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Theorem lindslinindsimp1 42918
Description: Implication 1 for lindslininds 42925. (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 4327 . . . 4 (𝑆 ∈ 𝒫 (Base‘𝑀) → 𝑆 ⊆ (Base‘𝑀))
21ad2antrl 719 . . 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 477 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑆𝑉𝑀 ∈ LMod) → 𝑀 ∈ LMod)
43anim2i 610 . . . . . . . . . . . . . . . . . . . . . . . 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 717 . . . . . . . . . . . . . . . . . . . . . 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 3896 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (𝐵 ∖ { 0 }) → 𝑦𝐵)
87adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })) → 𝑦𝐵)
98adantl 473 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑦𝐵)
109adantr 472 . . . . . . . . . . . . . . . . . . . . . . 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 787 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑠𝑆)
1211adantr 472 . . . . . . . . . . . . . . . . . . . . . . 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 787 . . . . . . . . . . . . . . . . . . . . . . 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 1158 . . . . . . . . . . . . . . . . . . . . . 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 799 . . . . . . . . . . . . . . . . . . . . . 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 2765 . . . . . . . . . . . . . . . . . . . . . . 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 2765 . . . . . . . . . . . . . . . . . . . . . . 23 (invg𝑅) = (invg𝑅)
22 eqid 2765 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))
2316, 17, 18, 19, 20, 21, 22lincext2 42916 . . . . . . . . . . . . . . . . . . . . . 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 1490 . . . . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . . . . . . . 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 42915 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑦𝐵𝑠𝑆𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠})))) → (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) ∈ (𝐵𝑚 𝑆))
2927, 14, 28syl2anc 579 . . . . . . . . . . . . . . . . . . . . . . 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 4814 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → (𝑓 finSupp 0 ↔ (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) finSupp 0 ))
31 oveq1 6851 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → (𝑓( linC ‘𝑀)𝑆) = ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))( linC ‘𝑀)𝑆))
3231eqeq1d 2767 . . . . . . . . . . . . . . . . . . . . . . . . . 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 624 . . . . . . . . . . . . . . . . . . . . . . . . 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 6376 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → (𝑓𝑥) = ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥))
3534eqeq1d 2767 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → ((𝑓𝑥) = 0 ↔ ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 ))
3635ralbidv 3133 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → (∀𝑥𝑆 (𝑓𝑥) = 0 ↔ ∀𝑥𝑆 ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 ))
3733, 36imbi12d 335 . . . . . . . . . . . . . . . . . . . . . . . 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 3458 . . . . . . . . . . . . . . . . . . . . . . 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 422 . . . . . . . . . . . . . . . . . . . . 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 789 . . . . . . . . . . . . . . . . . . . . . 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 42917 . . . . . . . . . . . . . . . . . . . . . 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 1490 . . . . . . . . . . . . . . . . . . . . 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 fveqeq2 6386 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑠 → (((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 ↔ ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑠) = 0 ))
4645rspcv 3458 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠𝑆 → (∀𝑥𝑆 ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 → ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑠) = 0 ))
4712, 46syl 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 ))
48 eqidd 2766 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))))
49 iftrue 4251 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 = 𝑠 → if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)) = ((invg𝑅)‘𝑦))
5049adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ 𝑧 = 𝑠) → if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)) = ((invg𝑅)‘𝑦))
51 fvexd 6392 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → ((invg𝑅)‘𝑦) ∈ V)
5248, 50, 11, 51fvmptd 6479 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑠) = ((invg𝑅)‘𝑦))
5352adantr 472 . . . . . . . . . . . . . . . . . . . . . . . 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))
5453eqeq1d 2767 . . . . . . . . . . . . . . . . . . . . . . 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 ))
5517lmodfgrp 19144 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑀 ∈ LMod → 𝑅 ∈ Grp)
5618, 19, 21grpinvnzcl 17757 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑅 ∈ Grp ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → ((invg𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }))
57 eldif 3744 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((invg𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }) ↔ (((invg𝑅)‘𝑦) ∈ 𝐵 ∧ ¬ ((invg𝑅)‘𝑦) ∈ { 0 }))
58 fvex 6390 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((invg𝑅)‘𝑦) ∈ V
5958elsn 4351 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((invg𝑅)‘𝑦) ∈ { 0 } ↔ ((invg𝑅)‘𝑦) = 0 )
60 pm2.21 121 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (¬ ((invg𝑅)‘𝑦) = 0 → (((invg𝑅)‘𝑦) = 0 → (𝑆𝑉 → (𝑠𝑆 → (𝑆 ∈ 𝒫 (Base‘𝑀) → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
6160com25 99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (¬ ((invg𝑅)‘𝑦) = 0 → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆𝑉 → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
6259, 61sylnbi 321 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (¬ ((invg𝑅)‘𝑦) ∈ { 0 } → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆𝑉 → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
6362adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((invg𝑅)‘𝑦) ∈ 𝐵 ∧ ¬ ((invg𝑅)‘𝑦) ∈ { 0 }) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆𝑉 → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
6457, 63sylbi 208 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((invg𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆𝑉 → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
6556, 64syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑅 ∈ Grp ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆𝑉 → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
6665ex 401 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑅 ∈ Grp → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆𝑉 → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))))
6755, 66syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑀 ∈ LMod → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆𝑉 → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))))
6867com24 95 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑀 ∈ LMod → (𝑆𝑉 → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))))
6968impcom 396 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑆𝑉𝑀 ∈ LMod) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
7069impcom 396 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
7170com13 88 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑠𝑆 → (𝑦 ∈ (𝐵 ∖ { 0 }) → ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
7271imp 395 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })) → ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
7372impcom 396 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
7473adantr 472 . . . . . . . . . . . . . . . . . . . . . . 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}))))
7554, 74sylbid 231 . . . . . . . . . . . . . . . . . . . . . 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}))))
7647, 75syld 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}))))
7744, 76embantd 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}))))
7841, 77syldc 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}))))
7978expd 404 . . . . . . . . . . . . . . . . . 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})))))
8079exp4c 423 . . . . . . . . . . . . . . . . 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})))))))
8180impcom 396 . . . . . . . . . . . . . . . 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}))))))
8281impcom 396 . . . . . . . . . . . . . . 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})))))
8382imp 395 . . . . . . . . . . . . . 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}))))
8483expdimp 444 . . . . . . . . . . . . 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}))))
8584expd 404 . . . . . . . . . . . 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})))))
8685impcom 396 . . . . . . . . . . 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}))))
8786pm2.01d 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})))
8887olcd 900 . . . . . . . . 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}))))
89 simpl 474 . . . . . . . . . 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 )
9089orcd 899 . . . . . . . . 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}))))
9188, 90pm2.61ian 846 . . . . . . . 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}))))
9291ralrimiva 3113 . . . . . . 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}))))
93 ralnex 3139 . . . . . . . 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}))))
94 ianor 1004 . . . . . . . . 9 (¬ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ (¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
9594ralbii 3127 . . . . . . . 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}))))
9693, 95bitr3i 268 . . . . . . 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}))))
9792, 96sylibr 225 . . . . . 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}))))
9897intnand 482 . . . . 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})))))
993ad2antrr 717 . . . . . . 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)
1001ssdifssd 3912 . . . . . . . . . 10 (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀))
101100ad2antrl 719 . . . . . . . . 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))
102 difexg 4971 . . . . . . . . . . 11 (𝑆𝑉 → (𝑆 ∖ {𝑠}) ∈ V)
103102ad2antrr 717 . . . . . . . . . 10 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) → (𝑆 ∖ {𝑠}) ∈ V)
104 elpwg 4325 . . . . . . . . . 10 ((𝑆 ∖ {𝑠}) ∈ V → ((𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀)))
105103, 104syl 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)))
106101, 105mpbird 248 . . . . . . . 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))
107106adantr 472 . . . . . . 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))
10816lspeqlco 42900 . . . . . . . . 9 ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo (𝑆 ∖ {𝑠})) = ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))
109108eleq2d 2830 . . . . . . . 8 ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → ((𝑦( ·𝑠𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠})) ↔ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠}))))
110109bicomd 214 . . . . . . 7 ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → ((𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})) ↔ (𝑦( ·𝑠𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠}))))
11199, 107, 110syl2anc 579 . . . . . 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}))))
1123adantr 472 . . . . . . . . 9 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) → 𝑀 ∈ LMod)
113 difexg 4971 . . . . . . . . . . . 12 (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ∈ V)
114113, 104syl 17 . . . . . . . . . . 11 (𝑆 ∈ 𝒫 (Base‘𝑀) → ((𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀)))
115100, 114mpbird 248 . . . . . . . . . 10 (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀))
116115ad2antrl 719 . . . . . . . . 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))
117112, 116jca 507 . . . . . . . 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)))
118117adantr 472 . . . . . . 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)))
11916, 17, 18lcoval 42873 . . . . . . . 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}))))))
12019eqcomi 2774 . . . . . . . . . . . 12 (0g𝑅) = 0
121120breq2i 4819 . . . . . . . . . . 11 (𝑔 finSupp (0g𝑅) ↔ 𝑔 finSupp 0 )
122121anbi1i 617 . . . . . . . . . 10 ((𝑔 finSupp (0g𝑅) ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
123122rexbii 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}))))
124123anbi2i 616 . . . . . . . 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})))))
125119, 124syl6bb 278 . . . . . . 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}))))))
126118, 125syl 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}))))))
127111, 126bitrd 270 . . . . 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}))))))
12898, 127mtbird 316 . . . 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})))
129128ralrimivva 3118 . . 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})))
1302, 129jca 507 . 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}))))
131130ex 401 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 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wral 3055  wrex 3056  Vcvv 3350  cdif 3731  wss 3734  ifcif 4245  𝒫 cpw 4317  {csn 4336   class class class wbr 4811  cmpt 4890  cfv 6070  (class class class)co 6844  𝑚 cmap 8062   finSupp cfsupp 8484  Basecbs 16133  Scalarcsca 16220   ·𝑠 cvsca 16221  0gc0g 16369  Grpcgrp 17692  invgcminusg 17693  LModclmod 19135  LSpanclspn 19246   linC clinc 42865   LinCo clinco 42866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-inf2 8755  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-iin 4681  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-of 7097  df-om 7266  df-1st 7368  df-2nd 7369  df-supp 7500  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-1o 7766  df-oadd 7770  df-er 7949  df-map 8064  df-en 8163  df-dom 8164  df-sdom 8165  df-fin 8166  df-fsupp 8485  df-oi 8624  df-card 9018  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-nn 11277  df-2 11337  df-n0 11541  df-z 11627  df-uz 11890  df-fz 12537  df-fzo 12677  df-seq 13012  df-hash 13325  df-ndx 16136  df-slot 16137  df-base 16139  df-sets 16140  df-ress 16141  df-plusg 16230  df-0g 16371  df-gsum 16372  df-mre 16515  df-mrc 16516  df-acs 16518  df-mgm 17511  df-sgrp 17553  df-mnd 17564  df-mhm 17604  df-submnd 17605  df-grp 17695  df-minusg 17696  df-sbg 17697  df-mulg 17811  df-subg 17858  df-ghm 17925  df-cntz 18016  df-cmn 18464  df-abl 18465  df-mgp 18760  df-ur 18772  df-ring 18819  df-lmod 19137  df-lss 19205  df-lsp 19247  df-linc 42867  df-lco 42868
This theorem is referenced by:  lindslininds  42925
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