Theorem lindslinindsimp2 48961

Description: Implication 2 for lindslininds 48962. (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 776	. . . 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 4539	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
3	2	ad2antrr 732	. . . 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 258	. . 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 774	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → 𝑀 ∈ LMod)
6		ssdifss 4077	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀))
7	6	adantl 482	. . . . . . . . . 10 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀))
8		difexg 5264	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∖ {𝑠}) ∈ V)
9	8	ad2antrr 732	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (𝑆 ∖ {𝑠}) ∈ V)
10		elpwg 4539	. . . . . . . . . . 11 ⊢ ((𝑆 ∖ {𝑠}) ∈ V → ((𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀)))
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → ((𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀)))
12	7, 11	mpbird 258	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀))
13		eqid 2740	. . . . . . . . . . . 12 ⊢ (Base‘𝑀) = (Base‘𝑀)
14	13	lspeqlco 48937	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo (𝑆 ∖ {𝑠})) = ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))
15	14	eleq2d 2826	. . . . . . . . . 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 224	. . . . . . . . 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 590	. . . . . . . 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 319	. . . . . . 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 48910	. . . . . . . . . . 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 2749	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝑅) = 0
24	23	breq2i 5087	. . . . . . . . . . . . . 14 ⊢ (𝑔 finSupp (0g‘𝑅) ↔ 𝑔 finSupp 0 )
25	24	anbi1i 630	. . . . . . . . . . . . 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 3087	. . . . . . . . . . . 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 629	. . . . . . . . . . 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 288	. . . . . . . . . 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 590	. . . . . . . . 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 319	. . . . . . . 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 989	. . . . . . . . 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 3066	. . . . . . . . . . 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 989	. . . . . . . . . . . 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 3086	. . . . . . . . . . 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 278	. . . . . . . . . 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 918	. . . . . . . . 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 276	. . . . . . . 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 288	. . . . . . 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 280	. . . . . 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 3204	. . . . 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 781	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑀 ∈ LMod)
42		eldifi 4068	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐵 ∖ { 0 }) → 𝑦 ∈ 𝐵)
43	42	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → 𝑦 ∈ 𝐵)
44	43	adantl 482	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑦 ∈ 𝐵)
45		ssel2 3917	. . . . . . . . . . . 12 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ (Base‘𝑀))
46	45	ad2ant2lr 754	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑠 ∈ (Base‘𝑀))
47		eqid 2740	. . . . . . . . . . . 12 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
48	13, 19, 47, 20	lmodvscl 20875	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ LMod ∧ 𝑦 ∈ 𝐵 ∧ 𝑠 ∈ (Base‘𝑀)) → (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀))
49	41, 44, 46, 48	syl3anc 1379	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀))
50	49	notnotd 144	. . . . . . . . 9 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → ¬ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀))
51		nbfal 1562	. . . . . . . . 9 ⊢ (¬ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ↔ (¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ↔ ⊥))
52	50, 51	sylib 219	. . . . . . . 8 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → (¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ↔ ⊥))
53	52	orbi1d 922	. . . . . . 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 3202	. . . . . 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 3173	. . . . . . . . 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 3086	. . . . . . . 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 3173	. . . . . . . 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 276	. . . . . . 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 1564	. . . . . . . . 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 4572	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑠 = 𝑥 → {𝑠} = {𝑥})
61	60	difeq2d 4064	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑠 = 𝑥 → (𝑆 ∖ {𝑠}) = (𝑆 ∖ {𝑥}))
62	61	oveq2d 7379	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = 𝑥 → (𝐵 ↑m (𝑆 ∖ {𝑠})) = (𝐵 ↑m (𝑆 ∖ {𝑥})))
63		oveq2 7371	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 = 𝑥 → (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑦( ·𝑠 ‘𝑀)𝑥))
64	61	oveq2d 7379	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 = 𝑥 → (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))
65	63, 64	eqeq12d 2756	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑠 = 𝑥 → ((𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})) ↔ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
66	65	notbid 319	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑠 = 𝑥 → (¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})) ↔ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
67	66	orbi2d 921	. . . . . . . . . . . . . . . . . . . . 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 3314	. . . . . . . . . . . . . . . . . . . 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 3163	. . . . . . . . . . . . . . . . . . 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 3565	. . . . . . . . . . . . . . . . . 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 48960	. . . . . . . . . . . . . . . . . . . 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 457	. . . . . . . . . . . . . . . . . . 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 96	. . . . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . 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 54	. . . . . . . . . . . . . . 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 96	. . . . . . . . . . . . . 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 407	. . . . . . . . . . . . 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 453	. . . . . . . . . . . 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 3138	. . . . . . . . . . 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 3132	. . . . . . . . . 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 414	. . . . . . . . 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 863	. . . . . . . 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 32	. . . . . . 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 243	. . . . . 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 241	. . . . 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 241	. . . 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 455	. . 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 516	. 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 413	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 207   ∧ wa 396   ∨ wo 853   = wceq 1547  ⊥wfal 1559   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064  Vcvv 3432   ∖ cdif 3887   ⊆ wss 3890  𝒫 cpw 4536  {csn 4562   class class class wbr 5079  ‘cfv 6492  (class class class)co 7363   ↑_m cmap 8770   finSupp cfsupp 9271  Basecbs 17177  Scalarcsca 17221   ·_𝑠 cvsca 17222  0_gc0g 17400  LModclmod 20857  LSpanclspn 20968   linC clinc 48902   LinCo clinco 48903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-iin 4931  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-of 7627  df-om 7814  df-1st 7938  df-2nd 7939  df-supp 8108  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fsupp 9272  df-oi 9422  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-n0 12436  df-z 12523  df-uz 12787  df-fz 13460  df-fzo 13607  df-seq 13962  df-hash 14291  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17178  df-ress 17199  df-plusg 17231  df-0g 17402  df-gsum 17403  df-mre 17546  df-mrc 17547  df-acs 17549  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-mhm 18749  df-submnd 18750  df-grp 18910  df-minusg 18911  df-sbg 18912  df-mulg 19042  df-subg 19097  df-ghm 19186  df-cntz 19290  df-cmn 19755  df-abl 19756  df-mgp 20120  df-rng 20132  df-ur 20161  df-ring 20214  df-lmod 20859  df-lss 20929  df-lsp 20969  df-linc 48904  df-lco 48905

This theorem is referenced by:  lindslininds  48962