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Theorem lindslinindsimp2 41566
Description: Implication 2 for lindslininds 41567. (Contributed by AV, 26-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
lindslinindsimp2 ((𝑆𝑉𝑀 ∈ LMod) → ((𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠}))) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))
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   𝑦,𝑅   𝑥,𝐵   𝑥,𝑀   𝑅,𝑠   𝑓,𝑉,𝑥   𝑥,𝑍

Proof of Theorem lindslinindsimp2
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 simprl 793 . . . 4 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))) → 𝑆 ⊆ (Base‘𝑀))
2 elpwg 4143 . . . . 5 (𝑆𝑉 → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
32ad2antrr 761 . . . 4 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))) → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
41, 3mpbird 247 . . 3 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))) → 𝑆 ∈ 𝒫 (Base‘𝑀))
5 simplr 791 . . . . . . . . 9 (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → 𝑀 ∈ LMod)
6 ssdifss 3724 . . . . . . . . . . 11 (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀))
76adantl 482 . . . . . . . . . 10 (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀))
8 difexg 4773 . . . . . . . . . . . 12 (𝑆𝑉 → (𝑆 ∖ {𝑠}) ∈ V)
98ad2antrr 761 . . . . . . . . . . 11 (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (𝑆 ∖ {𝑠}) ∈ V)
10 elpwg 4143 . . . . . . . . . . 11 ((𝑆 ∖ {𝑠}) ∈ V → ((𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀)))
119, 10syl 17 . . . . . . . . . 10 (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → ((𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑠}) ⊆ (Base‘𝑀)))
127, 11mpbird 247 . . . . . . . . 9 (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀))
13 eqid 2621 . . . . . . . . . . . 12 (Base‘𝑀) = (Base‘𝑀)
1413lspeqlco 41542 . . . . . . . . . . 11 ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo (𝑆 ∖ {𝑠})) = ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))
1514eleq2d 2684 . . . . . . . . . 10 ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → ((𝑦( ·𝑠𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠})) ↔ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠}))))
1615bicomd 213 . . . . . . . . 9 ((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑠}) ∈ 𝒫 (Base‘𝑀)) → ((𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})) ↔ (𝑦( ·𝑠𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠}))))
175, 12, 16syl2anc 692 . . . . . . . 8 (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → ((𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})) ↔ (𝑦( ·𝑠𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠}))))
1817notbid 308 . . . . . . 7 (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (¬ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})) ↔ ¬ (𝑦( ·𝑠𝑀)𝑠) ∈ (𝑀 LinCo (𝑆 ∖ {𝑠}))))
19 lindslinind.r . . . . . . . . . . . 12 𝑅 = (Scalar‘𝑀)
20 lindslinind.b . . . . . . . . . . . 12 𝐵 = (Base‘𝑅)
2113, 19, 20lcoval 41515 . . . . . . . . . . 11 ((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}))))))
22 lindslinind.0 . . . . . . . . . . . . . . . 16 0 = (0g𝑅)
2322eqcomi 2630 . . . . . . . . . . . . . . 15 (0g𝑅) = 0
2423breq2i 4626 . . . . . . . . . . . . . 14 (𝑔 finSupp (0g𝑅) ↔ 𝑔 finSupp 0 )
2524anbi1i 730 . . . . . . . . . . . . 13 ((𝑔 finSupp (0g𝑅) ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
2625rexbii 3035 . . . . . . . . . . . 12 (∃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}))))
2726anbi2i 729 . . . . . . . . . . 11 (((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})))))
2821, 27syl6bb 276 . . . . . . . . . 10 ((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}))))))
295, 12, 28syl2anc 692 . . . . . . . . 9 (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (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}))))))
3029notbid 308 . . . . . . . 8 (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (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}))))))
31 ianor 509 . . . . . . . . 9 (¬ ((𝑦( ·𝑠𝑀)𝑠) ∈ (Base‘𝑀) ∧ ∃𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (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})))))
32 ralnex 2987 . . . . . . . . . . 11 (∀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}))))
33 ianor 509 . . . . . . . . . . . 12 (¬ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ (¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))
3433ralbii 2975 . . . . . . . . . . 11 (∀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}))))
3532, 34bitr3i 266 . . . . . . . . . 10 (¬ ∃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}))))
3635orbi2i 541 . . . . . . . . 9 ((¬ (𝑦( ·𝑠𝑀)𝑠) ∈ (Base‘𝑀) ∨ ¬ ∃𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (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})))))
3731, 36bitri 264 . . . . . . . 8 (¬ ((𝑦( ·𝑠𝑀)𝑠) ∈ (Base‘𝑀) ∧ ∃𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(𝑔 finSupp 0 ∧ (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})))))
3830, 37syl6bb 276 . . . . . . 7 (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (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}))))))
3918, 38bitrd 268 . . . . . 6 (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (¬ (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}))))))
40392ralbidv 2984 . . . . 5 (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})) ↔ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })(¬ (𝑦( ·𝑠𝑀)𝑠) ∈ (Base‘𝑀) ∨ ∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
41 simpllr 798 . . . . . . . . . . 11 ((((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑀 ∈ LMod)
42 eldifi 3715 . . . . . . . . . . . . 13 (𝑦 ∈ (𝐵 ∖ { 0 }) → 𝑦𝐵)
4342adantl 482 . . . . . . . . . . . 12 ((𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })) → 𝑦𝐵)
4443adantl 482 . . . . . . . . . . 11 ((((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑦𝐵)
45 ssel2 3582 . . . . . . . . . . . 12 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑠𝑆) → 𝑠 ∈ (Base‘𝑀))
4645ad2ant2lr 783 . . . . . . . . . . 11 ((((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → 𝑠 ∈ (Base‘𝑀))
47 eqid 2621 . . . . . . . . . . . 12 ( ·𝑠𝑀) = ( ·𝑠𝑀)
4813, 19, 47, 20lmodvscl 18812 . . . . . . . . . . 11 ((𝑀 ∈ LMod ∧ 𝑦𝐵𝑠 ∈ (Base‘𝑀)) → (𝑦( ·𝑠𝑀)𝑠) ∈ (Base‘𝑀))
4941, 44, 46, 48syl3anc 1323 . . . . . . . . . 10 ((((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → (𝑦( ·𝑠𝑀)𝑠) ∈ (Base‘𝑀))
5049notnotd 138 . . . . . . . . 9 ((((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → ¬ ¬ (𝑦( ·𝑠𝑀)𝑠) ∈ (Base‘𝑀))
51 nbfal 1492 . . . . . . . . 9 (¬ ¬ (𝑦( ·𝑠𝑀)𝑠) ∈ (Base‘𝑀) ↔ (¬ (𝑦( ·𝑠𝑀)𝑠) ∈ (Base‘𝑀) ↔ ⊥))
5250, 51sylib 208 . . . . . . . 8 ((((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → (¬ (𝑦( ·𝑠𝑀)𝑠) ∈ (Base‘𝑀) ↔ ⊥))
5352orbi1d 738 . . . . . . 7 ((((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ (𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }))) → ((¬ (𝑦( ·𝑠𝑀)𝑠) ∈ (Base‘𝑀) ∨ ∀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}))))))
54532ralbidva 2983 . . . . . 6 (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })(¬ (𝑦( ·𝑠𝑀)𝑠) ∈ (Base‘𝑀) ∨ ∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) ↔ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })(⊥ ∨ ∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))))))
55 r19.32v 3076 . . . . . . . . 9 (∀𝑦 ∈ (𝐵 ∖ { 0 })(⊥ ∨ ∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) ↔ (⊥ ∨ ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
5655ralbii 2975 . . . . . . . 8 (∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })(⊥ ∨ ∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) ↔ ∀𝑠𝑆 (⊥ ∨ ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
57 r19.32v 3076 . . . . . . . 8 (∀𝑠𝑆 (⊥ ∨ ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) ↔ (⊥ ∨ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
5856, 57bitri 264 . . . . . . 7 (∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })(⊥ ∨ ∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) ↔ (⊥ ∨ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))))
59 falim 1495 . . . . . . . . 9 (⊥ → (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
60 sneq 4163 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠 = 𝑥 → {𝑠} = {𝑥})
6160difeq2d 3711 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 = 𝑥 → (𝑆 ∖ {𝑠}) = (𝑆 ∖ {𝑥}))
6261oveq2d 6626 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 = 𝑥 → (𝐵𝑚 (𝑆 ∖ {𝑠})) = (𝐵𝑚 (𝑆 ∖ {𝑥})))
63 oveq2 6618 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑠 = 𝑥 → (𝑦( ·𝑠𝑀)𝑠) = (𝑦( ·𝑠𝑀)𝑥))
6461oveq2d 6626 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑠 = 𝑥 → (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))
6563, 64eqeq12d 2636 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠 = 𝑥 → ((𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})) ↔ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
6665notbid 308 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 = 𝑥 → (¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})) ↔ ¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
6766orbi2d 737 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 = 𝑥 → ((¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ (¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))))
6862, 67raleqbidv 3144 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = 𝑥 → (∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ ∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))))
6968ralbidv 2981 . . . . . . . . . . . . . . . . . . 19 (𝑠 = 𝑥 → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) ↔ ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))))
7069rspcva 3296 . . . . . . . . . . . . . . . . . 18 ((𝑥𝑆 ∧ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) → ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
71 lindslinind.z . . . . . . . . . . . . . . . . . . . . 21 𝑍 = (0g𝑀)
7219, 20, 22, 71lindslinindsimp2lem5 41565 . . . . . . . . . . . . . . . . . . . 20 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓𝑥) = 0 )))
7372expr 642 . . . . . . . . . . . . . . . . . . 19 (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (𝑥𝑆 → ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓𝑥) = 0 ))))
7473com14 96 . . . . . . . . . . . . . . . . . 18 (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑥𝑆 → ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (𝑓𝑥) = 0 ))))
7570, 74syl 17 . . . . . . . . . . . . . . . . 17 ((𝑥𝑆 ∧ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) → (𝑥𝑆 → ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (𝑓𝑥) = 0 ))))
7675ex 450 . . . . . . . . . . . . . . . 16 (𝑥𝑆 → (∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) → (𝑥𝑆 → ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (𝑓𝑥) = 0 )))))
7776pm2.43a 54 . . . . . . . . . . . . . . 15 (𝑥𝑆 → (∀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)) → (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (𝑓𝑥) = 0 ))))
7877com14 96 . . . . . . . . . . . . . 14 (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (∀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 ))))
7978imp 445 . . . . . . . . . . . . 13 ((((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ ∀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 )))
8079expdimp 453 . . . . . . . . . . . 12 (((((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ ∀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 )))
8180ralrimdv 2963 . . . . . . . . . . 11 (((((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ ∀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 ))
8281ralrimiva 2961 . . . . . . . . . 10 ((((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) ∧ ∀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 ))
8382expcom 451 . . . . . . . . 9 (∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠}))) → (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
8459, 83jaoi 394 . . . . . . . 8 ((⊥ ∨ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑠) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑠})))) → (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
8584com12 32 . . . . . . 7 (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → ((⊥ ∨ ∀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 )))
8658, 85syl5bi 232 . . . . . 6 (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (∀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 )))
8754, 86sylbid 230 . . . . 5 (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 })(¬ (𝑦( ·𝑠𝑀)𝑠) ∈ (Base‘𝑀) ∨ ∀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 )))
8840, 87sylbid 230 . . . 4 (((𝑆𝑉𝑀 ∈ LMod) ∧ 𝑆 ⊆ (Base‘𝑀)) → (∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})) → ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
8988impr 648 . . 3 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))) → ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))
904, 89jca 554 . 2 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
9190ex 450 1 ((𝑆𝑉𝑀 ∈ LMod) → ((𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠}))) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wfal 1485  wcel 1987  wral 2907  wrex 2908  Vcvv 3189  cdif 3556  wss 3559  𝒫 cpw 4135  {csn 4153   class class class wbr 4618  cfv 5852  (class class class)co 6610  𝑚 cmap 7809   finSupp cfsupp 8227  Basecbs 15792  Scalarcsca 15876   ·𝑠 cvsca 15877  0gc0g 16032  LModclmod 18795  LSpanclspn 18903   linC clinc 41507   LinCo clinco 41508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-fsupp 8228  df-oi 8367  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-n0 11245  df-z 11330  df-uz 11640  df-fz 12277  df-fzo 12415  df-seq 12750  df-hash 13066  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-0g 16034  df-gsum 16035  df-mre 16178  df-mrc 16179  df-acs 16181  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-mhm 17267  df-submnd 17268  df-grp 17357  df-minusg 17358  df-sbg 17359  df-mulg 17473  df-subg 17523  df-ghm 17590  df-cntz 17682  df-cmn 18127  df-abl 18128  df-mgp 18422  df-ur 18434  df-ring 18481  df-lmod 18797  df-lss 18865  df-lsp 18904  df-linc 41509  df-lco 41510
This theorem is referenced by:  lindslininds  41567
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