Theorem lindslinindsimp2 49220

Description: Implication 2 for lindslininds 49221. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.)

Hypotheses

Ref	Expression
lindslinind.r	⊢ 𝑅 = (Scalar‘𝑀)
lindslinind.b	⊢ 𝐵 = (Base‘𝑅)
lindslinind.0	⊢ 0 = (0g‘𝑅)
lindslinind.z	⊢ 𝑍 = (0g‘𝑀)

Assertion

Ref	Expression
lindslinindsimp2	⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → ((𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠}))) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))))

Distinct variable groups:   𝐵,𝑓,𝑠,𝑦   𝑓,𝑀,𝑠,𝑦   𝑅,𝑓,𝑥   𝑆,𝑓,𝑠,𝑥,𝑦   𝑉,𝑠,𝑦   𝑓,𝑍,𝑠,𝑦   0 ,𝑓,𝑠,𝑥,𝑦   𝑦,𝑅   𝑥,𝐵   𝑥,𝑀   𝑅,𝑠   𝑓,𝑉,𝑥   𝑥,𝑍

Proof of Theorem lindslinindsimp2

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 782	. . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))) → 𝑆 ⊆ (Base‘𝑀))
2		elpwg 4566	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
3	2	ad2antrr 738	. . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))) → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
4	1, 3	mpbird 260	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))) → 𝑆 ∈ 𝒫 (Base‘𝑀))
5		simplr 780	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → 𝑀 ∈ LMod)
6		ssdifss 4095	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀))
7	6	adantl 486	. . . . . . . . . 10 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀))
8		difexg 5301	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∖ {𝑠}) ∈ V)
9	8	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (𝑆 ∖ {𝑠}) ∈ V)
10		elpwg 4566	. . . . . . . . . . 11 ⊢ ((𝑆 ∖ {𝑠}) ∈ V → ((𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀)))
11	9, 10	syl 18	. . . . . . . . . 10 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → ((𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀)))
12	7, 11	mpbird 260	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀))
13		eqid 2763	. . . . . . . . . . . 12 ⊢ (Base‘𝑀) = (Base‘𝑀)
14	13	lspeqlco 49196	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo (𝑆 ∖ {𝑠})) = ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))
15	14	eleq2d 2849	. . . . . . . . . 10 ⊢ ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → ((𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠})) ↔ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠}))))
16	15	bicomd 226	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → ((𝑦( ·𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})) ↔ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠}))))
17	5, 12, 16	syl2anc 595	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → ((𝑦( ·𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})) ↔ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠}))))
18	17	notbid 321	. . . . . . 7 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})) ↔ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠}))))
19		lindslinind.r	. . . . . . . . . . . 12 ⊢ 𝑅 = (Scalar‘𝑀)
20		lindslinind.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑅)
21	13, 19, 20	lcoval 49169	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → ((𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠})) ↔ ((𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ∧ ∃𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(𝑔 finSupp (0g‘𝑅) ∧ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
22		lindslinind.0	. . . . . . . . . . . . . . . 16 ⊢ 0 = (0g‘𝑅)
23	22	eqcomi 2772	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝑅) = 0
24	23	breq2i 5118	. . . . . . . . . . . . . 14 ⊢ (𝑔 finSupp (0g‘𝑅) ↔ 𝑔 finSupp 0 )
25	24	anbi1i 635	. . . . . . . . . . . . 13 ⊢ ((𝑔 finSupp (0g‘𝑅) ∧ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
26	25	rexbii 3112	. . . . . . . . . . . 12 ⊢ (∃𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(𝑔 finSupp (0g‘𝑅) ∧ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ ∃𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
27	26	anbi2i 634	. . . . . . . . . . 11 ⊢ (((𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ∧ ∃𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(𝑔 finSupp (0g‘𝑅) ∧ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) ↔ ((𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ∧ ∃𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
28	21, 27	bitrdi 290	. . . . . . . . . 10 ⊢ ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → ((𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠})) ↔ ((𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ∧ ∃𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
29	5, 12, 28	syl2anc 595	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → ((𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠})) ↔ ((𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ∧ ∃𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
30	29	notbid 321	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠})) ↔ ¬ ((𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ∧ ∃𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
31		ianor 997	. . . . . . . . 9 ⊢ (¬ ((𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ∧ ∃𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) ↔ (¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ∨ ¬ ∃𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
32		ralnex 3091	. . . . . . . . . . 11 ⊢ (∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠})) ¬ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ ¬ ∃𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
33		ianor 997	. . . . . . . . . . . 12 ⊢ (¬ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ (¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
34	33	ralbii 3111	. . . . . . . . . . 11 ⊢ (∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠})) ¬ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ ∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
35	32, 34	bitr3i 280	. . . . . . . . . 10 ⊢ (¬ ∃𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ ∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
36	35	orbi2i 925	. . . . . . . . 9 ⊢ ((¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ∨ ¬ ∃𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) ↔ (¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ∨ ∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
37	31, 36	bitri 278	. . . . . . . 8 ⊢ (¬ ((𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ∧ ∃𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) ↔ (¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ∨ ∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
38	30, 37	bitrdi 290	. . . . . . 7 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠})) ↔ (¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ∨ ∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
39	18, 38	bitrd 282	. . . . . 6 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})) ↔ (¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ∨ ∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
40	39	2ralbidv 3229	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})) ↔ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 })(¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ∨ ∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
41		simpllr 787	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑀 ∈ LMod)
42		eldifi 4086	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐵 ∖ { 0 }) → 𝑦 ∈ 𝐵)
43	42	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → 𝑦 ∈ 𝐵)
44	43	adantl 486	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑦 ∈ 𝐵)
45		ssel2 3933	. . . . . . . . . . . 12 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ (Base‘𝑀))
46	45	ad2ant2lr 760	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑠 ∈ (Base‘𝑀))
47		eqid 2763	. . . . . . . . . . . 12 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
48	13, 19, 47, 20	lmodvscl 20980	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ LMod ∧ 𝑦 ∈ 𝐵 ∧ 𝑠 ∈ (Base‘𝑀)) → (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀))
49	41, 44, 46, 48	syl3anc 1398	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀))
50	49	notnotd 145	. . . . . . . . 9 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → ¬ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀))
51		nbfal 1585	. . . . . . . . 9 ⊢ (¬ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ↔ (¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ↔ ⊥))
52	50, 51	sylib 221	. . . . . . . 8 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → (¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ↔ ⊥))
53	52	orbi1d 929	. . . . . . 7 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → ((¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ∨ ∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) ↔ (⊥ ∨ ∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
54	53	2ralbidva 3227	. . . . . 6 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 })(¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ∨ ∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) ↔ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 })(⊥ ∨ ∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
55		r19.32v 3198	. . . . . . . . 9 ⊢ (∀𝑦 ∈ (𝐵 ∖ { 0 })(⊥ ∨ ∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) ↔ (⊥ ∨ ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
56	55	ralbii 3111	. . . . . . . 8 ⊢ (∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 })(⊥ ∨ ∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) ↔ ∀𝑠 ∈ 𝑆 (⊥ ∨ ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
57		r19.32v 3198	. . . . . . . 8 ⊢ (∀𝑠 ∈ 𝑆 (⊥ ∨ ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) ↔ (⊥ ∨ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
58	56, 57	bitri 278	. . . . . . 7 ⊢ (∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 })(⊥ ∨ ∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) ↔ (⊥ ∨ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
59		falim 1587	. . . . . . . . 9 ⊢ (⊥ → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → ∀𝑓 ∈ (𝐵 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
60		sneq 4600	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑠 = 𝑥 → {𝑠} = {𝑥})
61	60	difeq2d 4082	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑠 = 𝑥 → (𝑆 ∖ {𝑠}) = (𝑆 ∖ {𝑥}))
62	61	oveq2d 7428	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = 𝑥 → (𝐵 ↑m (𝑆 ∖ {𝑠})) = (𝐵 ↑m (𝑆 ∖ {𝑥})))
63		oveq2 7420	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 = 𝑥 → (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑦( ·𝑠 ‘𝑀)𝑥))
64	61	oveq2d 7428	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 = 𝑥 → (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))
65	63, 64	eqeq12d 2779	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑠 = 𝑥 → ((𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})) ↔ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
66	65	notbid 321	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑠 = 𝑥 → (¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})) ↔ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
67	66	orbi2d 928	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = 𝑥 → ((¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ (¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))))
68	62, 67	raleqbidv 3338	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = 𝑥 → (∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ ∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))))
69	68	ralbidv 3188	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = 𝑥 → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))))
70	69	rspcva 3580	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝑆 ∧ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) → ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
71		lindslinind.z	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑍 = (0g‘𝑀)
72	19, 20, 22, 71	lindslinindsimp2lem5 49219	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓‘𝑥) = 0 )))
73	72	expr 461	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (𝑥 ∈ 𝑆 → ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓‘𝑥) = 0 ))))
74	73	com14 97	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑥 ∈ 𝑆 → ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (𝑓‘𝑥) = 0 ))))
75	70, 74	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝑆 ∧ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) → (𝑥 ∈ 𝑆 → ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (𝑓‘𝑥) = 0 ))))
76	75	ex 417	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑆 → (∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) → (𝑥 ∈ 𝑆 → ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (𝑓‘𝑥) = 0 )))))
77	76	pm2.43a 55	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑆 → (∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) → ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (𝑓‘𝑥) = 0 ))))
78	77	com14 97	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) → ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (𝑥 ∈ 𝑆 → (𝑓‘𝑥) = 0 ))))
79	78	imp 411	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) → ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (𝑥 ∈ 𝑆 → (𝑓‘𝑥) = 0 )))
80	79	expdimp 457	. . . . . . . . . . . 12 ⊢ (((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) ∧ 𝑓 ∈ (𝐵 ↑m 𝑆)) → ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → (𝑥 ∈ 𝑆 → (𝑓‘𝑥) = 0 )))
81	80	ralrimdv 3163	. . . . . . . . . . 11 ⊢ (((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) ∧ 𝑓 ∈ (𝐵 ↑m 𝑆)) → ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))
82	81	ralrimiva 3157	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) → ∀𝑓 ∈ (𝐵 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))
83	82	expcom 418	. . . . . . . . 9 ⊢ (∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → ∀𝑓 ∈ (𝐵 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
84	59, 83	jaoi 870	. . . . . . . 8 ⊢ ((⊥ ∨ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → ∀𝑓 ∈ (𝐵 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
85	84	com12 33	. . . . . . 7 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → ((⊥ ∨ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) → ∀𝑓 ∈ (𝐵 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
86	58, 85	biimtrid 245	. . . . . 6 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 })(⊥ ∨ ∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) → ∀𝑓 ∈ (𝐵 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
87	54, 86	sylbid 243	. . . . 5 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 })(¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ∨ ∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) → ∀𝑓 ∈ (𝐵 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
88	40, 87	sylbid 243	. . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})) → ∀𝑓 ∈ (𝐵 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
89	88	impr 459	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))) → ∀𝑓 ∈ (𝐵 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))
90	4, 89	jca 520	. 2 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
91	90	ex 417	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → ((𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠}))) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570  ⊥wfal 1582   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ∖ cdif 3903   ⊆ wss 3906  𝒫 cpw 4563  {csn 4590   class class class wbr 5110  ‘cfv 6538  (class class class)co 7412   ↑_m cmap 8825   finSupp cfsupp 9322  Basecbs 17270  Scalarcsca 17314   ·_𝑠 cvsca 17315  0_gc0g 17493  LModclmod 20962  LSpanclspn 21073   linC clinc 49161   LinCo clinco 49162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  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 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7676  df-om 7864  df-1st 7987  df-2nd 7988  df-supp 8158  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-er 8695  df-map 8827  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-fsupp 9323  df-oi 9473  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-n0 12506  df-z 12593  df-uz 12864  df-fz 13537  df-fzo 13685  df-seq 14040  df-hash 14369  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  df-plusg 17324  df-0g 17495  df-gsum 17496  df-mre 17639  df-mrc 17640  df-acs 17642  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-mhm 18842  df-submnd 18843  df-grp 19004  df-minusg 19005  df-sbg 19006  df-mulg 19135  df-subg 19190  df-ghm 19285  df-cntz 19388  df-cmn 19853  df-abl 19854  df-mgp 20218  df-rng 20232  df-ur 20265  df-ring 20318  df-lmod 20964  df-lss 21034  df-lsp 21074  df-linc 49163  df-lco 49164

This theorem is referenced by:  lindslininds  49221