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Theorem lindslinindsimp1 42764
Description: Implication 1 for lindslininds 42771. (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 4305 . . . 4 (𝑆 ∈ 𝒫 (Base‘𝑀) → 𝑆 ⊆ (Base‘𝑀))
21ad2antrl 699 . . 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 471 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑆𝑉𝑀 ∈ LMod) → 𝑀 ∈ LMod)
43anim2i 595 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod))
54ancomd 453 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)))
65ad2antrr 697 . . . . . . . . . . . . . . . . . . . . . 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 3881 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (𝐵 ∖ { 0 }) → 𝑦𝐵)
87adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })) → 𝑦𝐵)
98adantl 467 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑦𝐵)
109adantr 466 . . . . . . . . . . . . . . . . . . . . . . 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 746 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑠𝑆)
1211adantr 466 . . . . . . . . . . . . . . . . . . . . . . 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 746 . . . . . . . . . . . . . . . . . . . . . . 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 1121 . . . . . . . . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . . . . . . . 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 2770 . . . . . . . . . . . . . . . . . . . . . . 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 2770 . . . . . . . . . . . . . . . . . . . . . . 23 (invg𝑅) = (invg𝑅)
22 eqid 2770 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))
2316, 17, 18, 19, 20, 21, 22lincext2 42762 . . . . . . . . . . . . . . . . . . . . . 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 1475 . . . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod))
2625ancomd 453 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)))
2726adantr 466 . . . . . . . . . . . . . . . . . . . . . . . 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 42761 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑦𝐵𝑠𝑆𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})))) → (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) ∈ (𝐵𝑚 𝑆))
2927, 14, 28syl2anc 565 . . . . . . . . . . . . . . . . . . . . . . 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 4787 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → (𝑓 finSupp 0 ↔ (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) finSupp 0 ))
31 oveq1 6799 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → (𝑓( linC ‘𝑀)𝑆) = ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))( linC ‘𝑀)𝑆))
3231eqeq1d 2772 . . . . . . . . . . . . . . . . . . . . . . . . . 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 608 . . . . . . . . . . . . . . . . . . . . . . . . 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 6331 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → (𝑓𝑥) = ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥))
3534eqeq1d 2772 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → ((𝑓𝑥) = 0 ↔ ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 ))
3635ralbidv 3134 . . . . . . . . . . . . . . . . . . . . . . . . 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 3454 . . . . . . . . . . . . . . . . . . . . . . 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 418 . . . . . . . . . . . . . . . . . . . . 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 748 . . . . . . . . . . . . . . . . . . . . . 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 42763 . . . . . . . . . . . . . . . . . . . . . 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 1475 . . . . . . . . . . . . . . . . . . . . 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 6332 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑠 → ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑠))
4645eqeq1d 2772 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑠 → (((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 ↔ ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑠) = 0 ))
4746rspcv 3454 . . . . . . . . . . . . . . . . . . . . . . 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 2771 . . . . . . . . . . . . . . . . . . . . . . . . . 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 4229 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑠 → if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)) = ((invg𝑅)‘𝑦))
5150adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ 𝑧 = 𝑠) → if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)) = ((invg𝑅)‘𝑦))
52 fvexd 6344 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → ((invg𝑅)‘𝑦) ∈ V)
5349, 51, 11, 52fvmptd 6430 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑠) = ((invg𝑅)‘𝑦))
5453adantr 466 . . . . . . . . . . . . . . . . . . . . . . . 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 2772 . . . . . . . . . . . . . . . . . . . . . . 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 19081 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑀 ∈ LMod → 𝑅 ∈ Grp)
5718, 19, 21grpinvnzcl 17694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑅 ∈ Grp ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → ((invg𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }))
58 eldif 3731 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((invg𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }) ↔ (((invg𝑅)‘𝑦) ∈ 𝐵 ∧ ¬ ((invg𝑅)‘𝑦) ∈ { 0 }))
59 fvex 6342 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((invg𝑅)‘𝑦) ∈ V
6059elsn 4329 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((invg𝑅)‘𝑦) ∈ { 0 } ↔ ((invg𝑅)‘𝑦) = 0 )
61 pm2.21 121 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 397 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 394 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑆𝑉𝑀 ∈ LMod) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
7170impcom 394 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 393 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })) → ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
7473impcom 394 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
7574adantr 466 . . . . . . . . . . . . . . . . . . . . . . 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 400 . . . . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . . . . . 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 394 . . . . . . . . . . . . . . . 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 394 . . . . . . . . . . . . . . 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 393 . . . . . . . . . . . . . 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 440 . . . . . . . . . . . . 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 400 . . . . . . . . . . . 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 394 . . . . . . . . . . 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 854 . . . . . . . . 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 468 . . . . . . . . . 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 853 . . . . . . . . 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 795 . . . . . . . 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 3114 . . . . . . 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 3140 . . . . . . . 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 910 . . . . . . . . 9 (¬ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ (¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
9695ralbii 3128 . . . . . . . 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 998 . . . . 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 697 . . . . . . 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 3897 . . . . . . . . . 10 (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀))
102101ad2antrl 699 . . . . . . . . 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 4939 . . . . . . . . . . 11 (𝑆𝑉 → (𝑆 ∖ {𝑠}) ∈ V)
104103ad2antrr 697 . . . . . . . . . 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 4303 . . . . . . . . . 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 466 . . . . . . 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 42746 . . . . . . . . 9 ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo (𝑆 ∖ {𝑠})) = ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))
110109eleq2d 2835 . . . . . . . 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 565 . . . . . 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 466 . . . . . . . . 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 4939 . . . . . . . . . . . 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 699 . . . . . . . . 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 495 . . . . . . . 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 466 . . . . . . 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 42719 . . . . . . . 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 2779 . . . . . . . . . . . 12 (0g𝑅) = 0
122121breq2i 4792 . . . . . . . . . . 11 (𝑔 finSupp (0g𝑅) ↔ 𝑔 finSupp 0 )
123122anbi1i 602 . . . . . . . . . 10 ((𝑔 finSupp (0g𝑅) ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
124123rexbii 3188 . . . . . . . . 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 601 . . . . . . . 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 3119 . . 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 495 . 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 397 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  wa 382  wo 826  w3a 1070   = wceq 1630  wcel 2144  wral 3060  wrex 3061  Vcvv 3349  cdif 3718  wss 3721  ifcif 4223  𝒫 cpw 4295  {csn 4314   class class class wbr 4784  cmpt 4861  cfv 6031  (class class class)co 6792  𝑚 cmap 8008   finSupp cfsupp 8430  Basecbs 16063  Scalarcsca 16151   ·𝑠 cvsca 16152  0gc0g 16307  Grpcgrp 17629  invgcminusg 17630  LModclmod 19072  LSpanclspn 19183   linC clinc 42711   LinCo clinco 42712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-of 7043  df-om 7212  df-1st 7314  df-2nd 7315  df-supp 7446  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-fsupp 8431  df-oi 8570  df-card 8964  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-n0 11494  df-z 11579  df-uz 11888  df-fz 12533  df-fzo 12673  df-seq 13008  df-hash 13321  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-ress 16071  df-plusg 16161  df-0g 16309  df-gsum 16310  df-mre 16453  df-mrc 16454  df-acs 16456  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-mhm 17542  df-submnd 17543  df-grp 17632  df-minusg 17633  df-sbg 17634  df-mulg 17748  df-subg 17798  df-ghm 17865  df-cntz 17956  df-cmn 18401  df-abl 18402  df-mgp 18697  df-ur 18709  df-ring 18756  df-lmod 19074  df-lss 19142  df-lsp 19184  df-linc 42713  df-lco 42714
This theorem is referenced by:  lindslininds  42771
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