Theorem lindslinindimp2lem4 49274

Description: Lemma 4 for lindslinindsimp2 49276. (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 4567	. . . . . . . . . . . . . 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 2765	. . . . . . . . . . 11 ⊢ (Base‘𝑀) = (Base‘𝑀)
15		lindslinind.r	. . . . . . . . . . 11 ⊢ 𝑅 = (Scalar‘𝑀)
16		lindslinind.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅)
17		eqid 2765	. . . . . . . . . . 11 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
18		eqid 2765	. . . . . . . . . . 11 ⊢ (+g‘𝑀) = (+g‘𝑀)
19		lindslinind.0	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝑅)
20	14, 15, 16, 17, 18, 19	lincdifsn 49237	. . . . . . . . . 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 2767	. . . . . . . 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 21018	. . . . . . . . . . 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 8848	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐵 ↑m 𝑆) → 𝑓:𝑆⟶𝐵)
28		ffvelcdm 7080	. . . . . . . . . . . . . . 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 3933	. . . . . . . . . . 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 21029	. . . . . . . . . 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 5302	. . . . . . . . . . . . 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 4094	. . . . . . . . . . . . 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 49272	. . . . . . . . . . . 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 49273	. . . . . . . . . . . 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 49226	. . . . . . . . . 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 2765	. . . . . . . . . 10 ⊢ (invg‘𝑀) = (invg‘𝑀)
62	14, 18, 50, 61	grpinvid2 19083	. . . . . . . . 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 2772	. . . . . . . 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 6888	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑅) = (Base‘(Scalar‘𝑀))
67	16, 66	eqtri 2788	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘(Scalar‘𝑀))
68	67	oveq1i 7426	. . . . . . . . . . . 12 ⊢ (𝐵 ↑m (𝑆 ∖ {𝑥})) = ((Base‘(Scalar‘𝑀)) ↑m (𝑆 ∖ {𝑥}))
69	53, 68	eleqtrdi 2875	. . . . . . . . . . 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 4570	. . . . . . . . . . . 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 49222	. . . . . . . . . . 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 2767	. . . . . . . . 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 6886	. . . . . . . . . . . . . . . 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 6904	. . . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . . 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 7431	. . . . . . . . . . . . 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 5206	. . . . . . . . . . . 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 7432	. . . . . . . . . . 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 2765	. . . . . . . . . . . . 13 ⊢ (invg‘𝑅) = (invg‘𝑅)
84	14, 15, 17, 61, 16, 83, 26, 35, 33	lmodvsneg 21057	. . . . . . . . . . . 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 2774	. . . . . . . . . . . . . 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 7431	. . . . . . . . . . . 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 2800	. . . . . . . . . . 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 2781	. . . . . . . . . 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 2146  Vcvv 3457   ∖ cdif 3903   ⊆ wss 3906  𝒫 cpw 4564  {csn 4591   class class class wbr 5111   ↦ cmpt 5194   ↾ cres 5665  ⟶wf 6536  ‘cfv 6540  (class class class)co 7416   ↑_m cmap 8826   finSupp cfsupp 9324  Basecbs 17287  +_gcplusg 17328  Scalarcsca 17331   ·_𝑠 cvsca 17332  0_gc0g 17510   Σ_g cgsu 17511  Grpcgrp 19024  inv_gcminusg 19025  LModclmod 21011   linC clinc 49217

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7680  df-om 7865  df-1st 7988  df-2nd 7989  df-supp 8159  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-er 8696  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fsupp 9325  df-oi 9475  df-card 9937  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-nn 12245  df-2 12314  df-n0 12516  df-z 12603  df-uz 12875  df-fz 13548  df-fzo 13696  df-seq 14052  df-hash 14381  df-sets 17242  df-slot 17260  df-ndx 17272  df-base 17288  df-ress 17309  df-plusg 17341  df-0g 17512  df-gsum 17513  df-mre 17656  df-mrc 17657  df-acs 17659  df-mgm 18716  df-sgrp 18799  df-mnd 18815  df-submnd 18866  df-grp 19027  df-minusg 19028  df-mulg 19158  df-cntz 19411  df-cmn 19876  df-abl 19877  df-mgp 20241  df-rng 20255  df-ur 20288  df-ring 20341  df-lmod 21013  df-linc 49219

This theorem is used by:  lindslinindsimp2lem5  49275