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Theorem lincresunit3 42778
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 1132 . . . 4 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑀 ∈ LMod)
213ad2ant1 1128 . . 3 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → 𝑀 ∈ LMod)
3 simp1 1131 . . . . . . . 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 1143 . . . . . . . . 9 ((𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) → (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈))
543ad2ant2 1129 . . . . . . . 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 555 . . . . . . 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 3873 . . . . . . 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 42773 . . . . . . 7 ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈)) ∧ 𝑠𝑆) → ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)) ∈ 𝐸)
196, 7, 18syl2an 495 . . . . . 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 6353 . . . . . . 7 (Base‘𝑅) = (Base‘(Scalar‘𝑀))
2110, 20eqtri 2780 . . . . . 6 𝐸 = (Base‘(Scalar‘𝑀))
2219, 21syl6eleq 2847 . . . . 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 6546 . . . 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 6360 . . . . 5 (Base‘(Scalar‘𝑀)) ∈ V
25 difexg 4958 . . . . . . 7 (𝑆 ∈ 𝒫 𝐵 → (𝑆 ∖ {𝑋}) ∈ V)
26253ad2ant1 1128 . . . . . 6 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → (𝑆 ∖ {𝑋}) ∈ V)
27263ad2ant1 1128 . . . . 5 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝑆 ∖ {𝑋}) ∈ V)
28 elmapg 8034 . . . . 5 (((Base‘(Scalar‘𝑀)) ∈ V ∧ (𝑆 ∖ {𝑋}) ∈ V) → (𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 (𝑆 ∖ {𝑋})) ↔ 𝐺:(𝑆 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀))))
2924, 27, 28sylancr 698 . . . 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 4310 . . . . . . . . . . 11 (𝑆 ∈ 𝒫 (Base‘𝑀) → 𝑆 ⊆ (Base‘𝑀))
32 ssdifss 3882 . . . . . . . . . . . 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 445 . . . . . . . . 9 ((𝑋𝑆𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))
36 difexg 4958 . . . . . . . . . . 11 (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑋}) ∈ V)
3736adantl 473 . . . . . . . . . 10 ((𝑋𝑆𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 ∖ {𝑋}) ∈ V)
38 elpwg 4308 . . . . . . . . . 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 450 . . . . . . 7 (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑋𝑆 → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)))
428pweqi 4304 . . . . . . 7 𝒫 𝐵 = 𝒫 (Base‘𝑀)
4341, 42eleq2s 2855 . . . . . 6 (𝑆 ∈ 𝒫 𝐵 → (𝑋𝑆 → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)))
4443imp 444 . . . . 5 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
45443adant2 1126 . . . 4 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
46453ad2ant1 1128 . . 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 42706 . . 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 1477 . 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 1131 . . . . . . . 8 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑆 ∈ 𝒫 𝐵)
50 simp3 1133 . . . . . . . 8 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑋𝑆)
511, 49, 503jca 1123 . . . . . . 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 1145 . . . . . . 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 2759 . . . . . 6 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝐹 ↾ (𝑆 ∖ {𝑋})) = (𝐹 ↾ (𝑆 ∖ {𝑋})))
56 eqid 2758 . . . . . . 7 ( ·𝑠𝑀) = ( ·𝑠𝑀)
57 eqid 2758 . . . . . . 7 (+g𝑀) = (+g𝑀)
588, 9, 10, 56, 57, 12lincdifsn 42721 . . . . . 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 1477 . . . . 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 2760 . . . 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 6350 . . . . . . . . . . . . 13 (𝑠 = 𝑧 → (𝐺𝑠) = (𝐺𝑧))
62 id 22 . . . . . . . . . . . . 13 (𝑠 = 𝑧𝑠 = 𝑧)
6361, 62oveq12d 6829 . . . . . . . . . . . 12 (𝑠 = 𝑧 → ((𝐺𝑠)( ·𝑠𝑀)𝑠) = ((𝐺𝑧)( ·𝑠𝑀)𝑧))
6463cbvmptv 4900 . . . . . . . . . . 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 6827 . . . . . . . . 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 6827 . . . . . . . 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 42777 . . . . . . . 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 2793 . . . . . . 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 6826 . . . . . 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 2760 . . . . 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 19070 . . . . . . . . 9 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
73723ad2ant2 1129 . . . . . . . 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 8043 . . . . . . . . . 10 (𝐹 ∈ (𝐸𝑚 𝑆) → 𝐹:𝑆𝐸)
77763ad2ant1 1128 . . . . . . . . 9 ((𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) → 𝐹:𝑆𝐸)
78 ffvelrn 6518 . . . . . . . . 9 ((𝐹:𝑆𝐸𝑋𝑆) → (𝐹𝑋) ∈ 𝐸)
7977, 50, 78syl2anr 496 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝐹𝑋) ∈ 𝐸)
80 elpwi 4310 . . . . . . . . . . 11 (𝑆 ∈ 𝒫 𝐵𝑆𝐵)
8180sselda 3742 . . . . . . . . . 10 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → 𝑋𝐵)
82813adant2 1126 . . . . . . . . 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 19080 . . . . . . . 8 ((𝑀 ∈ LMod ∧ (𝐹𝑋) ∈ 𝐸𝑋𝐵) → ((𝐹𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵)
8575, 79, 83, 84syl3anc 1477 . . . . . . 7 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → ((𝐹𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵)
869lmodfgrp 19072 . . . . . . . . . . 11 (𝑀 ∈ LMod → 𝑅 ∈ Grp)
87863ad2ant2 1129 . . . . . . . . . 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 17666 . . . . . . . . 9 ((𝑅 ∈ Grp ∧ (𝐹𝑋) ∈ 𝐸) → (𝑁‘(𝐹𝑋)) ∈ 𝐸)
9088, 79, 89syl2anc 696 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑁‘(𝐹𝑋)) ∈ 𝐸)
91 lmodcmn 19111 . . . . . . . . . . 11 (𝑀 ∈ LMod → 𝑀 ∈ CMnd)
92913ad2ant2 1129 . . . . . . . . . 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 1257 . . . . . . . . . . 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 42774 . . . . . . . . . . . . . 14 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈)) → 𝐺 ∈ (𝐸𝑚 (𝑆 ∖ {𝑋})))
97963adantr3 1177 . . . . . . . . . . . . 13 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝐺 ∈ (𝐸𝑚 (𝑆 ∖ {𝑋})))
98 elmapi 8043 . . . . . . . . . . . . 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 6520 . . . . . . . . . . 11 ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → (𝐺𝑠) ∈ 𝐸)
101 ssel2 3737 . . . . . . . . . . . . . . . . 17 ((𝑆𝐵𝑠𝑆) → 𝑠𝐵)
102101expcom 450 . . . . . . . . . . . . . . . 16 (𝑠𝑆 → (𝑆𝐵𝑠𝐵))
1037, 102syl 17 . . . . . . . . . . . . . . 15 (𝑠 ∈ (𝑆 ∖ {𝑋}) → (𝑆𝐵𝑠𝐵))
10480, 103syl5com 31 . . . . . . . . . . . . . 14 (𝑆 ∈ 𝒫 𝐵 → (𝑠 ∈ (𝑆 ∖ {𝑋}) → 𝑠𝐵))
1051043ad2ant1 1128 . . . . . . . . . . . . 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 444 . . . . . . . . . . 11 ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → 𝑠𝐵)
1088, 9, 56, 10lmodvscl 19080 . . . . . . . . . . 11 ((𝑀 ∈ LMod ∧ (𝐺𝑠) ∈ 𝐸𝑠𝐵) → ((𝐺𝑠)( ·𝑠𝑀)𝑠) ∈ 𝐵)
10995, 100, 107, 108syl3anc 1477 . . . . . . . . . 10 ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → ((𝐺𝑠)( ·𝑠𝑀)𝑠) ∈ 𝐵)
110 eqid 2758 . . . . . . . . . 10 (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)) = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))
111109, 110fmptd 6546 . . . . . . . . 9 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)):(𝑆 ∖ {𝑋})⟶𝐵)
112 ssdifss 3882 . . . . . . . . . . . . . . . . . 18 (𝑆𝐵 → (𝑆 ∖ {𝑋}) ⊆ 𝐵)
11380, 112syl 17 . . . . . . . . . . . . . . . . 17 (𝑆 ∈ 𝒫 𝐵 → (𝑆 ∖ {𝑋}) ⊆ 𝐵)
114113adantr 472 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → (𝑆 ∖ {𝑋}) ⊆ 𝐵)
115114, 8syl6sseq 3790 . . . . . . . . . . . . . . 15 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))
11625adantr 472 . . . . . . . . . . . . . . . 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 1126 . . . . . . . . . . . . 13 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
1201, 119jca 555 . . . . . . . . . . . 12 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → (𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)))
121120adantr 472 . . . . . . . . . . 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 42775 . . . . . . . . . . . 12 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝐺 finSupp 0 )
123122, 12syl6breq 4843 . . . . . . . . . . 11 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝐺 finSupp (0g𝑅))
1249, 10scmfsupp 42667 . . . . . . . . . . 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 1477 . . . . . . . . . 10 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)) finSupp (0g𝑀))
126125, 13syl6breqr 4844 . . . . . . . . 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 18514 . . . . . . . 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 19080 . . . . . . . 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 1477 . . . . . . 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 2758 . . . . . . . 8 (invg𝑀) = (invg𝑀)
1318, 57, 13, 130grpinvid2 17670 . . . . . . 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 1477 . . . . . 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 19107 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → ((invg𝑀)‘((𝐹𝑋)( ·𝑠𝑀)𝑋)) = ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)𝑋))
134133eqeq1d 2760 . . . . . . 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 1236 . . . . . . . . 9 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝐹𝑋) ∈ 𝑈)
1368, 9, 10, 11, 14, 56lincresunit3lem3 42771 . . . . . . . . . 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 2765 . . . . . . . . . 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 1480 . . . . . . . 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 1110 . 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 2792 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 383  w3a 1072   = wceq 1630  wcel 2137  Vcvv 3338  cdif 3710  wss 3713  𝒫 cpw 4300  {csn 4319   class class class wbr 4802  cmpt 4879  cres 5266  wf 6043  cfv 6047  (class class class)co 6811  𝑚 cmap 8021   finSupp cfsupp 8438  Basecbs 16057  +gcplusg 16141  .rcmulr 16142  Scalarcsca 16144   ·𝑠 cvsca 16145  0gc0g 16300   Σg cgsu 16301  Grpcgrp 17621  invgcminusg 17622  CMndccmn 18391  Unitcui 18837  invrcinvr 18869  LModclmod 19063   linC clinc 42701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-inf2 8709  ax-cnex 10182  ax-resscn 10183  ax-1cn 10184  ax-icn 10185  ax-addcl 10186  ax-addrcl 10187  ax-mulcl 10188  ax-mulrcl 10189  ax-mulcom 10190  ax-addass 10191  ax-mulass 10192  ax-distr 10193  ax-i2m1 10194  ax-1ne0 10195  ax-1rid 10196  ax-rnegex 10197  ax-rrecex 10198  ax-cnre 10199  ax-pre-lttri 10200  ax-pre-lttrn 10201  ax-pre-ltadd 10202  ax-pre-mulgt0 10203
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-iin 4673  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-se 5224  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-isom 6056  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-of 7060  df-om 7229  df-1st 7331  df-2nd 7332  df-supp 7462  df-tpos 7519  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-1o 7727  df-oadd 7731  df-er 7909  df-map 8023  df-en 8120  df-dom 8121  df-sdom 8122  df-fin 8123  df-fsupp 8439  df-oi 8578  df-card 8953  df-pnf 10266  df-mnf 10267  df-xr 10268  df-ltxr 10269  df-le 10270  df-sub 10458  df-neg 10459  df-nn 11211  df-2 11269  df-3 11270  df-n0 11483  df-z 11568  df-uz 11878  df-fz 12518  df-fzo 12658  df-seq 12994  df-hash 13310  df-ndx 16060  df-slot 16061  df-base 16063  df-sets 16064  df-ress 16065  df-plusg 16154  df-mulr 16155  df-0g 16302  df-gsum 16303  df-mre 16446  df-mrc 16447  df-acs 16449  df-mgm 17441  df-sgrp 17483  df-mnd 17494  df-mhm 17534  df-submnd 17535  df-grp 17624  df-minusg 17625  df-mulg 17740  df-ghm 17857  df-cntz 17948  df-cmn 18393  df-abl 18394  df-mgp 18688  df-ur 18700  df-ring 18747  df-oppr 18821  df-dvdsr 18839  df-unit 18840  df-invr 18870  df-lmod 19065  df-linc 42703
This theorem is referenced by:  lincreslvec3  42779
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