Users' Mathboxes 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 42571
Description: Implication 1 for lindslininds 42578. (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 = (0g𝑅)
lindslinind.z 𝑍 = (0g𝑀)
Assertion
Ref Expression
lindslinindsimp1 ((𝑆𝑉𝑀 ∈ LMod) → ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )) → (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠𝑀)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.
StepHypRef Expression
1 elpwi 4201 . . . 4 (𝑆 ∈ 𝒫 (Base‘𝑀) → 𝑆 ⊆ (Base‘𝑀))
21ad2antrl 764 . . 3 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) → 𝑆 ⊆ (Base‘𝑀))
3 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑆𝑉𝑀 ∈ LMod) → 𝑀 ∈ LMod)
43anim2i 592 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod))
54ancomd 466 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)))
65ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)))
7 eldifi 3765 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (𝐵 ∖ { 0 }) → 𝑦𝐵)
87adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })) → 𝑦𝐵)
98adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑦𝐵)
109adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → 𝑦𝐵)
11 simprl 809 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑠𝑆)
1211adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → 𝑠𝑆)
13 simprl 809 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → 𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})))
1410, 12, 133jca 1261 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (𝑦𝐵𝑠𝑆𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))))
15 simprrl 821 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → 𝑔 finSupp 0 )
16 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . 23 (Base‘𝑀) = (Base‘𝑀)
17 lindslinind.r . . . . . . . . . . . . . . . . . . . . . . 23 𝑅 = (Scalar‘𝑀)
18 lindslinind.b . . . . . . . . . . . . . . . . . . . . . . 23 𝐵 = (Base‘𝑅)
19 lindslinind.0 . . . . . . . . . . . . . . . . . . . . . . 23 0 = (0g𝑅)
20 lindslinind.z . . . . . . . . . . . . . . . . . . . . . . 23 𝑍 = (0g𝑀)
21 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . 23 (invg𝑅) = (invg𝑅)
22 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))
2316, 17, 18, 19, 20, 21, 22lincext2 42569 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑦𝐵𝑠𝑆𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))) ∧ 𝑔 finSupp 0 ) → (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) finSupp 0 )
246, 14, 15, 23syl3anc 1366 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) finSupp 0 )
254adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod))
2625ancomd 466 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)))
2726adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)))
2816, 17, 18, 19, 20, 21, 22lincext1 42568 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑦𝐵𝑠𝑆𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})))) → (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) ∈ (𝐵𝑚 𝑆))
2927, 14, 28syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) ∈ (𝐵𝑚 𝑆))
30 breq1 4688 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → (𝑓 finSupp 0 ↔ (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) finSupp 0 ))
31 oveq1 6697 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → (𝑓( linC ‘𝑀)𝑆) = ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))( linC ‘𝑀)𝑆))
3231eqeq1d 2653 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 ↔ ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))( linC ‘𝑀)𝑆) = 𝑍))
3330, 32anbi12d 747 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ↔ ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) finSupp 0 ∧ ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))( linC ‘𝑀)𝑆) = 𝑍)))
34 fveq1 6228 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → (𝑓𝑥) = ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥))
3534eqeq1d 2653 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → ((𝑓𝑥) = 0 ↔ ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 ))
3635ralbidv 3015 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → (∀𝑥𝑆 (𝑓𝑥) = 0 ↔ ∀𝑥𝑆 ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 ))
3733, 36imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) ↔ (((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) finSupp 0 ∧ ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 )))
3837rspcv 3336 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) ∈ (𝐵𝑚 𝑆) → (∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → (((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) finSupp 0 ∧ ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 )))
3929, 38syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → (((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) finSupp 0 ∧ ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 )))
4039exp4a 632 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) finSupp 0 → (((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 ))))
4124, 40mpid 44 . . . . . . . . . . . . . . . . . . . 20 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → (((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 )))
42 simprr 811 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
4316, 17, 18, 19, 20, 21, 22lincext3 42570 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑦𝐵𝑠𝑆𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) → ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))( linC ‘𝑀)𝑆) = 𝑍)
446, 14, 42, 43syl3anc 1366 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))( linC ‘𝑀)𝑆) = 𝑍)
45 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑠 → ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑠))
4645eqeq1d 2653 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑠 → (((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 ↔ ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑠) = 0 ))
4746rspcv 3336 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠𝑆 → (∀𝑥𝑆 ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 → ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑠) = 0 ))
4812, 47syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (∀𝑥𝑆 ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 → ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑠) = 0 ))
49 eqidd 2652 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))))
50 iftrue 4125 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑠 → if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)) = ((invg𝑅)‘𝑦))
5150adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ 𝑧 = 𝑠) → if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)) = ((invg𝑅)‘𝑦))
52 fvexd 6241 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → ((invg𝑅)‘𝑦) ∈ V)
5349, 51, 11, 52fvmptd 6327 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑠) = ((invg𝑅)‘𝑦))
5453adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑠) = ((invg𝑅)‘𝑦))
5554eqeq1d 2653 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑠) = 0 ↔ ((invg𝑅)‘𝑦) = 0 ))
5617lmodfgrp 18920 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑀 ∈ LMod → 𝑅 ∈ Grp)
5718, 19, 21grpinvnzcl 17534 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑅 ∈ Grp ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → ((invg𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }))
58 eldif 3617 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((invg𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }) ↔ (((invg𝑅)‘𝑦) ∈ 𝐵 ∧ ¬ ((invg𝑅)‘𝑦) ∈ { 0 }))
59 fvex 6239 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((invg𝑅)‘𝑦) ∈ V
6059elsn 4225 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((invg𝑅)‘𝑦) ∈ { 0 } ↔ ((invg𝑅)‘𝑦) = 0 )
61 pm2.21 120 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (¬ ((invg𝑅)‘𝑦) = 0 → (((invg𝑅)‘𝑦) = 0 → (𝑆𝑉 → (𝑠𝑆 → (𝑆 ∈ 𝒫 (Base‘𝑀) → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
6261com25 99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (¬ ((invg𝑅)‘𝑦) = 0 → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆𝑉 → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
6360, 62sylnbi 319 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (¬ ((invg𝑅)‘𝑦) ∈ { 0 } → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆𝑉 → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
6463adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((invg𝑅)‘𝑦) ∈ 𝐵 ∧ ¬ ((invg𝑅)‘𝑦) ∈ { 0 }) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆𝑉 → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
6558, 64sylbi 207 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((invg𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆𝑉 → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
6657, 65syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑅 ∈ Grp ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆𝑉 → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
6766ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑅 ∈ Grp → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆𝑉 → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))))
6856, 67syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑀 ∈ LMod → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆𝑉 → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))))
6968com24 95 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑀 ∈ LMod → (𝑆𝑉 → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))))
7069impcom 445 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑆𝑉𝑀 ∈ LMod) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
7170impcom 445 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
7271com13 88 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑠𝑆 → (𝑦 ∈ (𝐵 ∖ { 0 }) → ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
7372imp 444 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })) → ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
7473impcom 445 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
7574adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
7655, 75sylbid 230 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑠) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
7748, 76syld 47 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → (∀𝑥𝑆 ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
7844, 77embantd 59 . . . . . . . . . . . . . . . . . . . 20 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → ((((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 ) → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
7941, 78syldc 48 . . . . . . . . . . . . . . . . . . 19 (∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ (𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))) → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
8079expd 451 . . . . . . . . . . . . . . . . . 18 (∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → ((𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
8180exp4c 635 . . . . . . . . . . . . . . . . 17 (∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → (𝑆 ∈ 𝒫 (Base‘𝑀) → ((𝑆𝑉𝑀 ∈ LMod) → ((𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })) → ((𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
8281impcom 445 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )) → ((𝑆𝑉𝑀 ∈ LMod) → ((𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })) → ((𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
8382impcom 445 . . . . . . . . . . . . . . 15 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) → ((𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })) → ((𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
8483imp 444 . . . . . . . . . . . . . 14 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → ((𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
8584expdimp 452 . . . . . . . . . . . . 13 (((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ 𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))) → ((𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
8685expd 451 . . . . . . . . . . . 12 (((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ 𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))) → (𝑔 finSupp 0 → ((𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})) → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
8786impcom 445 . . . . . . . . . . 11 ((𝑔 finSupp 0 ∧ ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ 𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})))) → ((𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})) → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
8887pm2.01d 181 . . . . . . . . . 10 ((𝑔 finSupp 0 ∧ ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ 𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})))) → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))
8988olcd 407 . . . . . . . . 9 ((𝑔 finSupp 0 ∧ ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ 𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})))) → (¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
90 simpl 472 . . . . . . . . . 10 ((¬ 𝑔 finSupp 0 ∧ ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ 𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})))) → ¬ 𝑔 finSupp 0 )
9190orcd 406 . . . . . . . . 9 ((¬ 𝑔 finSupp 0 ∧ ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ 𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})))) → (¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
9289, 91pm2.61ian 848 . . . . . . . 8 (((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ 𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))) → (¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
9392ralrimiva 2995 . . . . . . 7 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → ∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
94 ralnex 3021 . . . . . . . 8 (∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ¬ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ ¬ ∃𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
95 ianor 508 . . . . . . . . 9 (¬ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ (¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
9695ralbii 3009 . . . . . . . 8 (∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})) ¬ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ ∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
9794, 96bitr3i 266 . . . . . . 7 (¬ ∃𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ ∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
9893, 97sylibr 224 . . . . . 6 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → ¬ ∃𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
9998intnand 982 . . . . 5 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → ¬ ((𝑦( ·𝑠𝑀)𝑠) ∈ (Base‘𝑀) ∧ ∃𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
1003ad2antrr 762 . . . . . . 7 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑀 ∈ LMod)
1011ssdifssd 3781 . . . . . . . . . 10 (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀))
102101ad2antrl 764 . . . . . . . . 9 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) → (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀))
103 difexg 4841 . . . . . . . . . . 11 (𝑆𝑉 → (𝑆 ∖ {𝑠}) ∈ V)
104103ad2antrr 762 . . . . . . . . . 10 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) → (𝑆 ∖ {𝑠}) ∈ V)
105 elpwg 4199 . . . . . . . . . 10 ((𝑆 ∖ {𝑠}) ∈ V → ((𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀)))
106104, 105syl 17 . . . . . . . . 9 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) → ((𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀)))
107102, 106mpbird 247 . . . . . . . 8 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) → (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀))
108107adantr 480 . . . . . . 7 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀))
10916lspeqlco 42553 . . . . . . . . 9 ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo (𝑆 ∖ {𝑠})) = ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))
110109eleq2d 2716 . . . . . . . 8 ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → ((𝑦( ·𝑠𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠})) ↔ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠}))))
111110bicomd 213 . . . . . . 7 ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → ((𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})) ↔ (𝑦( ·𝑠𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠}))))
112100, 108, 111syl2anc 694 . . . . . 6 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → ((𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})) ↔ (𝑦( ·𝑠𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠}))))
1133adantr 480 . . . . . . . . 9 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) → 𝑀 ∈ LMod)
114 difexg 4841 . . . . . . . . . . . 12 (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ∈ V)
115114, 105syl 17 . . . . . . . . . . 11 (𝑆 ∈ 𝒫 (Base‘𝑀) → ((𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀)))
116101, 115mpbird 247 . . . . . . . . . 10 (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀))
117116ad2antrl 764 . . . . . . . . 9 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) → (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀))
118113, 117jca 553 . . . . . . . 8 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) → (𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)))
119118adantr 480 . . . . . . 7 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)))
12016, 17, 18lcoval 42526 . . . . . . . 8 ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → ((𝑦( ·𝑠𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠})) ↔ ((𝑦( ·𝑠𝑀)𝑠) ∈ (Base‘𝑀) ∧ ∃𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(𝑔 finSupp (0g𝑅) ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
12119eqcomi 2660 . . . . . . . . . . . 12 (0g𝑅) = 0
122121breq2i 4693 . . . . . . . . . . 11 (𝑔 finSupp (0g𝑅) ↔ 𝑔 finSupp 0 )
123122anbi1i 731 . . . . . . . . . 10 ((𝑔 finSupp (0g𝑅) ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
124123rexbii 3070 . . . . . . . . 9 (∃𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(𝑔 finSupp (0g𝑅) ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ ∃𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
125124anbi2i 730 . . . . . . . 8 (((𝑦( ·𝑠𝑀)𝑠) ∈ (Base‘𝑀) ∧ ∃𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(𝑔 finSupp (0g𝑅) ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) ↔ ((𝑦( ·𝑠𝑀)𝑠) ∈ (Base‘𝑀) ∧ ∃𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
126120, 125syl6bb 276 . . . . . . 7 ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → ((𝑦( ·𝑠𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠})) ↔ ((𝑦( ·𝑠𝑀)𝑠) ∈ (Base‘𝑀) ∧ ∃𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
127119, 126syl 17 . . . . . 6 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → ((𝑦( ·𝑠𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠})) ↔ ((𝑦( ·𝑠𝑀)𝑠) ∈ (Base‘𝑀) ∧ ∃𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
128112, 127bitrd 268 . . . . 5 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → ((𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})) ↔ ((𝑦( ·𝑠𝑀)𝑠) ∈ (Base‘𝑀) ∧ ∃𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
12999, 128mtbird 314 . . . 4 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → ¬ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))
130129ralrimivva 3000 . . 3 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) → ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))
1312, 130jca 553 . 2 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) → (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠}))))
132131ex 449 1 ((𝑆𝑉𝑀 ∈ LMod) → ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )) → (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  cdif 3604  wss 3607  ifcif 4119  𝒫 cpw 4191  {csn 4210   class class class wbr 4685  cmpt 4762  cfv 5926  (class class class)co 6690  𝑚 cmap 7899   finSupp cfsupp 8316  Basecbs 15904  Scalarcsca 15991   ·𝑠 cvsca 15992  0gc0g 16147  Grpcgrp 17469  invgcminusg 17470  LModclmod 18911  LSpanclspn 19019   linC clinc 42518   LinCo clinco 42519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-0g 16149  df-gsum 16150  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-subg 17638  df-ghm 17705  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-lmod 18913  df-lss 18981  df-lsp 19020  df-linc 42520  df-lco 42521
This theorem is referenced by:  lindslininds  42578
  Copyright terms: Public domain W3C validator