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Theorem lindslinindsimp1 48302
Description: Implication 1 for lindslininds 48309. (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 4611 . . . 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 4140 . . . . . . . . . . . . . . . . . . . . . . . . 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 1127 . . . . . . . . . . . . . . . . . . . . 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 2734 . . . . . . . . . . . . . . . . . . . . . 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 2734 . . . . . . . . . . . . . . . . . . . . . 22 (invg𝑅) = (invg𝑅)
22 eqid 2734 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))
2316, 17, 18, 19, 20, 21, 22lincext2 48300 . . . . . . . . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . . . . . . . . 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 48299 . . . . . . . . . . . . . . . . . . . . . . 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 5150 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → (𝑓 finSupp 0 ↔ (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) finSupp 0 ))
31 oveq1 7437 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → (𝑓( linC ‘𝑀)𝑆) = ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))( linC ‘𝑀)𝑆))
3231eqeq1d 2736 . . . . . . . . . . . . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 = (𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧))) → ((𝑓𝑥) = 0 ↔ ((𝑧𝑆 ↦ if(𝑧 = 𝑠, ((invg𝑅)‘𝑦), (𝑔𝑧)))‘𝑥) = 0 ))
3635ralbidv 3175 . . . . . . . . . . . . . . . . . . . . . . . 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 3617 . . . . . . . . . . . . . . . . . . . . . 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 48301 . . . . . . . . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . . . . . . . . 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 3617 . . . . . . . . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . . . . . . . . . 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 4536 . . . . . . . . . . . . . . . . . . . . . . . . . 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 7022 . . . . . . . . . . . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . . . . . . . . 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 20883 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑀 ∈ LMod → 𝑅 ∈ Grp)
5618, 19, 21grpinvnzcl 19041 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑅 ∈ Grp ∧ 𝑦 ∈ (𝐵 ∖ { 0 })) → ((invg𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }))
57 eldif 3972 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((invg𝑅)‘𝑦) ∈ (𝐵 ∖ { 0 }) ↔ (((invg𝑅)‘𝑦) ∈ 𝐵 ∧ ¬ ((invg𝑅)‘𝑦) ∈ { 0 }))
58 fvex 6919 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((invg𝑅)‘𝑦) ∈ V
5958elsn 4645 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 874 . . . . . . . . 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 979 . . . . . . . . 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 3143 . . . . . . 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 3069 . . . . . . . 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 983 . . . . . . . . 9 (¬ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ (¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
9291ralbii 3090 . . . . . . . 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 5334 . . . . . . . . . 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 4156 . . . . . . . . . 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 4610 . . . . . . . 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 48284 . . . . . . . . 9 ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo (𝑆 ∖ {𝑠})) = ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))
104103eleq2d 2824 . . . . . . . 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 5334 . . . . . . . . . . 11 (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑠}) ∈ V)
109108, 99elpwd 4610 . . . . . . . . . 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 48257 . . . . . . . 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 2743 . . . . . . . . . . . 12 (0g𝑅) = 0
115114breq2i 5155 . . . . . . . . . . 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 3091 . . . . . . . . 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 3199 . . 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 847  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wrex 3067  Vcvv 3477  cdif 3959  wss 3962  ifcif 4530  𝒫 cpw 4604  {csn 4630   class class class wbr 5147  cmpt 5230  cfv 6562  (class class class)co 7430  m cmap 8864   finSupp cfsupp 9398  Basecbs 17244  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17485  Grpcgrp 18963  invgcminusg 18964  LModclmod 20874  LSpanclspn 20986   linC clinc 48249   LinCo clinco 48250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691  df-seq 14039  df-hash 14366  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-0g 17487  df-gsum 17488  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-grp 18966  df-minusg 18967  df-sbg 18968  df-mulg 19098  df-subg 19153  df-ghm 19243  df-cntz 19347  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-lmod 20876  df-lss 20947  df-lsp 20987  df-linc 48251  df-lco 48252
This theorem is referenced by:  lindslininds  48309
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