝑀 Σ_g (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·_𝑠 ‘𝑀)𝑦))) = (𝑌( ·_𝑠 ‘𝑀)𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 Vcvv 3469 ∖ cdif 3941 ⊆ wss 3944 𝒫 cpw 4598 {csn 4624 class class class wbr 5142 ↦ cmpt 5225 ↾ cres 5674 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ↑_m cmap 8836 finSupp cfsupp 9377 Basecbs 17171 +_gcplusg 17224 Scalarcsca 17227 ·_𝑠 cvsca 17228 0_gc0g 17412 Σ_g cgsu 17413 Grpcgrp 18881 inv_gcminusg 18882 LModclmod 20732 linC clinc 47395

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-oi 9525 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-n0 12495 df-z 12581 df-uz 12845 df-fz 13509 df-fzo 13652 df-seq 13991 df-hash 14314 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-0g 17414 df-gsum 17415 df-mre 17557 df-mrc 17558 df-acs 17560 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-submnd 18732 df-grp 18884 df-minusg 18885 df-mulg 19015 df-cntz 19259 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-lmod 20734 df-linc 47397

This theorem is referenced by: lindslinindsimp2lem5 47453

W3C validator