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Theorem lincresunit3 42780
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 3875 . . . . . . 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 42775 . . . . . . 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 6355 . . . . . . 7 (Base‘𝑅) = (Base‘(Scalar‘𝑀))
2110, 20eqtri 2782 . . . . . 6 𝐸 = (Base‘(Scalar‘𝑀))
2219, 21syl6eleq 2849 . . . . 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 6548 . . . 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 6362 . . . . 5 (Base‘(Scalar‘𝑀)) ∈ V
25 difexg 4960 . . . . . . 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 8036 . . . . 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 4312 . . . . . . . . . . 11 (𝑆 ∈ 𝒫 (Base‘𝑀) → 𝑆 ⊆ (Base‘𝑀))
32 ssdifss 3884 . . . . . . . . . . . 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 4960 . . . . . . . . . . 11 (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑋}) ∈ V)
3736adantl 473 . . . . . . . . . 10 ((𝑋𝑆𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 ∖ {𝑋}) ∈ V)
38 elpwg 4310 . . . . . . . . . 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 4306 . . . . . . 7 𝒫 𝐵 = 𝒫 (Base‘𝑀)
4341, 42eleq2s 2857 . . . . . 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 42708 . . 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 2761 . . . . . 6 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝐹 ↾ (𝑆 ∖ {𝑋})) = (𝐹 ↾ (𝑆 ∖ {𝑋})))
56 eqid 2760 . . . . . . 7 ( ·𝑠𝑀) = ( ·𝑠𝑀)
57 eqid 2760 . . . . . . 7 (+g𝑀) = (+g𝑀)
588, 9, 10, 56, 57, 12lincdifsn 42723 . . . . . 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 2762 . . . 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 6352 . . . . . . . . . . . . 13 (𝑠 = 𝑧 → (𝐺𝑠) = (𝐺𝑧))
62 id 22 . . . . . . . . . . . . 13 (𝑠 = 𝑧𝑠 = 𝑧)
6361, 62oveq12d 6831 . . . . . . . . . . . 12 (𝑠 = 𝑧 → ((𝐺𝑠)( ·𝑠𝑀)𝑠) = ((𝐺𝑧)( ·𝑠𝑀)𝑧))
6463cbvmptv 4902 . . . . . . . . . . 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 6829 . . . . . . . . 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 6829 . . . . . . . 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 42779 . . . . . . . 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 2795 . . . . . . 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 6828 . . . . . 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 2762 . . . . 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 19072 . . . . . . . . 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 8045 . . . . . . . . . 10 (𝐹 ∈ (𝐸𝑚 𝑆) → 𝐹:𝑆𝐸)
77763ad2ant1 1128 . . . . . . . . 9 ((𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) → 𝐹:𝑆𝐸)
78 ffvelrn 6520 . . . . . . . . 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 4312 . . . . . . . . . . 11 (𝑆 ∈ 𝒫 𝐵𝑆𝐵)
8180sselda 3744 . . . . . . . . . 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 19082 . . . . . . . 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 19074 . . . . . . . . . . 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 17668 . . . . . . . . 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 19113 . . . . . . . . . . 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 42776 . . . . . . . . . . . . . 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 8045 . . . . . . . . . . . . 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 6522 . . . . . . . . . . 11 ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → (𝐺𝑠) ∈ 𝐸)
101 ssel2 3739 . . . . . . . . . . . . . . . . 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 19082 . . . . . . . . . . 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 2760 . . . . . . . . . 10 (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)) = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))
111109, 110fmptd 6548 . . . . . . . . 9 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)):(𝑆 ∖ {𝑋})⟶𝐵)
112 ssdifss 3884 . . . . . . . . . . . . . . . . . 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 3792 . . . . . . . . . . . . . . 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 42777 . . . . . . . . . . . 12 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝐺 finSupp 0 )
123122, 12syl6breq 4845 . . . . . . . . . . 11 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝐺 finSupp (0g𝑅))
1249, 10scmfsupp 42669 . . . . . . . . . . 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 4846 . . . . . . . . 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 18516 . . . . . . . 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 19082 . . . . . . . 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 2760 . . . . . . . 8 (invg𝑀) = (invg𝑀)
1318, 57, 13, 130grpinvid2 17672 . . . . . . 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 19109 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → ((invg𝑀)‘((𝐹𝑋)( ·𝑠𝑀)𝑋)) = ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)𝑋))
134133eqeq1d 2762 . . . . . . 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 42773 . . . . . . . . . 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 2767 . . . . . . . . . 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 2794 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 1632  wcel 2139  Vcvv 3340  cdif 3712  wss 3715  𝒫 cpw 4302  {csn 4321   class class class wbr 4804  cmpt 4881  cres 5268  wf 6045  cfv 6049  (class class class)co 6813  𝑚 cmap 8023   finSupp cfsupp 8440  Basecbs 16059  +gcplusg 16143  .rcmulr 16144  Scalarcsca 16146   ·𝑠 cvsca 16147  0gc0g 16302   Σg cgsu 16303  Grpcgrp 17623  invgcminusg 17624  CMndccmn 18393  Unitcui 18839  invrcinvr 18871  LModclmod 19065   linC clinc 42703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-supp 7464  df-tpos 7521  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fsupp 8441  df-oi 8580  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-z 11570  df-uz 11880  df-fz 12520  df-fzo 12660  df-seq 12996  df-hash 13312  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-mulr 16157  df-0g 16304  df-gsum 16305  df-mre 16448  df-mrc 16449  df-acs 16451  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-mhm 17536  df-submnd 17537  df-grp 17626  df-minusg 17627  df-mulg 17742  df-ghm 17859  df-cntz 17950  df-cmn 18395  df-abl 18396  df-mgp 18690  df-ur 18702  df-ring 18749  df-oppr 18823  df-dvdsr 18841  df-unit 18842  df-invr 18872  df-lmod 19067  df-linc 42705
This theorem is referenced by:  lincreslvec3  42781
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