Theorem lindslinindsimp1 42246

Description: Implication 1 for lindslininds 42253. (Contributed by AV, 25-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
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 4168	. . . 4 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → 𝑆 ⊆ (Base‘𝑀))
2	1	ad2antrl 764	. . 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 477	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → 𝑀 ∈ LMod)
4	3	anim2i 593	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod))
5	4	ancomd 467	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)))
6	5	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . 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}))))) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)))
7		eldifi 3732	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (𝐵 ∖ { 0 }) → 𝑦 ∈ 𝐵)
8	7	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → 𝑦 ∈ 𝐵)
9	8	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑦 ∈ 𝐵)
10	9	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}))))) → 𝑦 ∈ 𝐵)
11		simprl 794	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑠 ∈ 𝑆)
12	11	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}))))) → 𝑠 ∈ 𝑆)
13		simprl 794	. . . . . . . . . . . . . . . . . . . . . . 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}))))) → 𝑔 ∈ (𝐵 ↑𝑚 (𝑆 ∖ {𝑠})))
14	10, 12, 13	3jca 1242	. . . . . . . . . . . . . . . . . . . . . 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 ∧ 𝑠 ∈ 𝑆 ∧ 𝑔 ∈ (𝐵 ↑𝑚 (𝑆 ∖ {𝑠}))))
15		simprrl 804	. . . . . . . . . . . . . . . . . . . . . 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 finSupp 0 )
16		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Base‘𝑀) = (Base‘𝑀)
17		lindslinind.r	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑅 = (Scalar‘𝑀)
18		lindslinind.b	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐵 = (Base‘𝑅)
19		lindslinind.0	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 = (0g‘𝑅)
20		lindslinind.z	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑍 = (0g‘𝑀)
21		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (invg‘𝑅) = (invg‘𝑅)
22		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧))) = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))
23	16, 17, 18, 19, 20, 21, 22	lincext2 42244	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑠 ∈ 𝑆 ∧ 𝑔 ∈ (𝐵 ↑𝑚 (𝑆 ∖ {𝑠}))) ∧ 𝑔 finSupp 0 ) → (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧))) finSupp 0 )
24	6, 14, 15, 23	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . 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, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧))) finSupp 0 )
25	4	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod))
26	25	ancomd 467	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)))
27	26	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((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 42243	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑠 ∈ 𝑆 ∧ 𝑔 ∈ (𝐵 ↑𝑚 (𝑆 ∖ {𝑠})))) → (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧))) ∈ (𝐵 ↑𝑚 𝑆))
29	27, 14, 28	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . 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, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧))) ∈ (𝐵 ↑𝑚 𝑆))
30		breq1 4656	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧))) → (𝑓 finSupp 0 ↔ (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧))) finSupp 0 ))
31		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧))) → (𝑓( linC ‘𝑀)𝑆) = ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))( linC ‘𝑀)𝑆))
32	31	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧))) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 ↔ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))( linC ‘𝑀)𝑆) = 𝑍))
33	30, 32	anbi12d 747	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧))) → ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ↔ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧))) finSupp 0 ∧ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))( linC ‘𝑀)𝑆) = 𝑍)))
34		fveq1 6190	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧))) → (𝑓‘𝑥) = ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥))
35	34	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧))) → ((𝑓‘𝑥) = 0 ↔ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 ))
36	35	ralbidv 2986	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧))) → (∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ↔ ∀𝑥 ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 ))
37	33, 36	imbi12d 334	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧))) → (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) ↔ (((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧))) finSupp 0 ∧ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 )))
38	37	rspcv 3305	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘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, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧))) finSupp 0 ∧ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 )))
39	29, 38	syl 17	. . . . . . . . . . . . . . . . . . . . . 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}))))) → (∀𝑓 ∈ (𝐵 ↑𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) → (((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧))) finSupp 0 ∧ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 )))
40	39	exp4a 633	. . . . . . . . . . . . . . . . . . . . 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, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧))) finSupp 0 → (((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 ))))
41	24, 40	mpid 44	. . . . . . . . . . . . . . . . . . . 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, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 )))
42		simprr 796	. . . . . . . . . . . . . . . . . . . . . 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 finSupp 0 ∧ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
43	16, 17, 18, 19, 20, 21, 22	lincext3 42245	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((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, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))( linC ‘𝑀)𝑆) = 𝑍)
44	6, 14, 42, 43	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . 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, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))( linC ‘𝑀)𝑆) = 𝑍)
45		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = 𝑠 → ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑠))
46	45	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑠 → (((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 ↔ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑠) = 0 ))
47	46	rspcv 3305	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑠 ∈ 𝑆 → (∀𝑥 ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 → ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑠) = 0 ))
48	12, 47	syl 17	. . . . . . . . . . . . . . . . . . . . . 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}))))) → (∀𝑥 ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 → ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑠) = 0 ))
49		eqidd 2623	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧))) = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧))))
50		iftrue 4092	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 = 𝑠 → if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)) = ((invg‘𝑅)‘𝑦))
51	50	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ 𝑧 = 𝑠) → if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)) = ((invg‘𝑅)‘𝑦))
52		fvexd 6203	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → ((invg‘𝑅)‘𝑦) ∈ V)
53	49, 51, 11, 52	fvmptd 6288	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑠) = ((invg‘𝑅)‘𝑦))
54	53	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((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, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑠) = ((invg‘𝑅)‘𝑦))
55	54	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . . 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, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑠) = 0 ↔ ((invg‘𝑅)‘𝑦) = 0 ))
56	17	lmodfgrp 18872	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ Grp)
57	18, 19, 21	grpinvnzcl 17487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → ((invg‘𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }))
58		eldif 3584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((invg‘𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }) ↔ (((invg‘𝑅)‘𝑦) ∈ 𝐵 ∧ ¬ ((invg‘𝑅)‘𝑦) ∈ { 0 }))
59		fvex 6201	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((invg‘𝑅)‘𝑦) ∈ V
60	59	elsn 4192	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((invg‘𝑅)‘𝑦) ∈ { 0 } ↔ ((invg‘𝑅)‘𝑦) = 0 )
61		pm2.21 120	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (¬ ((invg‘𝑅)‘𝑦) = 0 → (((invg‘𝑅)‘𝑦) = 0 → (𝑆 ∈ 𝑉 → (𝑠 ∈ 𝑆 → (𝑆 ∈ 𝒫 (Base‘𝑀) → ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
62	61	com25 99	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (¬ ((invg‘𝑅)‘𝑦) = 0 → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∈ 𝑉 → (𝑠 ∈ 𝑆 → (((invg‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
63	60, 62	sylnbi 320	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (¬ ((invg‘𝑅)‘𝑦) ∈ { 0 } → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∈ 𝑉 → (𝑠 ∈ 𝑆 → (((invg‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
64	63	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((invg‘𝑅)‘𝑦) ∈ 𝐵 ∧ ¬ ((invg‘𝑅)‘𝑦) ∈ { 0 }) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∈ 𝑉 → (𝑠 ∈ 𝑆 → (((invg‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
65	58, 64	sylbi 207	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((invg‘𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∈ 𝑉 → (𝑠 ∈ 𝑆 → (((invg‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
66	57, 65	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∈ 𝑉 → (𝑠 ∈ 𝑆 → (((invg‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
67	66	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑅 ∈ Grp → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∈ 𝑉 → (𝑠 ∈ 𝑆 → (((invg‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))))
68	56, 67	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑀 ∈ LMod → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∈ 𝑉 → (𝑠 ∈ 𝑆 → (((invg‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))))
69	68	com24 95	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑀 ∈ LMod → (𝑆 ∈ 𝑉 → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑠 ∈ 𝑆 → (((invg‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))))
70	69	impcom 446	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑠 ∈ 𝑆 → (((invg‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
71	70	impcom 446	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑠 ∈ 𝑆 → (((invg‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
72	71	com13 88	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑠 ∈ 𝑆 → (𝑦 ∈ (𝐵 ∖ { 0 }) → ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) → (((invg‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
73	72	imp 445	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) → (((invg‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
74	73	impcom 446	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐵 ∖ { 0 }))) → (((invg‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
75	74	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}))))) → (((invg‘𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
76	55, 75	sylbid 230	. . . . . . . . . . . . . . . . . . . . . 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, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑠) = 0 → ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
77	48, 76	syld 47	. . . . . . . . . . . . . . . . . . . . 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, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 → ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
78	44, 77	embantd 59	. . . . . . . . . . . . . . . . . . . 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, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invg‘𝑅)‘𝑦), (𝑔‘𝑧)))‘𝑥) = 0 ) → ¬ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
79	41, 78	syldc 48	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀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}))))
80	79	expd 452	. . . . . . . . . . . . . . . . . 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})))))
81	80	exp4c 636	. . . . . . . . . . . . . . . . 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})))))))
82	81	impcom 446	. . . . . . . . . . . . . . . 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}))))))
83	82	impcom 446	. . . . . . . . . . . . . . 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})))))
84	83	imp 445	. . . . . . . . . . . . . 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}))))
85	84	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}))))
86	85	expd 452	. . . . . . . . . . . 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})))))
87	86	impcom 446	. . . . . . . . . . 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}))))
88	87	pm2.01d 181	. . . . . . . . . 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})))
89	88	olcd 408	. . . . . . . . 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}))))
90		simpl 473	. . . . . . . . . 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})))) → ¬ 𝑔 finSupp 0 )
91	90	orcd 407	. . . . . . . . 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}))))
92	89, 91	pm2.61ian 831	. . . . . . . 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}))))
93	92	ralrimiva 2966	. . . . . . 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}))))
94		ralnex 2992	. . . . . . . 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}))))
95		ianor 509	. . . . . . . . 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}))))
96	95	ralbii 2980	. . . . . . . 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}))))
97	94, 96	bitr3i 266	. . . . . . 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}))))
98	93, 97	sylibr 224	. . . . . 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}))))
99	98	intnand 962	. . . . 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})))))
100	3	ad2antrr 762	. . . . . . 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)
101	1	ssdifssd 3748	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀))
102	101	ad2antrl 764	. . . . . . . . 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))
103		difexg 4808	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∖ {𝑠}) ∈ V)
104	103	ad2antrr 762	. . . . . . . . . 10 ⊢ (((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)
105		elpwg 4166	. . . . . . . . . 10 ⊢ ((𝑆 ∖ {𝑠}) ∈ V → ((𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀)))
106	104, 105	syl 17	. . . . . . . . 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) ↔ (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀)))
107	102, 106	mpbird 247	. . . . . . . 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))
108	107	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))
109	16	lspeqlco 42228	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo (𝑆 ∖ {𝑠})) = ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))
110	109	eleq2d 2687	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → ((𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠})) ↔ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠}))))
111	110	bicomd 213	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → ((𝑦( ·𝑠 ‘𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})) ↔ (𝑦( ·𝑠 ‘𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠}))))
112	100, 108, 111	syl2anc 693	. . . . . 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}))))
113	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)
114		difexg 4808	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ∈ V)
115	114, 105	syl 17	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → ((𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀)))
116	101, 115	mpbird 247	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀))
117	116	ad2antrl 764	. . . . . . . . 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))
118	113, 117	jca 554	. . . . . . . 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)))
119	118	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)))
120	16, 17, 18	lcoval 42201	. . . . . . . 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 (0g‘𝑅) ∧ (𝑦( ·𝑠 ‘𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
121	19	eqcomi 2631	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = 0
122	121	breq2i 4661	. . . . . . . . . . 11 ⊢ (𝑔 finSupp (0g‘𝑅) ↔ 𝑔 finSupp 0 )
123	122	anbi1i 731	. . . . . . . . . 10 ⊢ ((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}))))
124	123	rexbii 3041	. . . . . . . . 9 ⊢ (∃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}))))
125	124	anbi2i 730	. . . . . . . 8 ⊢ (((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})))))
126	120, 125	syl6bb 276	. . . . . . 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}))))))
127	119, 126	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}))))))
128	112, 127	bitrd 268	. . . . 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}))))))
129	99, 128	mtbird 315	. . . 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})))
130	129	ralrimivva 2971	. . 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})))
131	2, 130	jca 554	. 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}))))
132	131	ex 450	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})))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ifcif 4086  𝒫 cpw 4158  {csn 4177   class class class wbr 4653   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857   finSupp cfsupp 8275  Basecbs 15857  Scalarcsca 15944   ·_𝑠 cvsca 15945  0_gc0g 16100  Grpcgrp 17422  inv_gcminusg 17423  LModclmod 18863  LSpanclspn 18971   linC clinc 42193   LinCo clinco 42194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lss 18933  df-lsp 18972  df-linc 42195  df-lco 42196

This theorem is referenced by:  lindslininds  42253