Theorem lindslinindsimp2 45804

Description: Implication 2 for lindslininds 45805. (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 768	. . . 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 4536	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
3	2	ad2antrr 723	. . . 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 256	. . 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 766	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → 𝑀 ∈ LMod)
6		ssdifss 4070	. . . . . . . . . . 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 5251	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∖ {𝑠}) ∈ V)
9	8	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (𝑆 ∖ {𝑠}) ∈ V)
10		elpwg 4536	. . . . . . . . . . 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 256	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀))
13		eqid 2738	. . . . . . . . . . . 12 ⊢ (Base‘𝑀) = (Base‘𝑀)
14	13	lspeqlco 45780	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo (𝑆 ∖ {𝑠})) = ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))
15	14	eleq2d 2824	. . . . . . . . . 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 222	. . . . . . . . 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 584	. . . . . . . 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 318	. . . . . . 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 45753	. . . . . . . . . . 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 2747	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝑅) = 0
24	23	breq2i 5082	. . . . . . . . . . . . . 14 ⊢ (𝑔 finSupp (0g‘𝑅) ↔ 𝑔 finSupp 0 )
25	24	anbi1i 624	. . . . . . . . . . . . 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 3181	. . . . . . . . . . . 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 623	. . . . . . . . . . 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 287	. . . . . . . . . 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 584	. . . . . . . . 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 318	. . . . . . . 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 979	. . . . . . . . 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 3167	. . . . . . . . . . 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 979	. . . . . . . . . . . 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 3092	. . . . . . . . . . 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 276	. . . . . . . . . 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 910	. . . . . . . . 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 274	. . . . . . . 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 287	. . . . . . 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 278	. . . . . 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 3129	. . . . 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 773	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑀 ∈ LMod)
42		eldifi 4061	. . . . . . . . . . . . 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 3916	. . . . . . . . . . . 12 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ (Base‘𝑀))
46	45	ad2ant2lr 745	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑠 ∈ (Base‘𝑀))
47		eqid 2738	. . . . . . . . . . . 12 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
48	13, 19, 47, 20	lmodvscl 20140	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ LMod ∧ 𝑦 ∈ 𝐵 ∧ 𝑠 ∈ (Base‘𝑀)) → (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀))
49	41, 44, 46, 48	syl3anc 1370	. . . . . . . . . 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 1554	. . . . . . . . 9 ⊢ (¬ ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ↔ (¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ↔ ⊥))
52	50, 51	sylib 217	. . . . . . . 8 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → (¬ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ↔ ⊥))
53	52	orbi1d 914	. . . . . . 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 3128	. . . . . 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 3270	. . . . . . . . 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 3092	. . . . . . . 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 3270	. . . . . . . 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 274	. . . . . . 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 1556	. . . . . . . . 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 4571	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑠 = 𝑥 → {𝑠} = {𝑥})
61	60	difeq2d 4057	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑠 = 𝑥 → (𝑆 ∖ {𝑠}) = (𝑆 ∖ {𝑥}))
62	61	oveq2d 7291	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = 𝑥 → (𝐵 ↑m (𝑆 ∖ {𝑠})) = (𝐵 ↑m (𝑆 ∖ {𝑥})))
63		oveq2 7283	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 = 𝑥 → (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑦( ·𝑠 ‘𝑀)𝑥))
64	61	oveq2d 7291	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 = 𝑥 → (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))
65	63, 64	eqeq12d 2754	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑠 = 𝑥 → ((𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})) ↔ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
66	65	notbid 318	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑠 = 𝑥 → (¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})) ↔ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
67	66	orbi2d 913	. . . . . . . . . . . . . . . . . . . . 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 3336	. . . . . . . . . . . . . . . . . . . 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 3112	. . . . . . . . . . . . . . . . . . 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 3559	. . . . . . . . . . . . . . . . . 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 45803	. . . . . . . . . . . . . . . . . . . 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 3105	. . . . . . . . . . 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 3103	. . . . . . . . . 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 854	. . . . . . . 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	syl5bi 241	. . . . . 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 239	. . . . 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 239	. . . 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 512	. 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 205   ∧ wa 396   ∨ wo 844   = wceq 1539  ⊥wfal 1551   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∖ cdif 3884   ⊆ wss 3887  𝒫 cpw 4533  {csn 4561   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615   finSupp cfsupp 9128  Basecbs 16912  Scalarcsca 16965   ·_𝑠 cvsca 16966  0_gc0g 17150  LModclmod 20123  LSpanclspn 20233   linC clinc 45745   LinCo clinco 45746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-seq 13722  df-hash 14045  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-0g 17152  df-gsum 17153  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-submnd 18431  df-grp 18580  df-minusg 18581  df-sbg 18582  df-mulg 18701  df-subg 18752  df-ghm 18832  df-cntz 18923  df-cmn 19388  df-abl 19389  df-mgp 19721  df-ur 19738  df-ring 19785  df-lmod 20125  df-lss 20194  df-lsp 20234  df-linc 45747  df-lco 45748

This theorem is referenced by:  lindslininds  45805