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Theorem lindslinindsimp1 48374
Description: Implication 1 for lindslininds 48381. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.) (Proof shortened by II, 16-Feb-2023.)
Hypotheses
Ref Expression
lindslinind.r 𝑅 = (Scalar‘𝑀)
lindslinind.b 𝐵 = (Base‘𝑅)
lindslinind.0 0 = (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 4607 . . . 4 (𝑆 ∈ 𝒫 (Base‘𝑀) → 𝑆 ⊆ (Base‘𝑀))
21ad2antrl 728 . . 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 484 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑆𝑉𝑀 ∈ LMod) → 𝑀 ∈ LMod)
43anim2i 617 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod))
54ancomd 461 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)))
65ad2antrr 726 . . . . . . . . . . . . . . . . . . . . 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}))))) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)))
7 eldifi 4131 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ (𝐵 ∖ { 0 }) → 𝑦𝐵)
87adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })) → 𝑦𝐵)
98adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑦𝐵)
109adantr 480 . . . . . . . . . . . . . . . . . . . . . 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}))))) → 𝑦𝐵)
11 simprl 771 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑠𝑆)
1211adantr 480 . . . . . . . . . . . . . . . . . . . . . 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}))))) → 𝑠𝑆)
13 simprl 771 . . . . . . . . . . . . . . . . . . . . . 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 ∈ (𝐵m (𝑆 ∖ {𝑠})))
1410, 12, 133jca 1129 . . . . . . . . . . . . . . . . . . . . 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}))))) → (𝑦𝐵𝑠𝑆𝑔 ∈ (𝐵m (𝑆 ∖ {𝑠}))))
15 simprrl 781 . . . . . . . . . . . . . . . . . . . . 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}))))) → 𝑔 finSupp 0 )
16 eqid 2737 . . . . . . . . . . . . . . . . . . . . . 22 (Base‘𝑀) = (Base‘𝑀)
17 lindslinind.r . . . . . . . . . . . . . . . . . . . . . 22 𝑅 = (Scalar‘𝑀)
18 lindslinind.b . . . . . . . . . . . . . . . . . . . . . 22 𝐵 = (Base‘𝑅)
19 lindslinind.0 . . . . . . . . . . . . . . . . . . . . . 22 0 = (0g𝑅)
20 lindslinind.z . . . . . . . . . . . . . . . . . . . . . 22 𝑍 = (0g𝑀)
21 eqid 2737 . . . . . . . . . . . . . . . . . . . . . 22 (invg𝑅) = (invg𝑅)
22 eqid 2737 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))
2316, 17, 18, 19, 20, 21, 22lincext2 48372 . . . . . . . . . . . . . . . . . . . . 21 (((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 1373 . . . . . . . . . . . . . . . . . . . 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))) finSupp 0 )
254adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑀 ∈ LMod))
2625ancomd 461 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)))
2726adantr 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}))))) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)))
2816, 17, 18, 19, 20, 21, 22lincext1 48371 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑦𝐵𝑠𝑆𝑔 ∈ (𝐵m (𝑆 ∖ {𝑠})))) → (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) ∈ (𝐵m 𝑆))
2927, 14, 28syl2anc 584 . . . . . . . . . . . . . . . . . . . . . 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))) ∈ (𝐵m 𝑆))
30 breq1 5146 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → (𝑓 finSupp 0 ↔ (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) finSupp 0 ))
31 oveq1 7438 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → (𝑓( linC ‘𝑀)𝑆) = ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))( linC ‘𝑀)𝑆))
3231eqeq1d 2739 . . . . . . . . . . . . . . . . . . . . . . . . 25 (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 632 . . . . . . . . . . . . . . . . . . . . . . . 24 (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 6905 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → (𝑓𝑥) = ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥))
3534eqeq1d 2739 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → ((𝑓𝑥) = 0 ↔ ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 ))
3635ralbidv 3178 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → (∀𝑥𝑆 (𝑓𝑥) = 0 ↔ ∀𝑥𝑆 ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 ))
3733, 36imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . 23 (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 3618 . . . . . . . . . . . . . . . . . . . . . 22 ((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 . . . . . . . . . . . . . . . . . . . . 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 )))
4039exp4a 431 . . . . . . . . . . . . . . . . . . . 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))) 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 . . . . . . . . . . . . . . . . . . 19 ((((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 773 . . . . . . . . . . . . . . . . . . . . 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}))))) → (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
4316, 17, 18, 19, 20, 21, 22lincext3 48373 . . . . . . . . . . . . . . . . . . . . 21 (((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 1373 . . . . . . . . . . . . . . . . . . . 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)
45 fveqeq2 6915 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑠 → (((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 ↔ ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑠) = 0 ))
4645rspcv 3618 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠𝑆 → (∀𝑥𝑆 ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 → ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑠) = 0 ))
4712, 46syl 17 . . . . . . . . . . . . . . . . . . . . 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 → ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑠) = 0 ))
48 eqidd 2738 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((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 4531 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 = 𝑠 → if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)) = ((invg𝑅)‘𝑦))
5049adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) ∧ 𝑧 = 𝑠) → if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)) = ((invg𝑅)‘𝑦))
51 fvexd 6921 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → ((invg𝑅)‘𝑦) ∈ V)
5248, 50, 11, 51fvmptd 7023 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑠) = ((invg𝑅)‘𝑦))
5352adantr 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}))))) → ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑠) = ((invg𝑅)‘𝑦))
5453eqeq1d 2739 . . . . . . . . . . . . . . . . . . . . . 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 ↔ ((invg𝑅)‘𝑦) = 0 ))
5517lmodfgrp 20867 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑀 ∈ LMod → 𝑅 ∈ Grp)
5618, 19, 21grpinvnzcl 19029 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑅 ∈ Grp ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → ((invg𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }))
57 eldif 3961 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((invg𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }) ↔ (((invg𝑅)‘𝑦) ∈ 𝐵 ∧ ¬ ((invg𝑅)‘𝑦) ∈ { 0 }))
58 fvex 6919 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((invg𝑅)‘𝑦) ∈ V
5958elsn 4641 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((invg𝑅)‘𝑦) ∈ { 0 } ↔ ((invg𝑅)‘𝑦) = 0 )
60 pm2.21 123 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (¬ ((invg𝑅)‘𝑦) = 0 → (((invg𝑅)‘𝑦) = 0 → (𝑆𝑉 → (𝑠𝑆 → (𝑆 ∈ 𝒫 (Base‘𝑀) → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
6160com25 99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (¬ ((invg𝑅)‘𝑦) = 0 → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆𝑉 → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
6259, 61sylnbi 330 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (¬ ((invg𝑅)‘𝑦) ∈ { 0 } → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆𝑉 → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
6357, 62simplbiim 504 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((invg𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆𝑉 → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
6456, 63syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑅 ∈ Grp ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆𝑉 → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
6564ex 412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑅 ∈ Grp → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆𝑉 → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))))
6655, 65syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑀 ∈ LMod → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆𝑉 → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))))
6766com24 95 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑀 ∈ LMod → (𝑆𝑉 → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))))
6867impcom 407 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑆𝑉𝑀 ∈ LMod) → (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))))
6968impcom 407 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) → (𝑦 ∈ (𝐵 ∖ { 0 }) → (𝑠𝑆 → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
7069com13 88 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑠𝑆 → (𝑦 ∈ (𝐵 ∖ { 0 }) → ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
7170imp 406 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })) → ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
7271impcom 407 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (𝑆𝑉𝑀 ∈ LMod)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
7372adantr 480 . . . . . . . . . . . . . . . . . . . . . 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}))))) → (((invg𝑅)‘𝑦) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
7454, 73sylbid 240 . . . . . . . . . . . . . . . . . . . . 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)))‘𝑠) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
7547, 74syld 47 . . . . . . . . . . . . . . . . . . . 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}))))) → (∀𝑥𝑆 ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 → ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
7644, 75embantd 59 . . . . . . . . . . . . . . . . . . 19 ((((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}))))
7741, 76syldc 48 . . . . . . . . . . . . . . . . . 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}))))
7877exp5j 445 . . . . . . . . . . . . . . . . 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})))))))
7978impcom 407 . . . . . . . . . . . . . . . 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}))))))
8079impcom 407 . . . . . . . . . . . . . . 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})))))
8180imp 406 . . . . . . . . . . . . . 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}))))
8281expdimp 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}))))
8382expd 415 . . . . . . . . . . . 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})))))
8483impcom 407 . . . . . . . . . . 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}))))
8584pm2.01d 190 . . . . . . . . . 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})))
8685olcd 875 . . . . . . . . 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}))))
87 animorl 980 . . . . . . . . 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}))))
8886, 87pm2.61ian 812 . . . . . . . 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}))))
8988ralrimiva 3146 . . . . . . 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}))))
90 ralnex 3072 . . . . . . . 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}))))
91 ianor 984 . . . . . . . . 9 (¬ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ (¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
9291ralbii 3093 . . . . . . . 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}))))
9390, 92bitr3i 277 . . . . . . 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}))))
9489, 93sylibr 234 . . . . . 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}))))
9594intnand 488 . . . . 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})))))
963ad2antrr 726 . . . . . . 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)
97 difexg 5329 . . . . . . . . . 10 (𝑆𝑉 → (𝑆 ∖ {𝑠}) ∈ V)
9897ad2antrr 726 . . . . . . . . 9 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) → (𝑆 ∖ {𝑠}) ∈ V)
991ssdifssd 4147 . . . . . . . . . 10 (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀))
10099ad2antrl 728 . . . . . . . . 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))
10198, 100elpwd 4606 . . . . . . . 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))
102101adantr 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))
10316lspeqlco 48356 . . . . . . . . 9 ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo (𝑆 ∖ {𝑠})) = ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))
104103eleq2d 2827 . . . . . . . 8 ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → ((𝑦( ·𝑠𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠})) ↔ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠}))))
105104bicomd 223 . . . . . . 7 ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → ((𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})) ↔ (𝑦( ·𝑠𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠}))))
10696, 102, 105syl2anc 584 . . . . . 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}))))
1073adantr 480 . . . . . . . . 9 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))) → 𝑀 ∈ LMod)
108 difexg 5329 . . . . . . . . . . 11 (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ∈ V)
109108, 99elpwd 4606 . . . . . . . . . 10 (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀))
110109ad2antrl 728 . . . . . . . . 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))
111107, 110jca 511 . . . . . . . 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)))
112111adantr 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)))
11316, 17, 18lcoval 48329 . . . . . . . 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}))))))
11419eqcomi 2746 . . . . . . . . . . . 12 (0g𝑅) = 0
115114breq2i 5151 . . . . . . . . . . 11 (𝑔 finSupp (0g𝑅) ↔ 𝑔 finSupp 0 )
116115anbi1i 624 . . . . . . . . . 10 ((𝑔 finSupp (0g𝑅) ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
117116rexbii 3094 . . . . . . . . 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}))))
118117anbi2i 623 . . . . . . . 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})))))
119113, 118bitrdi 287 . . . . . . 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}))))))
120112, 119syl 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}))))))
121106, 120bitrd 279 . . . . 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}))))))
12295, 121mtbird 325 . . . 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})))
123122ralrimivva 3202 . . 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})))
1242, 123jca 511 . 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}))))
125124ex 412 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 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  cdif 3948  wss 3951  ifcif 4525  𝒫 cpw 4600  {csn 4626   class class class wbr 5143  cmpt 5225  cfv 6561  (class class class)co 7431  m cmap 8866   finSupp cfsupp 9401  Basecbs 17247  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17484  Grpcgrp 18951  invgcminusg 18952  LModclmod 20858  LSpanclspn 20969   linC clinc 48321   LinCo clinco 48322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-gsum 17487  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-lmod 20860  df-lss 20930  df-lsp 20970  df-linc 48323  df-lco 48324
This theorem is referenced by:  lindslininds  48381
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