Theorem lindslinindimp2lem4 46532

Description: Lemma 4 for lindslinindsimp2 46534. (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 485	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → 𝑀 ∈ LMod)
2	1	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → 𝑀 ∈ LMod)
3		simprl 769	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → 𝑆 ⊆ (Base‘𝑀))
4		elpwg 4563	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
5	4	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
6	3, 5	mpbird 256	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → 𝑆 ∈ 𝒫 (Base‘𝑀))
7		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
8	7	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → 𝑥 ∈ 𝑆)
9	2, 6, 8	3jca 1128	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))
10	9	adantl 482	. . . . . . . . . 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 483	. . . . . . . . . 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 2736	. . . . . . . . . . 11 ⊢ (Base‘𝑀) = (Base‘𝑀)
15		lindslinind.r	. . . . . . . . . . 11 ⊢ 𝑅 = (Scalar‘𝑀)
16		lindslinind.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅)
17		eqid 2736	. . . . . . . . . . 11 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
18		eqid 2736	. . . . . . . . . . 11 ⊢ (+g‘𝑀) = (+g‘𝑀)
19		lindslinind.0	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝑅)
20	14, 15, 16, 17, 18, 19	lincdifsn 46495	. . . . . . . . . 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 1371	. . . . . . . . 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 2738	. . . . . . . 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 20329	. . . . . . . . . . 11 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
24	23	adantl 482	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → 𝑀 ∈ Grp)
25	24	ad2antrl 726	. . . . . . . . 9 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → 𝑀 ∈ Grp)
26	1	ad2antrl 726	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → 𝑀 ∈ LMod)
27		elmapi 8787	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐵 ↑m 𝑆) → 𝑓:𝑆⟶𝐵)
28		ffvelcdm 7032	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝑆⟶𝐵 ∧ 𝑥 ∈ 𝑆) → (𝑓‘𝑥) ∈ 𝐵)
29	28	expcom 414	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑆 → (𝑓:𝑆⟶𝐵 → (𝑓‘𝑥) ∈ 𝐵))
30	29	ad2antll 727	. . . . . . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑓‘𝑥) ∈ 𝐵))
33	32	imp 407	. . . . . . . . . 10 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (𝑓‘𝑥) ∈ 𝐵)
34		ssel2 3939	. . . . . . . . . . 11 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝑀))
35	34	ad2antll 727	. . . . . . . . . 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 20339	. . . . . . . . . 10 ⊢ ((𝑀 ∈ LMod ∧ (𝑓‘𝑥) ∈ 𝐵 ∧ 𝑥 ∈ (Base‘𝑀)) → ((𝑓‘𝑥)( ·𝑠 ‘𝑀)𝑥) ∈ (Base‘𝑀))
37	26, 33, 35, 36	syl3anc 1371	. . . . . . . . 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 5284	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∖ {𝑥}) ∈ V)
39	38	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∖ {𝑥}) ∈ V)
40		ssdifss 4095	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
41	40	ad2antrl 726	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
42	39, 41	jca 512	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → ((𝑆 ∖ {𝑥}) ∈ V ∧ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)))
43	42	adantl 482	. . . . . . . . . 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 769	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod))
45		simpl 483	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ (Base‘𝑀))
46	45	ad2antll 727	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → 𝑆 ⊆ (Base‘𝑀))
47	7	ad2antll 727	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → 𝑥 ∈ 𝑆)
48		simpl 483	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) → 𝑓 ∈ (𝐵 ↑m 𝑆))
49	48	adantr 481	. . . . . . . . . . . 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 46530	. . . . . . . . . . . 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 1372	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → 𝐺 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥})))
54		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))
55	54	adantl 482	. . . . . . . . . . . 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 46531	. . . . . . . . . . . 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 1371	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → 𝐺 finSupp 0 )
58	53, 57	jca 512	. . . . . . . . . 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 46484	. . . . . . . . . 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 1371	. . . . . . . . 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 2736	. . . . . . . . . 10 ⊢ (invg‘𝑀) = (invg‘𝑀)
62	14, 18, 50, 61	grpinvid2 18803	. . . . . . . . 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 1371	. . . . . . . 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 281	. . . . . . 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 2743	. . . . . . . 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 6845	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑅) = (Base‘(Scalar‘𝑀))
67	16, 66	eqtri 2764	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘(Scalar‘𝑀))
68	67	oveq1i 7367	. . . . . . . . . . . 12 ⊢ (𝐵 ↑m (𝑆 ∖ {𝑥})) = ((Base‘(Scalar‘𝑀)) ↑m (𝑆 ∖ {𝑥}))
69	53, 68	eleqtrdi 2848	. . . . . . . . . . 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 4566	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
71	70	adantl 482	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
72		lincval 46480	. . . . . . . . . . 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 1371	. . . . . . . . . 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 2738	. . . . . . . . 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 6843	. . . . . . . . . . . . . . . 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 6861	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (𝑆 ∖ {𝑥}) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑦) = (𝑓‘𝑦))
78	77	adantl 482	. . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . 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 7372	. . . . . . . . . . . . 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 5205	. . . . . . . . . . . 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 7373	. . . . . . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ (invg‘𝑅) = (invg‘𝑅)
84	14, 15, 17, 61, 16, 83, 26, 35, 33	lmodvsneg 20366	. . . . . . . . . . . 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 2745	. . . . . . . . . . . . . 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 7372	. . . . . . . . . . . 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 2776	. . . . . . . . . . 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 2752	. . . . . . . . . 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 228	. . . . . . . . 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 239	. . . . . . . 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 241	. . . . . . 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 239	. . . . . 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 413	. . . . 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 86	. . . 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 1117	. . 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 32	. 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 1117	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))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3445   ∖ cdif 3907   ⊆ wss 3910  𝒫 cpw 4560  {csn 4586   class class class wbr 5105   ↦ cmpt 5188   ↾ cres 5635  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ↑_m cmap 8765   finSupp cfsupp 9305  Basecbs 17083  +_gcplusg 17133  Scalarcsca 17136   ·_𝑠 cvsca 17137  0_gc0g 17321   Σ_g cgsu 17322  Grpcgrp 18748  inv_gcminusg 18749  LModclmod 20322   linC clinc 46475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-gsum 17324  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-grp 18751  df-minusg 18752  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-lmod 20324  df-linc 46477

This theorem is referenced by:  lindslinindsimp2lem5  46533