Theorem lindslinindimp2lem4 49542

Description: Lemma 4 for lindslinindsimp2 49544. (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 = (0g‘𝑅)
lindslinind.z	⊢ 𝑍 = (0g‘𝑀)
lindslinind.y	⊢ 𝑌 = ((invg‘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 Σg (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 490	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → 𝑀 ∈ LMod)
2	1	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → 𝑀 ∈ LMod)
3		simprl 783	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → 𝑆 ⊆ (Base‘𝑀))
4		elpwg 4560	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
5	4	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
6	3, 5	mpbird 260	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → 𝑆 ∈ 𝒫 (Base‘𝑀))
7		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
8	7	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → 𝑥 ∈ 𝑆)
9	2, 6, 8	3jca 1146	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))
10	9	adantl 487	. . . . . . . . . 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 488	. . . . . . . . . 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 2761	. . . . . . . . . . 11 ⊢ (Base‘𝑀) = (Base‘𝑀)
15		lindslinind.r	. . . . . . . . . . 11 ⊢ 𝑅 = (Scalar‘𝑀)
16		lindslinind.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅)
17		eqid 2761	. . . . . . . . . . 11 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
18		eqid 2761	. . . . . . . . . . 11 ⊢ (+g‘𝑀) = (+g‘𝑀)
19		lindslinind.0	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝑅)
20	14, 15, 16, 17, 18, 19	lincdifsn 49505	. . . . . . . . . 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 1398	. . . . . . . . 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 2763	. . . . . . . 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 21135	. . . . . . . . . . 11 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
24	23	adantl 487	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → 𝑀 ∈ Grp)
25	24	ad2antrl 741	. . . . . . . . 9 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → 𝑀 ∈ Grp)
26	1	ad2antrl 741	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → 𝑀 ∈ LMod)
27		elmapi 8862	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐵 ↑m 𝑆) → 𝑓:𝑆⟶𝐵)
28		ffvelcdm 7079	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝑆⟶𝐵 ∧ 𝑥 ∈ 𝑆) → (𝑓‘𝑥) ∈ 𝐵)
29	28	expcom 419	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑆 → (𝑓:𝑆⟶𝐵 → (𝑓‘𝑥) ∈ 𝐵))
30	29	ad2antll 742	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑓:𝑆⟶𝐵 → (𝑓‘𝑥) ∈ 𝐵))
31	27, 30	syl5com 32	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝐵 ↑m 𝑆) → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑓‘𝑥) ∈ 𝐵))
32	31	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑓‘𝑥) ∈ 𝐵))
33	32	imp 412	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (𝑓‘𝑥) ∈ 𝐵)
34		ssel2 3926	. . . . . . . . . . 11 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝑀))
35	34	ad2antll 742	. . . . . . . . . 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 21146	. . . . . . . . . 10 ⊢ ((𝑀 ∈ LMod ∧ (𝑓‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ (Base‘𝑀)) → ((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥) ∈ (Base‘𝑀))
37	26, 33, 35, 36	syl3anc 1398	. . . . . . . . 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 5291	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∖ {𝑥}) ∈ V)
39	38	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∖ {𝑥}) ∈ V)
40		ssdifss 4087	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
41	40	ad2antrl 741	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
42	39, 41	jca 521	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → ((𝑆 ∖ {𝑥}) ∈ V ∧ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)))
43	42	adantl 487	. . . . . . . . . 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 783	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod))
45		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ (Base‘𝑀))
46	45	ad2antll 742	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → 𝑆 ⊆ (Base‘𝑀))
47	7	ad2antll 742	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → 𝑥 ∈ 𝑆)
48		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) → 𝑓 ∈ (𝐵 ↑m 𝑆))
49	48	adantr 486	. . . . . . . . . . . 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 = (0g‘𝑀)
51		lindslinind.y	. . . . . . . . . . . . 13 ⊢ 𝑌 = ((invg‘𝑅)‘(𝑓‘𝑥))
52	15, 16, 19, 50, 51, 12	lindslinindimp2lem2 49540	. . . . . . . . . . . 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 1399	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → 𝐺 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥})))
54		simpr 490	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))
55	54	adantl 487	. . . . . . . . . . . 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 49541	. . . . . . . . . . . 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 1398	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → 𝐺 finSupp 0 )
58	53, 57	jca 521	. . . . . . . . . 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 49494	. . . . . . . . . 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 1398	. . . . . . . . 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 2761	. . . . . . . . . 10 ⊢ (invg‘𝑀) = (invg‘𝑀)
62	14, 18, 50, 61	grpinvid2 19196	. . . . . . . . 9 ⊢ ((𝑀 ∈ Grp ∧ ((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥) ∈ (Base‘𝑀) ∧ (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ∈ (Base‘𝑀)) → (((invg‘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 1398	. . . . . . . 8 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (((invg‘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 285	. . . . . . 7 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 ↔ ((invg‘𝑀)‘((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥)) = (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
65		eqcom 2768	. . . . . . . 8 ⊢ (((invg‘𝑀)‘((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥)) = (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) = ((invg‘𝑀)‘((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥)))
66	15	fveq2i 6886	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑅) = (Base‘(Scalar‘𝑀))
67	16, 66	eqtri 2784	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘(Scalar‘𝑀))
68	67	oveq1i 7428	. . . . . . . . . . . 12 ⊢ (𝐵 ↑m (𝑆 ∖ {𝑥})) = ((Base‘(Scalar‘𝑀)) ↑m (𝑆 ∖ {𝑥}))
69	53, 68	eleqtrdi 2871	. . . . . . . . . . 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 4563	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
71	70	adantl 487	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
72		lincval 49490	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ LMod ∧ 𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑m (𝑆 ∖ {𝑥})) ∧ (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀)) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) = (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝐺‘𝑦)( ·𝑠 ‘𝑀)𝑦))))
73	26, 69, 71, 72	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) = (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝐺‘𝑦)( ·𝑠 ‘𝑀)𝑦))))
74	73	eqeq1d 2763	. . . . . . . . 9 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → ((𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) = ((invg‘𝑀)‘((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥)) ↔ (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝐺‘𝑦)( ·𝑠 ‘𝑀)𝑦))) = ((invg‘𝑀)‘((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥))))
75	12	fveq1i 6884	. . . . . . . . . . . . . . . 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 6902	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (𝑆 ∖ {𝑥}) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑦) = (𝑓‘𝑦))
78	77	adantl 487	. . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . . 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 7433	. . . . . . . . . . . . 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 5198	. . . . . . . . . . . 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 7434	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝐺‘𝑦)( ·𝑠 ‘𝑀)𝑦))) = (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·𝑠 ‘𝑀)𝑦))))
83		eqid 2761	. . . . . . . . . . . . 13 ⊢ (invg‘𝑅) = (invg‘𝑅)
84	14, 15, 17, 61, 16, 83, 26, 35, 33	lmodvsneg 21174	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → ((invg‘𝑀)‘((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥)) = (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥))
85	51	eqcomi 2770	. . . . . . . . . . . . . 14 ⊢ ((invg‘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))) → ((invg‘𝑅)‘(𝑓‘𝑥)) = 𝑌)
87	86	oveq1d 7433	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥) = (𝑌( ·𝑠 ‘𝑀)𝑥))
88	84, 87	eqtrd 2796	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → ((invg‘𝑀)‘((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥)) = (𝑌( ·𝑠 ‘𝑀)𝑥))
89	82, 88	eqeq12d 2777	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → ((𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝐺‘𝑦)( ·𝑠 ‘𝑀)𝑦))) = ((invg‘𝑀)‘((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥)) ↔ (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·𝑠 ‘𝑀)𝑦))) = (𝑌( ·𝑠 ‘𝑀)𝑥)))
90	89	biimpd 232	. . . . . . . . 9 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → ((𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝐺‘𝑦)( ·𝑠 ‘𝑀)𝑦))) = ((invg‘𝑀)‘((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥)) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·𝑠 ‘𝑀)𝑦))) = (𝑌( ·𝑠 ‘𝑀)𝑥)))
91	74, 90	sylbid 243	. . . . . . . 8 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → ((𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) = ((invg‘𝑀)‘((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥)) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·𝑠 ‘𝑀)𝑦))) = (𝑌( ·𝑠 ‘𝑀)𝑥)))
92	65, 91	biimtrid 245	. . . . . . 7 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (((invg‘𝑀)‘((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥)) = (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·𝑠 ‘𝑀)𝑦))) = (𝑌( ·𝑠 ‘𝑀)𝑥)))
93	64, 92	sylbid 243	. . . . . 6 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·𝑠 ‘𝑀)𝑦))) = (𝑌( ·𝑠 ‘𝑀)𝑥)))
94	93	ex 418	. . . . 5 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·𝑠 ‘𝑀)𝑦))) = (𝑌( ·𝑠 ‘𝑀)𝑥))))
95	94	com23 87	. . . 4 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·𝑠 ‘𝑀)𝑦))) = (𝑌( ·𝑠 ‘𝑀)𝑥))))
96	95	3impia 1135	. . 3 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·𝑠 ‘𝑀)𝑦))) = (𝑌( ·𝑠 ‘𝑀)𝑥)))
97	96	com12 33	. 2 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·𝑠 ‘𝑀)𝑦))) = (𝑌( ·𝑠 ‘𝑀)𝑥)))
98	97	3impia 1135	1 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑦)( ·𝑠 ‘𝑀)𝑦))) = (𝑌( ·𝑠 ‘𝑀)𝑥))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ∖ cdif 3896   ⊆ wss 3899  𝒫 cpw 4557  {csn 4584   class class class wbr 5103   ↦ cmpt 5186   ↾ cres 5653  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418   ↑_m cmap 8840   finSupp cfsupp 9346  Basecbs 17380  +_gcplusg 17421  Scalarcsca 17424   ·_𝑠 cvsca 17425  0_gc0g 17603   Σ_g cgsu 17604  Grpcgrp 19137  inv_gcminusg 19138  LModclmod 21128   linC clinc 49485

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 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-of 7691  df-om 7876  df-1st 7999  df-2nd 8000  df-supp 8171  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-er 8710  df-map 8842  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-fsupp 9347  df-oi 9497  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-n0 12600  df-z 12687  df-uz 12959  df-fz 13633  df-fzo 13782  df-seq 14138  df-hash 14468  df-sets 17335  df-slot 17353  df-ndx 17365  df-base 17381  df-ress 17402  df-plusg 17434  df-0g 17605  df-gsum 17606  df-mre 17749  df-mrc 17750  df-acs 17752  df-mgm 18809  df-sgrp 18901  df-mnd 18917  df-submnd 18972  df-grp 19140  df-minusg 19141  df-mulg 19271  df-cntz 19524  df-cmn 19989  df-abl 19990  df-mgp 20354  df-rng 20368  df-ur 20401  df-ring 20454  df-lmod 21130  df-linc 49487

This theorem is used by:  lindslinindsimp2lem5  49543