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Theorem lindslinindsimp1 47091
Description: Implication 1 for lindslininds 47098. (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 = (Scalarβ€˜π‘€)
lindslinind.b 𝐡 = (Baseβ€˜π‘…)
lindslinind.0 0 = (0gβ€˜π‘…)
lindslinind.z 𝑍 = (0gβ€˜π‘€)
Assertion
Ref Expression
lindslinindsimp1 ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) β†’ ((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 )) β†’ (𝑆 βŠ† (Baseβ€˜π‘€) ∧ βˆ€π‘  ∈ 𝑆 βˆ€π‘¦ ∈ (𝐡 βˆ– { 0 }) Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) ∈ ((LSpanβ€˜π‘€)β€˜(𝑆 βˆ– {𝑠})))))
Distinct variable groups:   𝐡,𝑓,𝑠,𝑦   𝑓,𝑀,𝑠,𝑦   𝑅,𝑓,π‘₯   𝑆,𝑓,𝑠,π‘₯,𝑦   𝑉,𝑠,𝑦   𝑓,𝑍,𝑠,𝑦   0 ,𝑓,𝑠,π‘₯,𝑦
Allowed substitution hints:   𝐡(π‘₯)   𝑅(𝑦,𝑠)   𝑀(π‘₯)   𝑉(π‘₯,𝑓)   𝑍(π‘₯)
Proof of Theorem lindslinindsimp1
Dummy variables 𝑔 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpwi 4608 . . . 4 (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) β†’ 𝑆 βŠ† (Baseβ€˜π‘€))
21ad2antrl 726 . . 3 (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) β†’ 𝑆 βŠ† (Baseβ€˜π‘€))
3 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) β†’ 𝑀 ∈ LMod)
43anim2i 617 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) β†’ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝑀 ∈ LMod))
54ancomd 462 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) β†’ (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Baseβ€˜π‘€)))
65ad2antrr 724 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ (𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))) β†’ (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Baseβ€˜π‘€)))
7 eldifi 4125 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ (𝐡 βˆ– { 0 }) β†’ 𝑦 ∈ 𝐡)
87adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 })) β†’ 𝑦 ∈ 𝐡)
98adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) β†’ 𝑦 ∈ 𝐡)
109adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ (𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))) β†’ 𝑦 ∈ 𝐡)
11 simprl 769 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) β†’ 𝑠 ∈ 𝑆)
1211adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ (𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))) β†’ 𝑠 ∈ 𝑆)
13 simprl 769 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ (𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))) β†’ 𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})))
1410, 12, 133jca 1128 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ (𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))) β†’ (𝑦 ∈ 𝐡 ∧ 𝑠 ∈ 𝑆 ∧ 𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠}))))
15 simprrl 779 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ (𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))) β†’ 𝑔 finSupp 0 )
16 eqid 2732 . . . . . . . . . . . . . . . . . . . . . 22 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
17 lindslinind.r . . . . . . . . . . . . . . . . . . . . . 22 𝑅 = (Scalarβ€˜π‘€)
18 lindslinind.b . . . . . . . . . . . . . . . . . . . . . 22 𝐡 = (Baseβ€˜π‘…)
19 lindslinind.0 . . . . . . . . . . . . . . . . . . . . . 22 0 = (0gβ€˜π‘…)
20 lindslinind.z . . . . . . . . . . . . . . . . . . . . . 22 𝑍 = (0gβ€˜π‘€)
21 eqid 2732 . . . . . . . . . . . . . . . . . . . . . 22 (invgβ€˜π‘…) = (invgβ€˜π‘…)
22 eqid 2732 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§))) = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))
2316, 17, 18, 19, 20, 21, 22lincext2 47089 . . . . . . . . . . . . . . . . . . . . 21 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝑦 ∈ 𝐡 ∧ 𝑠 ∈ 𝑆 ∧ 𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠}))) ∧ 𝑔 finSupp 0 ) β†’ (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§))) finSupp 0 )
246, 14, 15, 23syl3anc 1371 . . . . . . . . . . . . . . . . . . . 20 ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ (𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))) β†’ (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§))) finSupp 0 )
254adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) β†’ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝑀 ∈ LMod))
2625ancomd 462 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) β†’ (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Baseβ€˜π‘€)))
2726adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ (𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))) β†’ (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Baseβ€˜π‘€)))
2816, 17, 18, 19, 20, 21, 22lincext1 47088 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝑦 ∈ 𝐡 ∧ 𝑠 ∈ 𝑆 ∧ 𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})))) β†’ (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§))) ∈ (𝐡 ↑m 𝑆))
2927, 14, 28syl2anc 584 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ (𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))) β†’ (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§))) ∈ (𝐡 ↑m 𝑆))
30 breq1 5150 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§))) β†’ (𝑓 finSupp 0 ↔ (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§))) finSupp 0 ))
31 oveq1 7412 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§))) β†’ (𝑓( linC β€˜π‘€)𝑆) = ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))( linC β€˜π‘€)𝑆))
3231eqeq1d 2734 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§))) β†’ ((𝑓( linC β€˜π‘€)𝑆) = 𝑍 ↔ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))( linC β€˜π‘€)𝑆) = 𝑍))
3330, 32anbi12d 631 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§))) β†’ ((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) ↔ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§))) finSupp 0 ∧ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))( linC β€˜π‘€)𝑆) = 𝑍)))
34 fveq1 6887 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§))) β†’ (π‘“β€˜π‘₯) = ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))β€˜π‘₯))
3534eqeq1d 2734 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§))) β†’ ((π‘“β€˜π‘₯) = 0 ↔ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))β€˜π‘₯) = 0 ))
3635ralbidv 3177 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§))) β†’ (βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ↔ βˆ€π‘₯ ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))β€˜π‘₯) = 0 ))
3733, 36imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§))) β†’ (((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ) ↔ (((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§))) finSupp 0 ∧ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))β€˜π‘₯) = 0 )))
3837rspcv 3608 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§))) ∈ (𝐡 ↑m 𝑆) β†’ (βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ) β†’ (((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§))) finSupp 0 ∧ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))β€˜π‘₯) = 0 )))
3929, 38syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ (𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))) β†’ (βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ) β†’ (((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§))) finSupp 0 ∧ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))β€˜π‘₯) = 0 )))
4039exp4a 432 . . . . . . . . . . . . . . . . . . . 20 ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ (𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))) β†’ (βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ) β†’ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§))) finSupp 0 β†’ (((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))( linC β€˜π‘€)𝑆) = 𝑍 β†’ βˆ€π‘₯ ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))β€˜π‘₯) = 0 ))))
4124, 40mpid 44 . . . . . . . . . . . . . . . . . . 19 ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ (𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))) β†’ (βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ) β†’ (((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))( linC β€˜π‘€)𝑆) = 𝑍 β†’ βˆ€π‘₯ ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))β€˜π‘₯) = 0 )))
42 simprr 771 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ (𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))) β†’ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))
4316, 17, 18, 19, 20, 21, 22lincext3 47090 . . . . . . . . . . . . . . . . . . . . 21 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝑦 ∈ 𝐡 ∧ 𝑠 ∈ 𝑆 ∧ 𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠}))) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠})))) β†’ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))( linC β€˜π‘€)𝑆) = 𝑍)
446, 14, 42, 43syl3anc 1371 . . . . . . . . . . . . . . . . . . . 20 ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ (𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))) β†’ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))( linC β€˜π‘€)𝑆) = 𝑍)
45 fveqeq2 6897 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘₯ = 𝑠 β†’ (((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))β€˜π‘₯) = 0 ↔ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))β€˜π‘ ) = 0 ))
4645rspcv 3608 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 ∈ 𝑆 β†’ (βˆ€π‘₯ ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))β€˜π‘₯) = 0 β†’ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))β€˜π‘ ) = 0 ))
4712, 46syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ (𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))) β†’ (βˆ€π‘₯ ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))β€˜π‘₯) = 0 β†’ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))β€˜π‘ ) = 0 ))
48 eqidd 2733 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) β†’ (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§))) = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§))))
49 iftrue 4533 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 = 𝑠 β†’ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)) = ((invgβ€˜π‘…)β€˜π‘¦))
5049adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ 𝑧 = 𝑠) β†’ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)) = ((invgβ€˜π‘…)β€˜π‘¦))
51 fvexd 6903 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) β†’ ((invgβ€˜π‘…)β€˜π‘¦) ∈ V)
5248, 50, 11, 51fvmptd 7002 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) β†’ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))β€˜π‘ ) = ((invgβ€˜π‘…)β€˜π‘¦))
5352adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ (𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))) β†’ ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))β€˜π‘ ) = ((invgβ€˜π‘…)β€˜π‘¦))
5453eqeq1d 2734 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ (𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))) β†’ (((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))β€˜π‘ ) = 0 ↔ ((invgβ€˜π‘…)β€˜π‘¦) = 0 ))
5517lmodfgrp 20472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑀 ∈ LMod β†’ 𝑅 ∈ Grp)
5618, 19, 21grpinvnzcl 18891 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑅 ∈ Grp ∧ 𝑦 ∈ (𝐡 βˆ– { 0 })) β†’ ((invgβ€˜π‘…)β€˜π‘¦) ∈ (𝐡 βˆ– { 0 }))
57 eldif 3957 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((invgβ€˜π‘…)β€˜π‘¦) ∈ (𝐡 βˆ– { 0 }) ↔ (((invgβ€˜π‘…)β€˜π‘¦) ∈ 𝐡 ∧ Β¬ ((invgβ€˜π‘…)β€˜π‘¦) ∈ { 0 }))
58 fvex 6901 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((invgβ€˜π‘…)β€˜π‘¦) ∈ V
5958elsn 4642 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((invgβ€˜π‘…)β€˜π‘¦) ∈ { 0 } ↔ ((invgβ€˜π‘…)β€˜π‘¦) = 0 )
60 pm2.21 123 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (Β¬ ((invgβ€˜π‘…)β€˜π‘¦) = 0 β†’ (((invgβ€˜π‘…)β€˜π‘¦) = 0 β†’ (𝑆 ∈ 𝑉 β†’ (𝑠 ∈ 𝑆 β†’ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠})))))))
6160com25 99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (Β¬ ((invgβ€˜π‘…)β€˜π‘¦) = 0 β†’ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) β†’ (𝑆 ∈ 𝑉 β†’ (𝑠 ∈ 𝑆 β†’ (((invgβ€˜π‘…)β€˜π‘¦) = 0 β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠})))))))
6259, 61sylnbi 329 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (Β¬ ((invgβ€˜π‘…)β€˜π‘¦) ∈ { 0 } β†’ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) β†’ (𝑆 ∈ 𝑉 β†’ (𝑠 ∈ 𝑆 β†’ (((invgβ€˜π‘…)β€˜π‘¦) = 0 β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠})))))))
6357, 62simplbiim 505 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((invgβ€˜π‘…)β€˜π‘¦) ∈ (𝐡 βˆ– { 0 }) β†’ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) β†’ (𝑆 ∈ 𝑉 β†’ (𝑠 ∈ 𝑆 β†’ (((invgβ€˜π‘…)β€˜π‘¦) = 0 β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠})))))))
6456, 63syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑅 ∈ Grp ∧ 𝑦 ∈ (𝐡 βˆ– { 0 })) β†’ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) β†’ (𝑆 ∈ 𝑉 β†’ (𝑠 ∈ 𝑆 β†’ (((invgβ€˜π‘…)β€˜π‘¦) = 0 β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠})))))))
6564ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑅 ∈ Grp β†’ (𝑦 ∈ (𝐡 βˆ– { 0 }) β†’ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) β†’ (𝑆 ∈ 𝑉 β†’ (𝑠 ∈ 𝑆 β†’ (((invgβ€˜π‘…)β€˜π‘¦) = 0 β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))))))
6655, 65syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑀 ∈ LMod β†’ (𝑦 ∈ (𝐡 βˆ– { 0 }) β†’ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) β†’ (𝑆 ∈ 𝑉 β†’ (𝑠 ∈ 𝑆 β†’ (((invgβ€˜π‘…)β€˜π‘¦) = 0 β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))))))
6766com24 95 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑀 ∈ LMod β†’ (𝑆 ∈ 𝑉 β†’ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) β†’ (𝑦 ∈ (𝐡 βˆ– { 0 }) β†’ (𝑠 ∈ 𝑆 β†’ (((invgβ€˜π‘…)β€˜π‘¦) = 0 β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))))))
6867impcom 408 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) β†’ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) β†’ (𝑦 ∈ (𝐡 βˆ– { 0 }) β†’ (𝑠 ∈ 𝑆 β†’ (((invgβ€˜π‘…)β€˜π‘¦) = 0 β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠})))))))
6968impcom 408 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) β†’ (𝑦 ∈ (𝐡 βˆ– { 0 }) β†’ (𝑠 ∈ 𝑆 β†’ (((invgβ€˜π‘…)β€˜π‘¦) = 0 β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))))
7069com13 88 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑠 ∈ 𝑆 β†’ (𝑦 ∈ (𝐡 βˆ– { 0 }) β†’ ((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) β†’ (((invgβ€˜π‘…)β€˜π‘¦) = 0 β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))))
7170imp 407 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 })) β†’ ((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) β†’ (((invgβ€˜π‘…)β€˜π‘¦) = 0 β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠})))))
7271impcom 408 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) β†’ (((invgβ€˜π‘…)β€˜π‘¦) = 0 β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))
7372adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ (𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))) β†’ (((invgβ€˜π‘…)β€˜π‘¦) = 0 β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))
7454, 73sylbid 239 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ (𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))) β†’ (((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))β€˜π‘ ) = 0 β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))
7547, 74syld 47 . . . . . . . . . . . . . . . . . . . 20 ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ (𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))) β†’ (βˆ€π‘₯ ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))β€˜π‘₯) = 0 β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))
7644, 75embantd 59 . . . . . . . . . . . . . . . . . . 19 ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ (𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))) β†’ ((((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))( linC β€˜π‘€)𝑆) = 𝑍 β†’ βˆ€π‘₯ ∈ 𝑆 ((𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑠, ((invgβ€˜π‘…)β€˜π‘¦), (π‘”β€˜π‘§)))β€˜π‘₯) = 0 ) β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))
7741, 76syldc 48 . . . . . . . . . . . . . . . . . 18 (βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ) β†’ ((((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod)) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ (𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))) β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))
7877exp5j 446 . . . . . . . . . . . . . . . . 17 (βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ) β†’ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) β†’ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) β†’ ((𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 })) β†’ ((𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠})))) β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠})))))))
7978impcom 408 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 )) β†’ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) β†’ ((𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 })) β†’ ((𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠})))) β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))))
8079impcom 408 . . . . . . . . . . . . . . 15 (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) β†’ ((𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 })) β†’ ((𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠})))) β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠})))))
8180imp 407 . . . . . . . . . . . . . 14 ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) β†’ ((𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) ∧ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠})))) β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))
8281expdimp 453 . . . . . . . . . . . . 13 (((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ 𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠}))) β†’ ((𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))) β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))
8382expd 416 . . . . . . . . . . . 12 (((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ 𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠}))) β†’ (𝑔 finSupp 0 β†’ ((𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠})) β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠})))))
8483impcom 408 . . . . . . . . . . 11 ((𝑔 finSupp 0 ∧ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ 𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})))) β†’ ((𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠})) β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))
8584pm2.01d 189 . . . . . . . . . 10 ((𝑔 finSupp 0 ∧ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ 𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})))) β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠})))
8685olcd 872 . . . . . . . . 9 ((𝑔 finSupp 0 ∧ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ 𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})))) β†’ (Β¬ 𝑔 finSupp 0 ∨ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))
87 animorl 976 . . . . . . . . 9 ((Β¬ 𝑔 finSupp 0 ∧ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ 𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})))) β†’ (Β¬ 𝑔 finSupp 0 ∨ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))
8886, 87pm2.61ian 810 . . . . . . . 8 (((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) ∧ 𝑔 ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠}))) β†’ (Β¬ 𝑔 finSupp 0 ∨ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))
8988ralrimiva 3146 . . . . . . 7 ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) β†’ βˆ€π‘” ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠}))(Β¬ 𝑔 finSupp 0 ∨ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))
90 ralnex 3072 . . . . . . . 8 (βˆ€π‘” ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) Β¬ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))) ↔ Β¬ βˆƒπ‘” ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))
91 ianor 980 . . . . . . . . 9 (Β¬ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))) ↔ (Β¬ 𝑔 finSupp 0 ∨ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))
9291ralbii 3093 . . . . . . . 8 (βˆ€π‘” ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠})) Β¬ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))) ↔ βˆ€π‘” ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠}))(Β¬ 𝑔 finSupp 0 ∨ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))
9390, 92bitr3i 276 . . . . . . 7 (Β¬ βˆƒπ‘” ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))) ↔ βˆ€π‘” ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠}))(Β¬ 𝑔 finSupp 0 ∨ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))
9489, 93sylibr 233 . . . . . 6 ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) β†’ Β¬ βˆƒπ‘” ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))
9594intnand 489 . . . . 5 ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) β†’ Β¬ ((𝑦( ·𝑠 β€˜π‘€)𝑠) ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘” ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠})))))
963ad2antrr 724 . . . . . . 7 ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) β†’ 𝑀 ∈ LMod)
97 difexg 5326 . . . . . . . . . 10 (𝑆 ∈ 𝑉 β†’ (𝑆 βˆ– {𝑠}) ∈ V)
9897ad2antrr 724 . . . . . . . . 9 (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) β†’ (𝑆 βˆ– {𝑠}) ∈ V)
991ssdifssd 4141 . . . . . . . . . 10 (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) β†’ (𝑆 βˆ– {𝑠}) βŠ† (Baseβ€˜π‘€))
10099ad2antrl 726 . . . . . . . . 9 (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) β†’ (𝑆 βˆ– {𝑠}) βŠ† (Baseβ€˜π‘€))
10198, 100elpwd 4607 . . . . . . . 8 (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) β†’ (𝑆 βˆ– {𝑠}) ∈ 𝒫 (Baseβ€˜π‘€))
102101adantr 481 . . . . . . 7 ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) β†’ (𝑆 βˆ– {𝑠}) ∈ 𝒫 (Baseβ€˜π‘€))
10316lspeqlco 47073 . . . . . . . . 9 ((𝑀 ∈ LMod ∧ (𝑆 βˆ– {𝑠}) ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (𝑀 LinCo (𝑆 βˆ– {𝑠})) = ((LSpanβ€˜π‘€)β€˜(𝑆 βˆ– {𝑠})))
104103eleq2d 2819 . . . . . . . 8 ((𝑀 ∈ LMod ∧ (𝑆 βˆ– {𝑠}) ∈ 𝒫 (Baseβ€˜π‘€)) β†’ ((𝑦( ·𝑠 β€˜π‘€)𝑠) ∈ (𝑀 LinCo (𝑆 βˆ– {𝑠})) ↔ (𝑦( ·𝑠 β€˜π‘€)𝑠) ∈ ((LSpanβ€˜π‘€)β€˜(𝑆 βˆ– {𝑠}))))
105104bicomd 222 . . . . . . 7 ((𝑀 ∈ LMod ∧ (𝑆 βˆ– {𝑠}) ∈ 𝒫 (Baseβ€˜π‘€)) β†’ ((𝑦( ·𝑠 β€˜π‘€)𝑠) ∈ ((LSpanβ€˜π‘€)β€˜(𝑆 βˆ– {𝑠})) ↔ (𝑦( ·𝑠 β€˜π‘€)𝑠) ∈ (𝑀 LinCo (𝑆 βˆ– {𝑠}))))
10696, 102, 105syl2anc 584 . . . . . 6 ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) β†’ ((𝑦( ·𝑠 β€˜π‘€)𝑠) ∈ ((LSpanβ€˜π‘€)β€˜(𝑆 βˆ– {𝑠})) ↔ (𝑦( ·𝑠 β€˜π‘€)𝑠) ∈ (𝑀 LinCo (𝑆 βˆ– {𝑠}))))
1073adantr 481 . . . . . . . . 9 (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) β†’ 𝑀 ∈ LMod)
108 difexg 5326 . . . . . . . . . . 11 (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) β†’ (𝑆 βˆ– {𝑠}) ∈ V)
109108, 99elpwd 4607 . . . . . . . . . 10 (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) β†’ (𝑆 βˆ– {𝑠}) ∈ 𝒫 (Baseβ€˜π‘€))
110109ad2antrl 726 . . . . . . . . 9 (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) β†’ (𝑆 βˆ– {𝑠}) ∈ 𝒫 (Baseβ€˜π‘€))
111107, 110jca 512 . . . . . . . 8 (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) β†’ (𝑀 ∈ LMod ∧ (𝑆 βˆ– {𝑠}) ∈ 𝒫 (Baseβ€˜π‘€)))
112111adantr 481 . . . . . . 7 ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) β†’ (𝑀 ∈ LMod ∧ (𝑆 βˆ– {𝑠}) ∈ 𝒫 (Baseβ€˜π‘€)))
11316, 17, 18lcoval 47046 . . . . . . . 8 ((𝑀 ∈ LMod ∧ (𝑆 βˆ– {𝑠}) ∈ 𝒫 (Baseβ€˜π‘€)) β†’ ((𝑦( ·𝑠 β€˜π‘€)𝑠) ∈ (𝑀 LinCo (𝑆 βˆ– {𝑠})) ↔ ((𝑦( ·𝑠 β€˜π‘€)𝑠) ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘” ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠}))(𝑔 finSupp (0gβ€˜π‘…) ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))))
11419eqcomi 2741 . . . . . . . . . . . 12 (0gβ€˜π‘…) = 0
115114breq2i 5155 . . . . . . . . . . 11 (𝑔 finSupp (0gβ€˜π‘…) ↔ 𝑔 finSupp 0 )
116115anbi1i 624 . . . . . . . . . 10 ((𝑔 finSupp (0gβ€˜π‘…) ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))) ↔ (𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))
117116rexbii 3094 . . . . . . . . 9 (βˆƒπ‘” ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠}))(𝑔 finSupp (0gβ€˜π‘…) ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))) ↔ βˆƒπ‘” ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))
118117anbi2i 623 . . . . . . . 8 (((𝑦( ·𝑠 β€˜π‘€)𝑠) ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘” ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠}))(𝑔 finSupp (0gβ€˜π‘…) ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠})))) ↔ ((𝑦( ·𝑠 β€˜π‘€)𝑠) ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘” ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠})))))
119113, 118bitrdi 286 . . . . . . 7 ((𝑀 ∈ LMod ∧ (𝑆 βˆ– {𝑠}) ∈ 𝒫 (Baseβ€˜π‘€)) β†’ ((𝑦( ·𝑠 β€˜π‘€)𝑠) ∈ (𝑀 LinCo (𝑆 βˆ– {𝑠})) ↔ ((𝑦( ·𝑠 β€˜π‘€)𝑠) ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘” ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))))
120112, 119syl 17 . . . . . 6 ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) β†’ ((𝑦( ·𝑠 β€˜π‘€)𝑠) ∈ (𝑀 LinCo (𝑆 βˆ– {𝑠})) ↔ ((𝑦( ·𝑠 β€˜π‘€)𝑠) ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘” ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))))
121106, 120bitrd 278 . . . . 5 ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) β†’ ((𝑦( ·𝑠 β€˜π‘€)𝑠) ∈ ((LSpanβ€˜π‘€)β€˜(𝑆 βˆ– {𝑠})) ↔ ((𝑦( ·𝑠 β€˜π‘€)𝑠) ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘” ∈ (𝐡 ↑m (𝑆 βˆ– {𝑠}))(𝑔 finSupp 0 ∧ (𝑦( ·𝑠 β€˜π‘€)𝑠) = (𝑔( linC β€˜π‘€)(𝑆 βˆ– {𝑠}))))))
12295, 121mtbird 324 . . . 4 ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) ∧ (𝑠 ∈ 𝑆 ∧ 𝑦 ∈ (𝐡 βˆ– { 0 }))) β†’ Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) ∈ ((LSpanβ€˜π‘€)β€˜(𝑆 βˆ– {𝑠})))
123122ralrimivva 3200 . . 3 (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) β†’ βˆ€π‘  ∈ 𝑆 βˆ€π‘¦ ∈ (𝐡 βˆ– { 0 }) Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) ∈ ((LSpanβ€˜π‘€)β€˜(𝑆 βˆ– {𝑠})))
1242, 123jca 512 . 2 (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))) β†’ (𝑆 βŠ† (Baseβ€˜π‘€) ∧ βˆ€π‘  ∈ 𝑆 βˆ€π‘¦ ∈ (𝐡 βˆ– { 0 }) Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) ∈ ((LSpanβ€˜π‘€)β€˜(𝑆 βˆ– {𝑠}))))
125124ex 413 1 ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) β†’ ((𝑆 ∈ 𝒫 (Baseβ€˜π‘€) ∧ βˆ€π‘“ ∈ (𝐡 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 )) β†’ (𝑆 βŠ† (Baseβ€˜π‘€) ∧ βˆ€π‘  ∈ 𝑆 βˆ€π‘¦ ∈ (𝐡 βˆ– { 0 }) Β¬ (𝑦( ·𝑠 β€˜π‘€)𝑠) ∈ ((LSpanβ€˜π‘€)β€˜(𝑆 βˆ– {𝑠})))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   βˆ– cdif 3944   βŠ† wss 3947  ifcif 4527  π’« cpw 4601  {csn 4627   class class class wbr 5147   ↦ cmpt 5230  β€˜cfv 6540  (class class class)co 7405   ↑m cmap 8816   finSupp cfsupp 9357  Basecbs 17140  Scalarcsca 17196   ·𝑠 cvsca 17197  0gc0g 17381  Grpcgrp 18815  invgcminusg 18816  LModclmod 20463  LSpanclspn 20574   linC clinc 47038   LinCo clinco 47039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-seq 13963  df-hash 14287  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-0g 17383  df-gsum 17384  df-mre 17526  df-mrc 17527  df-acs 17529  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-mhm 18667  df-submnd 18668  df-grp 18818  df-minusg 18819  df-sbg 18820  df-mulg 18945  df-subg 18997  df-ghm 19084  df-cntz 19175  df-cmn 19644  df-abl 19645  df-mgp 19982  df-ur 19999  df-ring 20051  df-lmod 20465  df-lss 20535  df-lsp 20575  df-linc 47040  df-lco 47041
This theorem is referenced by:  lindslininds  47098
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