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Theorem lincresunit3 41555
Description: Property 3 of a specially modified restriction of a linear combination in a vector space. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.)
Hypotheses
Ref Expression
lincresunit.b 𝐵 = (Base‘𝑀)
lincresunit.r 𝑅 = (Scalar‘𝑀)
lincresunit.e 𝐸 = (Base‘𝑅)
lincresunit.u 𝑈 = (Unit‘𝑅)
lincresunit.0 0 = (0g𝑅)
lincresunit.z 𝑍 = (0g𝑀)
lincresunit.n 𝑁 = (invg𝑅)
lincresunit.i 𝐼 = (invr𝑅)
lincresunit.t · = (.r𝑅)
lincresunit.g 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))
Assertion
Ref Expression
lincresunit3 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = 𝑋)
Distinct variable groups:   𝐵,𝑠   𝐸,𝑠   𝐹,𝑠   𝑀,𝑠   𝑆,𝑠   𝑋,𝑠   𝑈,𝑠   𝐼,𝑠   𝑁,𝑠   · ,𝑠   0 ,𝑠   𝐺,𝑠   𝑅,𝑠   𝑍,𝑠

Proof of Theorem lincresunit3
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 simp2 1060 . . . 4 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑀 ∈ LMod)
213ad2ant1 1080 . . 3 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → 𝑀 ∈ LMod)
3 simp1 1059 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆))
4 3simpa 1056 . . . . . . . . 9 ((𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) → (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈))
543ad2ant2 1081 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈))
63, 5jca 554 . . . . . . 7 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈)))
7 eldifi 3710 . . . . . . 7 (𝑠 ∈ (𝑆 ∖ {𝑋}) → 𝑠𝑆)
8 lincresunit.b . . . . . . . 8 𝐵 = (Base‘𝑀)
9 lincresunit.r . . . . . . . 8 𝑅 = (Scalar‘𝑀)
10 lincresunit.e . . . . . . . 8 𝐸 = (Base‘𝑅)
11 lincresunit.u . . . . . . . 8 𝑈 = (Unit‘𝑅)
12 lincresunit.0 . . . . . . . 8 0 = (0g𝑅)
13 lincresunit.z . . . . . . . 8 𝑍 = (0g𝑀)
14 lincresunit.n . . . . . . . 8 𝑁 = (invg𝑅)
15 lincresunit.i . . . . . . . 8 𝐼 = (invr𝑅)
16 lincresunit.t . . . . . . . 8 · = (.r𝑅)
17 lincresunit.g . . . . . . . 8 𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))
188, 9, 10, 11, 12, 13, 14, 15, 16, 17lincresunitlem2 41550 . . . . . . 7 ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈)) ∧ 𝑠𝑆) → ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)) ∈ 𝐸)
196, 7, 18syl2an 494 . . . . . 6 ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)) ∈ 𝐸)
209fveq2i 6151 . . . . . . 7 (Base‘𝑅) = (Base‘(Scalar‘𝑀))
2110, 20eqtri 2643 . . . . . 6 𝐸 = (Base‘(Scalar‘𝑀))
2219, 21syl6eleq 2708 . . . . 5 ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)) ∈ (Base‘(Scalar‘𝑀)))
2322, 17fmptd 6340 . . . 4 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → 𝐺:(𝑆 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀)))
24 fvex 6158 . . . . 5 (Base‘(Scalar‘𝑀)) ∈ V
25 difexg 4768 . . . . . . 7 (𝑆 ∈ 𝒫 𝐵 → (𝑆 ∖ {𝑋}) ∈ V)
26253ad2ant1 1080 . . . . . 6 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → (𝑆 ∖ {𝑋}) ∈ V)
27263ad2ant1 1080 . . . . 5 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝑆 ∖ {𝑋}) ∈ V)
28 elmapg 7815 . . . . 5 (((Base‘(Scalar‘𝑀)) ∈ V ∧ (𝑆 ∖ {𝑋}) ∈ V) → (𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 (𝑆 ∖ {𝑋})) ↔ 𝐺:(𝑆 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀))))
2924, 27, 28sylancr 694 . . . 4 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 (𝑆 ∖ {𝑋})) ↔ 𝐺:(𝑆 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀))))
3023, 29mpbird 247 . . 3 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → 𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 (𝑆 ∖ {𝑋})))
31 elpwi 4140 . . . . . . . . . . 11 (𝑆 ∈ 𝒫 (Base‘𝑀) → 𝑆 ⊆ (Base‘𝑀))
32 ssdifss 3719 . . . . . . . . . . . 12 (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))
3332a1i 11 . . . . . . . . . . 11 (𝑋𝑆 → (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀)))
3431, 33syl5com 31 . . . . . . . . . 10 (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑋𝑆 → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀)))
3534impcom 446 . . . . . . . . 9 ((𝑋𝑆𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))
36 difexg 4768 . . . . . . . . . . 11 (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑋}) ∈ V)
3736adantl 482 . . . . . . . . . 10 ((𝑋𝑆𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 ∖ {𝑋}) ∈ V)
38 elpwg 4138 . . . . . . . . . 10 ((𝑆 ∖ {𝑋}) ∈ V → ((𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀)))
3937, 38syl 17 . . . . . . . . 9 ((𝑋𝑆𝑆 ∈ 𝒫 (Base‘𝑀)) → ((𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀)))
4035, 39mpbird 247 . . . . . . . 8 ((𝑋𝑆𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
4140expcom 451 . . . . . . 7 (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑋𝑆 → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)))
428pweqi 4134 . . . . . . 7 𝒫 𝐵 = 𝒫 (Base‘𝑀)
4341, 42eleq2s 2716 . . . . . 6 (𝑆 ∈ 𝒫 𝐵 → (𝑋𝑆 → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)))
4443imp 445 . . . . 5 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
45443adant2 1078 . . . 4 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
46453ad2ant1 1080 . . 3 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
47 lincval 41483 . . 3 ((𝑀 ∈ LMod ∧ 𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 (𝑆 ∖ {𝑋})) ∧ (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))))
482, 30, 46, 47syl3anc 1323 . 2 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))))
49 simp1 1059 . . . . . . . 8 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑆 ∈ 𝒫 𝐵)
50 simp3 1061 . . . . . . . 8 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑋𝑆)
511, 49, 503jca 1240 . . . . . . 7 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵𝑋𝑆))
5251adantr 481 . . . . . 6 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵𝑋𝑆))
53 3simpb 1057 . . . . . . 7 ((𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) → (𝐹 ∈ (𝐸𝑚 𝑆) ∧ 𝐹 finSupp 0 ))
5453adantl 482 . . . . . 6 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝐹 ∈ (𝐸𝑚 𝑆) ∧ 𝐹 finSupp 0 ))
55 eqidd 2622 . . . . . 6 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝐹 ↾ (𝑆 ∖ {𝑋})) = (𝐹 ↾ (𝑆 ∖ {𝑋})))
56 eqid 2621 . . . . . . 7 ( ·𝑠𝑀) = ( ·𝑠𝑀)
57 eqid 2621 . . . . . . 7 (+g𝑀) = (+g𝑀)
588, 9, 10, 56, 57, 12lincdifsn 41498 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ 𝐹 finSupp 0 ) ∧ (𝐹 ↾ (𝑆 ∖ {𝑋})) = (𝐹 ↾ (𝑆 ∖ {𝑋}))) → (𝐹( linC ‘𝑀)𝑆) = (((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋}))(+g𝑀)((𝐹𝑋)( ·𝑠𝑀)𝑋)))
5952, 54, 55, 58syl3anc 1323 . . . . 5 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝐹( linC ‘𝑀)𝑆) = (((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋}))(+g𝑀)((𝐹𝑋)( ·𝑠𝑀)𝑋)))
6059eqeq1d 2623 . . . 4 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → ((𝐹( linC ‘𝑀)𝑆) = 𝑍 ↔ (((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋}))(+g𝑀)((𝐹𝑋)( ·𝑠𝑀)𝑋)) = 𝑍))
61 fveq2 6148 . . . . . . . . . . . . 13 (𝑠 = 𝑧 → (𝐺𝑠) = (𝐺𝑧))
62 id 22 . . . . . . . . . . . . 13 (𝑠 = 𝑧𝑠 = 𝑧)
6361, 62oveq12d 6622 . . . . . . . . . . . 12 (𝑠 = 𝑧 → ((𝐺𝑠)( ·𝑠𝑀)𝑠) = ((𝐺𝑧)( ·𝑠𝑀)𝑧))
6463cbvmptv 4710 . . . . . . . . . . 11 (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)) = (𝑧 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑧)( ·𝑠𝑀)𝑧))
6564a1i 11 . . . . . . . . . 10 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)) = (𝑧 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑧)( ·𝑠𝑀)𝑧)))
6665oveq2d 6620 . . . . . . . . 9 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))) = (𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑧)( ·𝑠𝑀)𝑧))))
6766oveq2d 6620 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)))) = ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑧)( ·𝑠𝑀)𝑧)))))
688, 9, 10, 11, 12, 13, 14, 15, 16, 17lincresunit3lem2 41554 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑧)( ·𝑠𝑀)𝑧)))) = ((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋})))
6967, 68eqtr2d 2656 . . . . . . 7 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → ((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋})) = ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)))))
7069oveq1d 6619 . . . . . 6 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋}))(+g𝑀)((𝐹𝑋)( ·𝑠𝑀)𝑋)) = (((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))))(+g𝑀)((𝐹𝑋)( ·𝑠𝑀)𝑋)))
7170eqeq1d 2623 . . . . 5 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → ((((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋}))(+g𝑀)((𝐹𝑋)( ·𝑠𝑀)𝑋)) = 𝑍 ↔ (((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))))(+g𝑀)((𝐹𝑋)( ·𝑠𝑀)𝑋)) = 𝑍))
72 lmodgrp 18791 . . . . . . . . 9 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
73723ad2ant2 1081 . . . . . . . 8 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑀 ∈ Grp)
7473adantr 481 . . . . . . 7 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝑀 ∈ Grp)
751adantr 481 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝑀 ∈ LMod)
76 elmapi 7823 . . . . . . . . . 10 (𝐹 ∈ (𝐸𝑚 𝑆) → 𝐹:𝑆𝐸)
77763ad2ant1 1080 . . . . . . . . 9 ((𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) → 𝐹:𝑆𝐸)
78 ffvelrn 6313 . . . . . . . . 9 ((𝐹:𝑆𝐸𝑋𝑆) → (𝐹𝑋) ∈ 𝐸)
7977, 50, 78syl2anr 495 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝐹𝑋) ∈ 𝐸)
80 elpwi 4140 . . . . . . . . . . 11 (𝑆 ∈ 𝒫 𝐵𝑆𝐵)
8180sselda 3583 . . . . . . . . . 10 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → 𝑋𝐵)
82813adant2 1078 . . . . . . . . 9 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑋𝐵)
8382adantr 481 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝑋𝐵)
848, 9, 56, 10lmodvscl 18801 . . . . . . . 8 ((𝑀 ∈ LMod ∧ (𝐹𝑋) ∈ 𝐸𝑋𝐵) → ((𝐹𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵)
8575, 79, 83, 84syl3anc 1323 . . . . . . 7 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → ((𝐹𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵)
869lmodfgrp 18793 . . . . . . . . . . 11 (𝑀 ∈ LMod → 𝑅 ∈ Grp)
87863ad2ant2 1081 . . . . . . . . . 10 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑅 ∈ Grp)
8887adantr 481 . . . . . . . . 9 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝑅 ∈ Grp)
8910, 14grpinvcl 17388 . . . . . . . . 9 ((𝑅 ∈ Grp ∧ (𝐹𝑋) ∈ 𝐸) → (𝑁‘(𝐹𝑋)) ∈ 𝐸)
9088, 79, 89syl2anc 692 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑁‘(𝐹𝑋)) ∈ 𝐸)
91 lmodcmn 18832 . . . . . . . . . . 11 (𝑀 ∈ LMod → 𝑀 ∈ CMnd)
92913ad2ant2 1081 . . . . . . . . . 10 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑀 ∈ CMnd)
9392adantr 481 . . . . . . . . 9 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝑀 ∈ CMnd)
9426adantr 481 . . . . . . . . 9 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑆 ∖ {𝑋}) ∈ V)
95 simpll2 1099 . . . . . . . . . . 11 ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → 𝑀 ∈ LMod)
968, 9, 10, 11, 12, 13, 14, 15, 16, 17lincresunit1 41551 . . . . . . . . . . . . . 14 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈)) → 𝐺 ∈ (𝐸𝑚 (𝑆 ∖ {𝑋})))
97963adantr3 1220 . . . . . . . . . . . . 13 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝐺 ∈ (𝐸𝑚 (𝑆 ∖ {𝑋})))
98 elmapi 7823 . . . . . . . . . . . . 13 (𝐺 ∈ (𝐸𝑚 (𝑆 ∖ {𝑋})) → 𝐺:(𝑆 ∖ {𝑋})⟶𝐸)
9997, 98syl 17 . . . . . . . . . . . 12 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝐺:(𝑆 ∖ {𝑋})⟶𝐸)
10099ffvelrnda 6315 . . . . . . . . . . 11 ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → (𝐺𝑠) ∈ 𝐸)
101 ssel2 3578 . . . . . . . . . . . . . . . . 17 ((𝑆𝐵𝑠𝑆) → 𝑠𝐵)
102101expcom 451 . . . . . . . . . . . . . . . 16 (𝑠𝑆 → (𝑆𝐵𝑠𝐵))
1037, 102syl 17 . . . . . . . . . . . . . . 15 (𝑠 ∈ (𝑆 ∖ {𝑋}) → (𝑆𝐵𝑠𝐵))
10480, 103syl5com 31 . . . . . . . . . . . . . 14 (𝑆 ∈ 𝒫 𝐵 → (𝑠 ∈ (𝑆 ∖ {𝑋}) → 𝑠𝐵))
1051043ad2ant1 1080 . . . . . . . . . . . . 13 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → (𝑠 ∈ (𝑆 ∖ {𝑋}) → 𝑠𝐵))
106105adantr 481 . . . . . . . . . . . 12 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) → 𝑠𝐵))
107106imp 445 . . . . . . . . . . 11 ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → 𝑠𝐵)
1088, 9, 56, 10lmodvscl 18801 . . . . . . . . . . 11 ((𝑀 ∈ LMod ∧ (𝐺𝑠) ∈ 𝐸𝑠𝐵) → ((𝐺𝑠)( ·𝑠𝑀)𝑠) ∈ 𝐵)
10995, 100, 107, 108syl3anc 1323 . . . . . . . . . 10 ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → ((𝐺𝑠)( ·𝑠𝑀)𝑠) ∈ 𝐵)
110 eqid 2621 . . . . . . . . . 10 (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)) = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))
111109, 110fmptd 6340 . . . . . . . . 9 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)):(𝑆 ∖ {𝑋})⟶𝐵)
112 ssdifss 3719 . . . . . . . . . . . . . . . . . 18 (𝑆𝐵 → (𝑆 ∖ {𝑋}) ⊆ 𝐵)
11380, 112syl 17 . . . . . . . . . . . . . . . . 17 (𝑆 ∈ 𝒫 𝐵 → (𝑆 ∖ {𝑋}) ⊆ 𝐵)
114113adantr 481 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → (𝑆 ∖ {𝑋}) ⊆ 𝐵)
115114, 8syl6sseq 3630 . . . . . . . . . . . . . . 15 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))
11625adantr 481 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → (𝑆 ∖ {𝑋}) ∈ V)
117116, 38syl 17 . . . . . . . . . . . . . . 15 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → ((𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀)))
118115, 117mpbird 247 . . . . . . . . . . . . . 14 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
1191183adant2 1078 . . . . . . . . . . . . 13 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
1201, 119jca 554 . . . . . . . . . . . 12 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → (𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)))
121120adantr 481 . . . . . . . . . . 11 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)))
1228, 9, 10, 11, 12, 13, 14, 15, 16, 17lincresunit2 41552 . . . . . . . . . . . 12 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝐺 finSupp 0 )
123122, 12syl6breq 4654 . . . . . . . . . . 11 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝐺 finSupp (0g𝑅))
1249, 10scmfsupp 41444 . . . . . . . . . . 11 (((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)) ∧ 𝐺 ∈ (𝐸𝑚 (𝑆 ∖ {𝑋})) ∧ 𝐺 finSupp (0g𝑅)) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)) finSupp (0g𝑀))
125121, 97, 123, 124syl3anc 1323 . . . . . . . . . 10 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)) finSupp (0g𝑀))
126125, 13syl6breqr 4655 . . . . . . . . 9 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)) finSupp 𝑍)
1278, 13, 93, 94, 111, 126gsumcl 18237 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))) ∈ 𝐵)
1288, 9, 56, 10lmodvscl 18801 . . . . . . . 8 ((𝑀 ∈ LMod ∧ (𝑁‘(𝐹𝑋)) ∈ 𝐸 ∧ (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))) ∈ 𝐵) → ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)))) ∈ 𝐵)
12975, 90, 127, 128syl3anc 1323 . . . . . . 7 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)))) ∈ 𝐵)
130 eqid 2621 . . . . . . . 8 (invg𝑀) = (invg𝑀)
1318, 57, 13, 130grpinvid2 17392 . . . . . . 7 ((𝑀 ∈ Grp ∧ ((𝐹𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵 ∧ ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)))) ∈ 𝐵) → (((invg𝑀)‘((𝐹𝑋)( ·𝑠𝑀)𝑋)) = ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)))) ↔ (((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))))(+g𝑀)((𝐹𝑋)( ·𝑠𝑀)𝑋)) = 𝑍))
13274, 85, 129, 131syl3anc 1323 . . . . . 6 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (((invg𝑀)‘((𝐹𝑋)( ·𝑠𝑀)𝑋)) = ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)))) ↔ (((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))))(+g𝑀)((𝐹𝑋)( ·𝑠𝑀)𝑋)) = 𝑍))
1338, 9, 56, 130, 10, 14, 75, 83, 79lmodvsneg 18828 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → ((invg𝑀)‘((𝐹𝑋)( ·𝑠𝑀)𝑋)) = ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)𝑋))
134133eqeq1d 2623 . . . . . . 7 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (((invg𝑀)‘((𝐹𝑋)( ·𝑠𝑀)𝑋)) = ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)))) ↔ ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)𝑋) = ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))))))
135 simpr2 1066 . . . . . . . . 9 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝐹𝑋) ∈ 𝑈)
1368, 9, 10, 11, 14, 56lincresunit3lem3 41548 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑋𝐵 ∧ (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))) ∈ 𝐵) ∧ (𝐹𝑋) ∈ 𝑈) → (((𝑁‘(𝐹𝑋))( ·𝑠𝑀)𝑋) = ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)))) ↔ 𝑋 = (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)))))
137 eqcom 2628 . . . . . . . . . 10 (𝑋 = (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))) ↔ (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))) = 𝑋)
138136, 137syl6bb 276 . . . . . . . . 9 (((𝑀 ∈ LMod ∧ 𝑋𝐵 ∧ (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))) ∈ 𝐵) ∧ (𝐹𝑋) ∈ 𝑈) → (((𝑁‘(𝐹𝑋))( ·𝑠𝑀)𝑋) = ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)))) ↔ (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))) = 𝑋))
13975, 83, 127, 135, 138syl31anc 1326 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (((𝑁‘(𝐹𝑋))( ·𝑠𝑀)𝑋) = ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)))) ↔ (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))) = 𝑋))
140139biimpd 219 . . . . . . 7 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (((𝑁‘(𝐹𝑋))( ·𝑠𝑀)𝑋) = ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)))) → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))) = 𝑋))
141134, 140sylbid 230 . . . . . 6 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (((invg𝑀)‘((𝐹𝑋)( ·𝑠𝑀)𝑋)) = ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)))) → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))) = 𝑋))
142132, 141sylbird 250 . . . . 5 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → ((((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))))(+g𝑀)((𝐹𝑋)( ·𝑠𝑀)𝑋)) = 𝑍 → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))) = 𝑋))
14371, 142sylbid 230 . . . 4 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → ((((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋}))(+g𝑀)((𝐹𝑋)( ·𝑠𝑀)𝑋)) = 𝑍 → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))) = 𝑋))
14460, 143sylbid 230 . . 3 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → ((𝐹( linC ‘𝑀)𝑆) = 𝑍 → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))) = 𝑋))
1451443impia 1258 . 2 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))) = 𝑋)
14648, 145eqtrd 2655 1 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3186  cdif 3552  wss 3555  𝒫 cpw 4130  {csn 4148   class class class wbr 4613  cmpt 4673  cres 5076  wf 5843  cfv 5847  (class class class)co 6604  𝑚 cmap 7802   finSupp cfsupp 8219  Basecbs 15781  +gcplusg 15862  .rcmulr 15863  Scalarcsca 15865   ·𝑠 cvsca 15866  0gc0g 16021   Σg cgsu 16022  Grpcgrp 17343  invgcminusg 17344  CMndccmn 18114  Unitcui 18560  invrcinvr 18592  LModclmod 18784   linC clinc 41478
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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  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 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-tpos 7297  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-oi 8359  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-seq 12742  df-hash 13058  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-0g 16023  df-gsum 16024  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-submnd 17257  df-grp 17346  df-minusg 17347  df-mulg 17462  df-ghm 17579  df-cntz 17671  df-cmn 18116  df-abl 18117  df-mgp 18411  df-ur 18423  df-ring 18470  df-oppr 18544  df-dvdsr 18562  df-unit 18563  df-invr 18593  df-lmod 18786  df-linc 41480
This theorem is referenced by:  lincreslvec3  41556
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