lindslinindsimp1 - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > lindslinindsimp1		Structured version Visualization version GIF version

Theorem lindslinindsimp1 45798

Description: Implication 1 for lindslininds 45805. (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 4542	. . . 4 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → 𝑆 ⊆ (Base‘𝑀))
2	1	ad2antrl 725	. . 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 723	. . . . . . . . . . . . . . . . . . . . 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 4061	. . . . . . . . . . . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . . . . . . 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 1127	. . . . . . . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (inv_g‘𝑅) = (inv_g‘𝑅)
22		eqid 2738	. . . . . . . . . . . . . . . . . . . . . 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 45796	. . . . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . . . 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 45795	. . . . . . . . . . . . . . . . . . . . . . 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 5077	. . . . . . . . . . . . . . . . . . . . . . . . 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 7282	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . . . . . . . . . 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 6773	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . . . . . . . . . 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 3112	. . . . . . . . . . . . . . . . . . . . . . . 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 345	. . . . . . . . . . . . . . . . . . . . . . 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 3557	. . . . . . . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . . . . . 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 45797	. . . . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . . . 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 6783	. . . . . . . . . . . . . . . . . . . . . . 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 3557	. . . . . . . . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . . . . . . . . . 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 4465	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6789	. . . . . . . . . . . . . . . . . . . . . . . . 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 6882	. . . . . . . . . . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . . . . . . 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 20132	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ Grp)
56	18, 19, 21	grpinvnzcl 18647	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → ((inv_g‘𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }))
57		eldif 3897	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((inv_g‘𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }) ↔ (((inv_g‘𝑅)‘𝑦) ∈ 𝐵 ∧ ¬ ((inv_g‘𝑅)‘𝑦) ∈ { 0 }))
58		fvex 6787	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((inv_g‘𝑅)‘𝑦) ∈ V
59	58	elsn 4576	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 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 871	. . . . . . . . 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 975	. . . . . . . . 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 809	. . . . . . . 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 3103	. . . . . . 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 3167	. . . . . . . 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 979	. . . . . . . . 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 3092	. . . . . . . 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 723	. . . . . . 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 5251	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∖ {𝑠}) ∈ V)
98	97	ad2antrr 723	. . . . . . . . 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 4077	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀))
100	99	ad2antrl 725	. . . . . . . . 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 4541	. . . . . . . 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 45780	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo (𝑆 ∖ {𝑠})) = ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))
104	103	eleq2d 2824	. . . . . . . 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 5251	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ∈ V)
109	108, 99	elpwd 4541	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀))
110	109	ad2antrl 725	. . . . . . . . 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 45753	. . . . . . . 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 2747	. . . . . . . . . . . 12 ⊢ (0_g‘𝑅) = 0
115	114	breq2i 5082	. . . . . . . . . . 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 3181	. . . . . . . . 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 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 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 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 3123	. . 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 844 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 Vcvv 3432 ∖ cdif 3884 ⊆ wss 3887 ifcif 4459 𝒫 cpw 4533 {csn 4561 class class class wbr 5074 ↦ cmpt 5157 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615 finSupp cfsupp 9128 Basecbs 16912 Scalarcsca 16965 ·_𝑠 cvsca 16966 0_gc0g 17150 Grpcgrp 18577 inv_gcminusg 18578 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

W3C validator