lindslinindsimp1 - Mathbox for Alexander van der Vekens

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Theorem lindslinindsimp1 46528

Description: Implication 1 for lindslininds 46535. (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 4567	. . . 4 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → 𝑆 ⊆ (Base‘𝑀))
2	1	ad2antrl 726	. . 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 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → 𝑀 ∈ LMod)
4	3	anim2i 617	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod))
5	4	ancomd 462	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)))
6	5	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 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 4086	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ (𝐵 ∖ { 0 }) → 𝑦 ∈ 𝐵)
8	7	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → 𝑦 ∈ 𝐵)
9	8	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑦 ∈ 𝐵)
10	9	adantr 481	. . . . . . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑠 ∈ 𝑆)
12	11	adantr 481	. . . . . . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . . . . . . . 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 46526	. . . . . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod))
26	25	ancomd 462	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)))
27	26	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 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 46525	. . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . 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 5108	. . . . . . . . . . . . . . . . . . . . . . . . 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 7364	. . . . . . . . . . . . . . . . . . . . . . . . . 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 631	. . . . . . . . . . . . . . . . . . . . . . . 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 6841	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3174	. . . . . . . . . . . . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . . . . . . . . . . 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 432	. . . . . . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . . . . . 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 46527	. . . . . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . . . . 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 6851	. . . . . . . . . . . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . . . . . . . . . . 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 4492	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 = 𝑠 → if(𝑧 = 𝑠, ((inv_g‘𝑅)‘𝑦), (𝑔‘𝑧)) = ((inv_g‘𝑅)‘𝑦))
50	49	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 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 6857	. . . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . 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 20331	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ Grp)
56	18, 19, 21	grpinvnzcl 18819	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → ((inv_g‘𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }))
57		eldif 3920	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((inv_g‘𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }) ↔ (((inv_g‘𝑅)‘𝑦) ∈ 𝐵 ∧ ¬ ((inv_g‘𝑅)‘𝑦) ∈ { 0 }))
58		fvex 6855	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((inv_g‘𝑅)‘𝑦) ∈ V
59	58	elsn 4601	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 329	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 505	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . 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 239	. . . . . . . . . . . . . . . . . . . . 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 446	. . . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . 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 453	. . . . . . . . . . . . 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 416	. . . . . . . . . . . 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 408	. . . . . . . . . . 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 189	. . . . . . . . . 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 872	. . . . . . . . 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 976	. . . . . . . . 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 810	. . . . . . . 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 3143	. . . . . . 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 3075	. . . . . . . 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 980	. . . . . . . . 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 3096	. . . . . . . 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 276	. . . . . . 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 233	. . . . . 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 489	. . . . 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 724	. . . . . . 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 5284	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∖ {𝑠}) ∈ V)
98	97	ad2antrr 724	. . . . . . . . 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 4102	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀))
100	99	ad2antrl 726	. . . . . . . . 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 4566	. . . . . . . 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 481	. . . . . . 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 46510	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo (𝑆 ∖ {𝑠})) = ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))
104	103	eleq2d 2823	. . . . . . . 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 222	. . . . . . 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 584	. . . . . 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 481	. . . . . . . . 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 5284	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ∈ V)
109	108, 99	elpwd 4566	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀))
110	109	ad2antrl 726	. . . . . . . . 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 512	. . . . . . . 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 481	. . . . . . 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 46483	. . . . . . . 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 5113	. . . . . . . . . . 11 ⊢ (𝑔 finSupp (0_g‘𝑅) ↔ 𝑔 finSupp 0 )
116	115	anbi1i 624	. . . . . . . . . 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 3097	. . . . . . . . 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 623	. . . . . . . 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 286	. . . . . . 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 278	. . . . 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 324	. . . 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 3197	. . 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 512	. 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 413	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 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∃wrex 3073 Vcvv 3445 ∖ cdif 3907 ⊆ wss 3910 ifcif 4486 𝒫 cpw 4560 {csn 4586 class class class wbr 5105 ↦ cmpt 5188 ‘cfv 6496 (class class class)co 7357 ↑_m cmap 8765 finSupp cfsupp 9305 Basecbs 17083 Scalarcsca 17136 ·_𝑠 cvsca 17137 0_gc0g 17321 Grpcgrp 18748 inv_gcminusg 18749 LModclmod 20322 LSpanclspn 20432 linC clinc 46475 LinCo clinco 46476

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-mhm 18601 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-ghm 19006 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-lmod 20324 df-lss 20393 df-lsp 20433 df-linc 46477 df-lco 46478

This theorem is referenced by: lindslininds 46535

W3C validator