lindslinindsimp1 - Mathbox for Alexander van der Vekens

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Theorem lindslinindsimp1 48933

Description: Implication 1 for lindslininds 48940. (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 = (0_g‘𝑅)
lindslinind.z	⊢ 𝑍 = (0_g‘𝑀)

**Assertion**
Ref	Expression
lindslinindsimp1	⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )) → (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))))

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 Allowed substitution hints: 𝐵(𝑥) 𝑅(𝑦,𝑠) 𝑀(𝑥) 𝑉(𝑥,𝑓) 𝑍(𝑥)

Proof of Theorem lindslinindsimp1 Dummy variables 𝑔 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpwi 4548	. . . 4 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → 𝑆 ⊆ (Base‘𝑀))
2	1	ad2antrl 729	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) → 𝑆 ⊆ (Base‘𝑀))
3		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → 𝑀 ∈ LMod)
4	3	anim2i 618	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod))
5	4	ancomd 461	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)))
6	5	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)))
7		eldifi 4071	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ (𝐵 ∖ { 0 }) → 𝑦 ∈ 𝐵)
8	7	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → 𝑦 ∈ 𝐵)
9	8	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑦 ∈ 𝐵)
10	9	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → 𝑦 ∈ 𝐵)
11		simprl 771	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑠 ∈ 𝑆)
12	11	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → 𝑠 ∈ 𝑆)
13		simprl 771	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → 𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})))
14	10, 12, 13	3jca 1129	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (𝑦 ∈ 𝐵 ∧ 𝑠 ∈ 𝑆 ∧ 𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠}))))
15		simprrl 781	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → 𝑔 finSupp 0 )
16		eqid 2736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘𝑀) = (Base‘𝑀)
17		lindslinind.r	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑅 = (Scalar‘𝑀)
18		lindslinind.b	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐵 = (Base‘𝑅)
19		lindslinind.0	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 = (0_g‘𝑅)
20		lindslinind.z	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑍 = (0_g‘𝑀)
21		eqid 2736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (inv_g‘𝑅) = (inv_g‘𝑅)
22		eqid 2736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧))) = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))
23	16, 17, 18, 19, 20, 21, 22	lincext2 48931	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑠 ∈ 𝑆 ∧ 𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠}))) ∧ 𝑔 finSupp 0 ) → (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧))) finSupp 0 )
24	6, 14, 15, 23	syl3anc 1374	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧))) finSupp 0 )
25	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod))
26	25	ancomd 461	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)))
27	26	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)))
28	16, 17, 18, 19, 20, 21, 22	lincext1 48930	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑠 ∈ 𝑆 ∧ 𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})))) → (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧))) ∈ (𝐵 ↑_m 𝑆))
29	27, 14, 28	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧))) ∈ (𝐵 ↑_m 𝑆))
30		breq1 5088	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧))) → (𝑓 finSupp 0 ↔ (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧))) finSupp 0 ))
31		oveq1 7374	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧))) → (𝑓( linC ‘𝑀)𝑆) = ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))( linC ‘𝑀)𝑆))
32	31	eqeq1d 2738	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧))) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 ↔ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))( linC ‘𝑀)𝑆) = 𝑍))
33	30, 32	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧))) → ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ↔ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧))) finSupp 0 ∧ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))( linC ‘𝑀)𝑆) = 𝑍)))
34		fveq1 6839	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧))) → (𝑓‘𝑥) = ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥))
35	34	eqeq1d 2738	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧))) → ((𝑓‘𝑥) = 0 ↔ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 ))
36	35	ralbidv 3160	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧))) → (∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ↔ ∀𝑥 ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 ))
37	33, 36	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧))) → (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) ↔ (((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧))) finSupp 0 ∧ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 )))
38	37	rspcv 3560	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧))) ∈ (𝐵 ↑_m 𝑆) → (∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) → (((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧))) finSupp 0 ∧ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 )))
39	29, 38	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (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 ) → (((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧))) finSupp 0 ∧ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 )))
40	39	exp4a 431	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (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 ) → ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧))) finSupp 0 → (((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 ))))
41	24, 40	mpid 44	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (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 ) → (((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 )))
42		simprr 773	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
43	16, 17, 18, 19, 20, 21, 22	lincext3 48932	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑠 ∈ 𝑆 ∧ 𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠}))) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) → ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))( linC ‘𝑀)𝑆) = 𝑍)
44	6, 14, 42, 43	syl3anc 1374	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))( linC ‘𝑀)𝑆) = 𝑍)
45		fveqeq2 6849	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑠 → (((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 ↔ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑠) = 0 ))
46	45	rspcv 3560	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑠 ∈ 𝑆 → (∀𝑥 ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 → ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑠) = 0 ))
47	12, 46	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (∀𝑥 ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 → ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑠) = 0 ))
48		eqidd 2737	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧))) = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧))))
49		iftrue 4472	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 = 𝑠 → if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)) = ((inv_g‘𝑅)‘𝑦))
50	49	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ 𝑧 = 𝑠) → if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)) = ((inv_g‘𝑅)‘𝑦))
51		fvexd 6855	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → ((inv_g‘𝑅)‘𝑦) ∈ V)
52	48, 50, 11, 51	fvmptd 6955	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑠) = ((inv_g‘𝑅)‘𝑦))
53	52	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑠) = ((inv_g‘𝑅)‘𝑦))
54	53	eqeq1d 2738	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑠) = 0 ↔ ((inv_g‘𝑅)‘𝑦) = 0 ))
55	17	lmodfgrp 20864	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ Grp)
56	18, 19, 21	grpinvnzcl 18987	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → ((inv_g‘𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }))
57		eldif 3899	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((inv_g‘𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }) ↔ (((inv_g‘𝑅)‘𝑦) ∈ 𝐵 ∧ ¬ ((inv_g‘𝑅)‘𝑦) ∈ { 0 }))
58		fvex 6853	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((inv_g‘𝑅)‘𝑦) ∈ V
59	58	elsn 4582	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((inv_g‘𝑅)‘𝑦) ∈ { 0 } ↔ ((inv_g‘𝑅)‘𝑦) = 0 )
60		pm2.21 123	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (¬ ((inv_g‘𝑅)‘𝑦) = 0 → (((inv_g‘𝑅)‘𝑦) = 0 → (𝑆 ∈ 𝑉 → (𝑠 ∈ 𝑆 → (𝑆 ∈ 𝒫 (Base‘𝑀) → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
61	60	com25 99	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (¬ ((inv_g‘𝑅)‘𝑦) = 0 → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∈ 𝑉 → (𝑠 ∈ 𝑆 → (((inv_g‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
62	59, 61	sylnbi 330	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (¬ ((inv_g‘𝑅)‘𝑦) ∈ { 0 } → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∈ 𝑉 → (𝑠 ∈ 𝑆 → (((inv_g‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
63	57, 62	simplbiim 504	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((inv_g‘𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∈ 𝑉 → (𝑠 ∈ 𝑆 → (((inv_g‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
64	56, 63	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∈ 𝑉 → (𝑠 ∈ 𝑆 → (((inv_g‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
65	64	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑅 ∈ Grp → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∈ 𝑉 → (𝑠 ∈ 𝑆 → (((inv_g‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))))
66	55, 65	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑀 ∈ LMod → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∈ 𝑉 → (𝑠 ∈ 𝑆 → (((inv_g‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))))
67	66	com24 95	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑀 ∈ LMod → (𝑆 ∈ 𝑉 → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑠 ∈ 𝑆 → (((inv_g‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))))
68	67	impcom 407	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑠 ∈ 𝑆 → (((inv_g‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
69	68	impcom 407	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑠 ∈ 𝑆 → (((inv_g‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
70	69	com13 88	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑠 ∈ 𝑆 → (𝑦 ∈ (𝐵 ∖ { 0 }) → ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) → (((inv_g‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
71	70	imp 406	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) → (((inv_g‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
72	71	impcom 407	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → (((inv_g‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
73	72	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (((inv_g‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
74	54, 73	sylbid 240	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑠) = 0 → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
75	47, 74	syld 47	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (∀𝑥 ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
76	44, 75	embantd 59	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → ((((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 ) → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
77	41, 76	syldc 48	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) → ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
78	77	exp5j 445	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) → (𝑆 ∈ 𝒫 (Base‘𝑀) → ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → ((𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → ((𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
79	78	impcom 407	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )) → ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → ((𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → ((𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
80	79	impcom 407	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) → ((𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → ((𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
81	80	imp 406	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → ((𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
82	81	expdimp 452	. . . . . . . . . . . . 13 ⊢ (((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ 𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠}))) → ((𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
83	82	expd 415	. . . . . . . . . . . 12 ⊢ (((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ 𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠}))) → (𝑔 finSupp 0 → ((𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})) → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
84	83	impcom 407	. . . . . . . . . . 11 ⊢ ((𝑔 finSupp 0 ∧ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ 𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})))) → ((𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})) → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
85	84	pm2.01d 190	. . . . . . . . . 10 ⊢ ((𝑔 finSupp 0 ∧ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ 𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})))) → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))
86	85	olcd 875	. . . . . . . . 9 ⊢ ((𝑔 finSupp 0 ∧ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ 𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})))) → (¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
87		animorl 980	. . . . . . . . 9 ⊢ ((¬ 𝑔 finSupp 0 ∧ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ 𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠})))) → (¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
88	86, 87	pm2.61ian 812	. . . . . . . 8 ⊢ (((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ 𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠}))) → (¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
89	88	ralrimiva 3129	. . . . . . 7 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → ∀𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
90		ralnex 3063	. . . . . . . 8 ⊢ (∀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}))))
91		ianor 984	. . . . . . . . 9 ⊢ (¬ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ (¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
92	91	ralbii 3083	. . . . . . . 8 ⊢ (∀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}))))
93	90, 92	bitr3i 277	. . . . . . 7 ⊢ (¬ ∃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}))))
94	89, 93	sylibr 234	. . . . . 6 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → ¬ ∃𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
95	94	intnand 488	. . . . 5 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) ∧ (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})))))
96	3	ad2antrr 727	. . . . . . 7 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑀 ∈ LMod)
97		difexg 5270	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∖ {𝑠}) ∈ V)
98	97	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) → (𝑆 ∖ {𝑠}) ∈ V)
99	1	ssdifssd 4087	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀))
100	99	ad2antrl 729	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) → (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀))
101	98, 100	elpwd 4547	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) → (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀))
102	101	adantr 480	. . . . . . 7 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀))
103	16	lspeqlco 48915	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo (𝑆 ∖ {𝑠})) = ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))
104	103	eleq2d 2822	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → ((𝑦( ·_𝑠 ‘𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠})) ↔ (𝑦( ·_𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠}))))
105	104	bicomd 223	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → ((𝑦( ·_𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})) ↔ (𝑦( ·_𝑠 ‘𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠}))))
106	96, 102, 105	syl2anc 585	. . . . . 6 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → ((𝑦( ·_𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})) ↔ (𝑦( ·_𝑠 ‘𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠}))))
107	3	adantr 480	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) → 𝑀 ∈ LMod)
108		difexg 5270	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ∈ V)
109	108, 99	elpwd 4547	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀))
110	109	ad2antrl 729	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) → (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀))
111	107, 110	jca 511	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) → (𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)))
112	111	adantr 480	. . . . . . 7 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)))
113	16, 17, 18	lcoval 48888	. . . . . . . 8 ⊢ ((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_g‘𝑅) ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
114	19	eqcomi 2745	. . . . . . . . . . . 12 ⊢ (0_g‘𝑅) = 0
115	114	breq2i 5093	. . . . . . . . . . 11 ⊢ (𝑔 finSupp (0_g‘𝑅) ↔ 𝑔 finSupp 0 )
116	115	anbi1i 625	. . . . . . . . . 10 ⊢ ((𝑔 finSupp (0_g‘𝑅) ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ (𝑔 finSupp 0 ∧ (𝑦( ·_𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
117	116	rexbii 3084	. . . . . . . . 9 ⊢ (∃𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠}))(𝑔 finSupp (0_g‘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}))))
118	117	anbi2i 624	. . . . . . . 8 ⊢ (((𝑦( ·_𝑠 ‘𝑀)𝑠) ∈ (Base‘𝑀) ∧ ∃𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑠}))(𝑔 finSupp (0_g‘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})))))
119	113, 118	bitrdi 287	. . . . . . 7 ⊢ ((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}))))))
120	112, 119	syl 17	. . . . . 6 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → ((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}))))))
121	106, 120	bitrd 279	. . . . 5 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → ((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}))))))
122	95, 121	mtbird 325	. . . 4 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))
123	122	ralrimivva 3180	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) → ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))
124	2, 123	jca 511	. 2 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))) → (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠}))))
125	124	ex 412	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵 ↑_m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )) → (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠 ∈ 𝑆 ∀𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·_𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3051 ∃wrex 3061 Vcvv 3429 ∖ cdif 3886 ⊆ wss 3889 ifcif 4466 𝒫 cpw 4541 {csn 4567 class class class wbr 5085 ↦ cmpt 5166 ‘cfv 6498 (class class class)co 7367 ↑_m cmap 8773 finSupp cfsupp 9274 Basecbs 17179 Scalarcsca 17223 ·_𝑠 cvsca 17224 0_gc0g 17402 Grpcgrp 18909 inv_gcminusg 18910 LModclmod 20855 LSpanclspn 20966 linC clinc 48880 LinCo clinco 48881

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-fzo 13609 df-seq 13964 df-hash 14293 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-0g 17404 df-gsum 17405 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-mulg 19044 df-subg 19099 df-ghm 19188 df-cntz 19292 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-lmod 20857 df-lss 20927 df-lsp 20967 df-linc 48882 df-lco 48883

This theorem is referenced by: lindslininds 48940

W3C validator