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Theorem lincresunit3 42939
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 1167 . . . 4 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑀 ∈ LMod)
213ad2ant1 1163 . . 3 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → 𝑀 ∈ LMod)
3 simp1 1166 . . . . . . . 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 1178 . . . . . . . . 9 ((𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) → (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈))
543ad2ant2 1164 . . . . . . . 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 507 . . . . . . 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 3894 . . . . . . 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 42934 . . . . . . 7 ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈)) ∧ 𝑠𝑆) → ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)) ∈ 𝐸)
196, 7, 18syl2an 589 . . . . . 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 6378 . . . . . . 7 (Base‘𝑅) = (Base‘(Scalar‘𝑀))
2110, 20eqtri 2787 . . . . . 6 𝐸 = (Base‘(Scalar‘𝑀))
2219, 21syl6eleq 2854 . . . . 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 6574 . . . 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 6388 . . . . 5 (Base‘(Scalar‘𝑀)) ∈ V
25 difexg 4969 . . . . . . 7 (𝑆 ∈ 𝒫 𝐵 → (𝑆 ∖ {𝑋}) ∈ V)
26253ad2ant1 1163 . . . . . 6 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → (𝑆 ∖ {𝑋}) ∈ V)
27263ad2ant1 1163 . . . . 5 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝑆 ∖ {𝑋}) ∈ V)
28 elmapg 8073 . . . . 5 (((Base‘(Scalar‘𝑀)) ∈ V ∧ (𝑆 ∖ {𝑋}) ∈ V) → (𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 (𝑆 ∖ {𝑋})) ↔ 𝐺:(𝑆 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀))))
2924, 27, 28sylancr 581 . . . 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 248 . . 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 4325 . . . . . . . . . . 11 (𝑆 ∈ 𝒫 (Base‘𝑀) → 𝑆 ⊆ (Base‘𝑀))
32 ssdifss 3903 . . . . . . . . . . . 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 396 . . . . . . . . 9 ((𝑋𝑆𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))
36 difexg 4969 . . . . . . . . . . 11 (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑋}) ∈ V)
3736adantl 473 . . . . . . . . . 10 ((𝑋𝑆𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 ∖ {𝑋}) ∈ V)
38 elpwg 4323 . . . . . . . . . 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 248 . . . . . . . 8 ((𝑋𝑆𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
4140expcom 402 . . . . . . 7 (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑋𝑆 → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)))
428pweqi 4319 . . . . . . 7 𝒫 𝐵 = 𝒫 (Base‘𝑀)
4341, 42eleq2s 2862 . . . . . 6 (𝑆 ∈ 𝒫 𝐵 → (𝑋𝑆 → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)))
4443imp 395 . . . . 5 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
45443adant2 1161 . . . 4 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
46453ad2ant1 1163 . . 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 42867 . . 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 1490 . 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 1166 . . . . . . . 8 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑆 ∈ 𝒫 𝐵)
50 simp3 1168 . . . . . . . 8 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑋𝑆)
511, 49, 503jca 1158 . . . . . . 7 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵𝑋𝑆))
5251adantr 472 . . . . . 6 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵𝑋𝑆))
53 3simpb 1180 . . . . . . 7 ((𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) → (𝐹 ∈ (𝐸𝑚 𝑆) ∧ 𝐹 finSupp 0 ))
5453adantl 473 . . . . . 6 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝐹 ∈ (𝐸𝑚 𝑆) ∧ 𝐹 finSupp 0 ))
55 eqidd 2766 . . . . . 6 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝐹 ↾ (𝑆 ∖ {𝑋})) = (𝐹 ↾ (𝑆 ∖ {𝑋})))
56 eqid 2765 . . . . . . 7 ( ·𝑠𝑀) = ( ·𝑠𝑀)
57 eqid 2765 . . . . . . 7 (+g𝑀) = (+g𝑀)
588, 9, 10, 56, 57, 12lincdifsn 42882 . . . . . 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 1490 . . . . 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 2767 . . . 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 6375 . . . . . . . . . . . . 13 (𝑠 = 𝑧 → (𝐺𝑠) = (𝐺𝑧))
62 id 22 . . . . . . . . . . . . 13 (𝑠 = 𝑧𝑠 = 𝑧)
6361, 62oveq12d 6860 . . . . . . . . . . . 12 (𝑠 = 𝑧 → ((𝐺𝑠)( ·𝑠𝑀)𝑠) = ((𝐺𝑧)( ·𝑠𝑀)𝑧))
6463cbvmptv 4909 . . . . . . . . . . 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 6858 . . . . . . . . 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 6858 . . . . . . . 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 42938 . . . . . . . 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 2800 . . . . . . 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 6857 . . . . . 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 2767 . . . . 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 19139 . . . . . . . . 9 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
73723ad2ant2 1164 . . . . . . . 8 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑀 ∈ Grp)
7473adantr 472 . . . . . . 7 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝑀 ∈ Grp)
751adantr 472 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝑀 ∈ LMod)
76 elmapi 8082 . . . . . . . . . 10 (𝐹 ∈ (𝐸𝑚 𝑆) → 𝐹:𝑆𝐸)
77763ad2ant1 1163 . . . . . . . . 9 ((𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) → 𝐹:𝑆𝐸)
78 ffvelrn 6547 . . . . . . . . 9 ((𝐹:𝑆𝐸𝑋𝑆) → (𝐹𝑋) ∈ 𝐸)
7977, 50, 78syl2anr 590 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝐹𝑋) ∈ 𝐸)
80 elpwi 4325 . . . . . . . . . . 11 (𝑆 ∈ 𝒫 𝐵𝑆𝐵)
8180sselda 3761 . . . . . . . . . 10 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → 𝑋𝐵)
82813adant2 1161 . . . . . . . . 9 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑋𝐵)
8382adantr 472 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝑋𝐵)
848, 9, 56, 10lmodvscl 19149 . . . . . . . 8 ((𝑀 ∈ LMod ∧ (𝐹𝑋) ∈ 𝐸𝑋𝐵) → ((𝐹𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵)
8575, 79, 83, 84syl3anc 1490 . . . . . . 7 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → ((𝐹𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵)
869lmodfgrp 19141 . . . . . . . . . . 11 (𝑀 ∈ LMod → 𝑅 ∈ Grp)
87863ad2ant2 1164 . . . . . . . . . 10 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑅 ∈ Grp)
8887adantr 472 . . . . . . . . 9 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝑅 ∈ Grp)
8910, 14grpinvcl 17734 . . . . . . . . 9 ((𝑅 ∈ Grp ∧ (𝐹𝑋) ∈ 𝐸) → (𝑁‘(𝐹𝑋)) ∈ 𝐸)
9088, 79, 89syl2anc 579 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑁‘(𝐹𝑋)) ∈ 𝐸)
91 lmodcmn 19180 . . . . . . . . . . 11 (𝑀 ∈ LMod → 𝑀 ∈ CMnd)
92913ad2ant2 1164 . . . . . . . . . 10 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑀 ∈ CMnd)
9392adantr 472 . . . . . . . . 9 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝑀 ∈ CMnd)
9426adantr 472 . . . . . . . . 9 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑆 ∖ {𝑋}) ∈ V)
95 simpll2 1271 . . . . . . . . . . 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 42935 . . . . . . . . . . . . . 14 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈)) → 𝐺 ∈ (𝐸𝑚 (𝑆 ∖ {𝑋})))
97963adantr3 1212 . . . . . . . . . . . . 13 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝐺 ∈ (𝐸𝑚 (𝑆 ∖ {𝑋})))
98 elmapi 8082 . . . . . . . . . . . . 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 6549 . . . . . . . . . . 11 ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → (𝐺𝑠) ∈ 𝐸)
101 ssel2 3756 . . . . . . . . . . . . . . . . 17 ((𝑆𝐵𝑠𝑆) → 𝑠𝐵)
102101expcom 402 . . . . . . . . . . . . . . . 16 (𝑠𝑆 → (𝑆𝐵𝑠𝐵))
1037, 102syl 17 . . . . . . . . . . . . . . 15 (𝑠 ∈ (𝑆 ∖ {𝑋}) → (𝑆𝐵𝑠𝐵))
10480, 103syl5com 31 . . . . . . . . . . . . . 14 (𝑆 ∈ 𝒫 𝐵 → (𝑠 ∈ (𝑆 ∖ {𝑋}) → 𝑠𝐵))
1051043ad2ant1 1163 . . . . . . . . . . . . 13 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → (𝑠 ∈ (𝑆 ∖ {𝑋}) → 𝑠𝐵))
106105adantr 472 . . . . . . . . . . . 12 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) → 𝑠𝐵))
107106imp 395 . . . . . . . . . . 11 ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → 𝑠𝐵)
1088, 9, 56, 10lmodvscl 19149 . . . . . . . . . . 11 ((𝑀 ∈ LMod ∧ (𝐺𝑠) ∈ 𝐸𝑠𝐵) → ((𝐺𝑠)( ·𝑠𝑀)𝑠) ∈ 𝐵)
10995, 100, 107, 108syl3anc 1490 . . . . . . . . . 10 ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → ((𝐺𝑠)( ·𝑠𝑀)𝑠) ∈ 𝐵)
110109fmpttd 6575 . . . . . . . . 9 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)):(𝑆 ∖ {𝑋})⟶𝐵)
111 ssdifss 3903 . . . . . . . . . . . . . . . . . 18 (𝑆𝐵 → (𝑆 ∖ {𝑋}) ⊆ 𝐵)
11280, 111syl 17 . . . . . . . . . . . . . . . . 17 (𝑆 ∈ 𝒫 𝐵 → (𝑆 ∖ {𝑋}) ⊆ 𝐵)
113112adantr 472 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → (𝑆 ∖ {𝑋}) ⊆ 𝐵)
114113, 8syl6sseq 3811 . . . . . . . . . . . . . . 15 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))
11525adantr 472 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → (𝑆 ∖ {𝑋}) ∈ V)
116115, 38syl 17 . . . . . . . . . . . . . . 15 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → ((𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀)))
117114, 116mpbird 248 . . . . . . . . . . . . . 14 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
1181173adant2 1161 . . . . . . . . . . . . 13 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
1191, 118jca 507 . . . . . . . . . . . 12 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → (𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)))
120119adantr 472 . . . . . . . . . . 11 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)))
1218, 9, 10, 11, 12, 13, 14, 15, 16, 17lincresunit2 42936 . . . . . . . . . . . 12 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝐺 finSupp 0 )
122121, 12syl6breq 4850 . . . . . . . . . . 11 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝐺 finSupp (0g𝑅))
1239, 10scmfsupp 42828 . . . . . . . . . . 11 (((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)) ∧ 𝐺 ∈ (𝐸𝑚 (𝑆 ∖ {𝑋})) ∧ 𝐺 finSupp (0g𝑅)) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)) finSupp (0g𝑀))
124120, 97, 122, 123syl3anc 1490 . . . . . . . . . 10 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)) finSupp (0g𝑀))
125124, 13syl6breqr 4851 . . . . . . . . 9 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)) finSupp 𝑍)
1268, 13, 93, 94, 110, 125gsumcl 18582 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))) ∈ 𝐵)
1278, 9, 56, 10lmodvscl 19149 . . . . . . . 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)
12875, 90, 126, 127syl3anc 1490 . . . . . . 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)
129 eqid 2765 . . . . . . . 8 (invg𝑀) = (invg𝑀)
1308, 57, 13, 129grpinvid2 17738 . . . . . . 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))
13174, 85, 128, 130syl3anc 1490 . . . . . 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))
1328, 9, 56, 129, 10, 14, 75, 83, 79lmodvsneg 19176 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → ((invg𝑀)‘((𝐹𝑋)( ·𝑠𝑀)𝑋)) = ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)𝑋))
133132eqeq1d 2767 . . . . . . 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))))))
134 simpr2 1250 . . . . . . . . 9 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝐹𝑋) ∈ 𝑈)
1358, 9, 10, 11, 14, 56lincresunit3lem3 42932 . . . . . . . . . 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)))))
136 eqcom 2772 . . . . . . . . . 10 (𝑋 = (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))) ↔ (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))) = 𝑋)
137135, 136syl6bb 278 . . . . . . . . 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))
13875, 83, 126, 134, 137syl31anc 1492 . . . . . . . 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))
139138biimpd 220 . . . . . . 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))
140133, 139sylbid 231 . . . . . 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))
141131, 140sylbird 251 . . . . 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))
14271, 141sylbid 231 . . . 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))
14360, 142sylbid 231 . . 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))
1441433impia 1145 . 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)
14548, 144eqtrd 2799 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  Vcvv 3350  cdif 3729  wss 3732  𝒫 cpw 4315  {csn 4334   class class class wbr 4809  cmpt 4888  cres 5279  wf 6064  cfv 6068  (class class class)co 6842  𝑚 cmap 8060   finSupp cfsupp 8482  Basecbs 16130  +gcplusg 16214  .rcmulr 16215  Scalarcsca 16217   ·𝑠 cvsca 16218  0gc0g 16366   Σg cgsu 16367  Grpcgrp 17689  invgcminusg 17690  CMndccmn 18459  Unitcui 18906  invrcinvr 18938  LModclmod 19132   linC clinc 42862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-tpos 7555  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-ndx 16133  df-slot 16134  df-base 16136  df-sets 16137  df-ress 16138  df-plusg 16227  df-mulr 16228  df-0g 16368  df-gsum 16369  df-mre 16512  df-mrc 16513  df-acs 16515  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-mhm 17601  df-submnd 17602  df-grp 17692  df-minusg 17693  df-mulg 17808  df-ghm 17922  df-cntz 18013  df-cmn 18461  df-abl 18462  df-mgp 18757  df-ur 18769  df-ring 18816  df-oppr 18890  df-dvdsr 18908  df-unit 18909  df-invr 18939  df-lmod 19134  df-linc 42864
This theorem is referenced by:  lincreslvec3  42940
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