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Theorem lincresunit3 46552
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 1137 . . . 4 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑀 ∈ LMod)
213ad2ant1 1133 . . 3 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → 𝑀 ∈ LMod)
3 simp1 1136 . . . . . . . 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 1148 . . . . . . . . 9 ((𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) → (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈))
543ad2ant2 1134 . . . . . . . 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 512 . . . . . . 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 4086 . . . . . . 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 46547 . . . . . . 7 ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈)) ∧ 𝑠𝑆) → ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)) ∈ 𝐸)
196, 7, 18syl2an 596 . . . . . 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 6845 . . . . . . 7 (Base‘𝑅) = (Base‘(Scalar‘𝑀))
2110, 20eqtri 2764 . . . . . 6 𝐸 = (Base‘(Scalar‘𝑀))
2219, 21eleqtrdi 2848 . . . . 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 7062 . . . 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 6855 . . . . 5 (Base‘(Scalar‘𝑀)) ∈ V
25 difexg 5284 . . . . . . 7 (𝑆 ∈ 𝒫 𝐵 → (𝑆 ∖ {𝑋}) ∈ V)
26253ad2ant1 1133 . . . . . 6 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → (𝑆 ∖ {𝑋}) ∈ V)
27263ad2ant1 1133 . . . . 5 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝑆 ∖ {𝑋}) ∈ V)
28 elmapg 8778 . . . . 5 (((Base‘(Scalar‘𝑀)) ∈ V ∧ (𝑆 ∖ {𝑋}) ∈ V) → (𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑m (𝑆 ∖ {𝑋})) ↔ 𝐺:(𝑆 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀))))
2924, 27, 28sylancr 587 . . . 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 256 . . 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 difexg 5284 . . . . . . . . . 10 (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑆 ∖ {𝑋}) ∈ V)
3231adantl 482 . . . . . . . . 9 ((𝑋𝑆𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 ∖ {𝑋}) ∈ V)
33 ssdifss 4095 . . . . . . . . . . 11 (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))
3433a1i 11 . . . . . . . . . 10 (𝑋𝑆 → (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀)))
35 elpwi 4567 . . . . . . . . . 10 (𝑆 ∈ 𝒫 (Base‘𝑀) → 𝑆 ⊆ (Base‘𝑀))
3634, 35impel 506 . . . . . . . . 9 ((𝑋𝑆𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))
3732, 36elpwd 4566 . . . . . . . 8 ((𝑋𝑆𝑆 ∈ 𝒫 (Base‘𝑀)) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
3837expcom 414 . . . . . . 7 (𝑆 ∈ 𝒫 (Base‘𝑀) → (𝑋𝑆 → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)))
398pweqi 4576 . . . . . . 7 𝒫 𝐵 = 𝒫 (Base‘𝑀)
4038, 39eleq2s 2856 . . . . . 6 (𝑆 ∈ 𝒫 𝐵 → (𝑋𝑆 → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)))
4140imp 407 . . . . 5 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
42413adant2 1131 . . . 4 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
43423ad2ant1 1133 . . 3 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
44 lincval 46480 . . 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))))
452, 30, 43, 44syl3anc 1371 . 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))))
46 simp1 1136 . . . . . . . 8 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑆 ∈ 𝒫 𝐵)
47 simp3 1138 . . . . . . . 8 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑋𝑆)
481, 46, 473jca 1128 . . . . . . 7 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵𝑋𝑆))
4948adantr 481 . . . . . 6 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵𝑋𝑆))
50 3simpb 1149 . . . . . . 7 ((𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) → (𝐹 ∈ (𝐸m 𝑆) ∧ 𝐹 finSupp 0 ))
5150adantl 482 . . . . . 6 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝐹 ∈ (𝐸m 𝑆) ∧ 𝐹 finSupp 0 ))
52 eqidd 2737 . . . . . 6 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝐹 ↾ (𝑆 ∖ {𝑋})) = (𝐹 ↾ (𝑆 ∖ {𝑋})))
53 eqid 2736 . . . . . . 7 ( ·𝑠𝑀) = ( ·𝑠𝑀)
54 eqid 2736 . . . . . . 7 (+g𝑀) = (+g𝑀)
558, 9, 10, 53, 54, 12lincdifsn 46495 . . . . . 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)))
5649, 51, 52, 55syl3anc 1371 . . . . 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)))
5756eqeq1d 2738 . . . 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))
58 fveq2 6842 . . . . . . . . . . . . 13 (𝑠 = 𝑧 → (𝐺𝑠) = (𝐺𝑧))
59 id 22 . . . . . . . . . . . . 13 (𝑠 = 𝑧𝑠 = 𝑧)
6058, 59oveq12d 7375 . . . . . . . . . . . 12 (𝑠 = 𝑧 → ((𝐺𝑠)( ·𝑠𝑀)𝑠) = ((𝐺𝑧)( ·𝑠𝑀)𝑧))
6160cbvmptv 5218 . . . . . . . . . . 11 (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)) = (𝑧 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑧)( ·𝑠𝑀)𝑧))
6261a1i 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)))
6362oveq2d 7373 . . . . . . . . 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))))
6463oveq2d 7373 . . . . . . . 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)))))
658, 9, 10, 11, 12, 13, 14, 15, 16, 17lincresunit3lem2 46551 . . . . . . . 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})))
6664, 65eqtr2d 2777 . . . . . . 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)))))
6766oveq1d 7372 . . . . . 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)))
6867eqeq1d 2738 . . . . 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))
69 lmodgrp 20329 . . . . . . . . 9 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
70693ad2ant2 1134 . . . . . . . 8 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑀 ∈ Grp)
7170adantr 481 . . . . . . 7 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝑀 ∈ Grp)
721adantr 481 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝑀 ∈ LMod)
73 elmapi 8787 . . . . . . . . . 10 (𝐹 ∈ (𝐸m 𝑆) → 𝐹:𝑆𝐸)
74733ad2ant1 1133 . . . . . . . . 9 ((𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) → 𝐹:𝑆𝐸)
75 ffvelcdm 7032 . . . . . . . . 9 ((𝐹:𝑆𝐸𝑋𝑆) → (𝐹𝑋) ∈ 𝐸)
7674, 47, 75syl2anr 597 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝐹𝑋) ∈ 𝐸)
77 elpwi 4567 . . . . . . . . . . 11 (𝑆 ∈ 𝒫 𝐵𝑆𝐵)
7877sselda 3944 . . . . . . . . . 10 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → 𝑋𝐵)
79783adant2 1131 . . . . . . . . 9 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑋𝐵)
8079adantr 481 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝑋𝐵)
818, 9, 53, 10lmodvscl 20339 . . . . . . . 8 ((𝑀 ∈ LMod ∧ (𝐹𝑋) ∈ 𝐸𝑋𝐵) → ((𝐹𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵)
8272, 76, 80, 81syl3anc 1371 . . . . . . 7 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → ((𝐹𝑋)( ·𝑠𝑀)𝑋) ∈ 𝐵)
839lmodfgrp 20331 . . . . . . . . . 10 (𝑀 ∈ LMod → 𝑅 ∈ Grp)
84833ad2ant2 1134 . . . . . . . . 9 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑅 ∈ Grp)
8510, 14grpinvcl 18798 . . . . . . . . 9 ((𝑅 ∈ Grp ∧ (𝐹𝑋) ∈ 𝐸) → (𝑁‘(𝐹𝑋)) ∈ 𝐸)
8684, 76, 85syl2an2r 683 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑁‘(𝐹𝑋)) ∈ 𝐸)
87 lmodcmn 20370 . . . . . . . . . . 11 (𝑀 ∈ LMod → 𝑀 ∈ CMnd)
88873ad2ant2 1134 . . . . . . . . . 10 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → 𝑀 ∈ CMnd)
8988adantr 481 . . . . . . . . 9 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝑀 ∈ CMnd)
9026adantr 481 . . . . . . . . 9 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑆 ∖ {𝑋}) ∈ V)
91 simpll2 1213 . . . . . . . . . . 11 ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → 𝑀 ∈ LMod)
928, 9, 10, 11, 12, 13, 14, 15, 16, 17lincresunit1 46548 . . . . . . . . . . . . . 14 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈)) → 𝐺 ∈ (𝐸m (𝑆 ∖ {𝑋})))
93923adantr3 1171 . . . . . . . . . . . . 13 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝐺 ∈ (𝐸m (𝑆 ∖ {𝑋})))
94 elmapi 8787 . . . . . . . . . . . . 13 (𝐺 ∈ (𝐸m (𝑆 ∖ {𝑋})) → 𝐺:(𝑆 ∖ {𝑋})⟶𝐸)
9593, 94syl 17 . . . . . . . . . . . 12 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝐺:(𝑆 ∖ {𝑋})⟶𝐸)
9695ffvelcdmda 7035 . . . . . . . . . . 11 ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → (𝐺𝑠) ∈ 𝐸)
97 ssel2 3939 . . . . . . . . . . . . . . . 16 ((𝑆𝐵𝑠𝑆) → 𝑠𝐵)
9897expcom 414 . . . . . . . . . . . . . . 15 (𝑠𝑆 → (𝑆𝐵𝑠𝐵))
997, 77, 98syl2imc 41 . . . . . . . . . . . . . 14 (𝑆 ∈ 𝒫 𝐵 → (𝑠 ∈ (𝑆 ∖ {𝑋}) → 𝑠𝐵))
100993ad2ant1 1133 . . . . . . . . . . . . 13 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → (𝑠 ∈ (𝑆 ∖ {𝑋}) → 𝑠𝐵))
101100adantr 481 . . . . . . . . . . . 12 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) → 𝑠𝐵))
102101imp 407 . . . . . . . . . . 11 ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → 𝑠𝐵)
1038, 9, 53, 10lmodvscl 20339 . . . . . . . . . . 11 ((𝑀 ∈ LMod ∧ (𝐺𝑠) ∈ 𝐸𝑠𝐵) → ((𝐺𝑠)( ·𝑠𝑀)𝑠) ∈ 𝐵)
10491, 96, 102, 103syl3anc 1371 . . . . . . . . . 10 ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) ∧ 𝑠 ∈ (𝑆 ∖ {𝑋})) → ((𝐺𝑠)( ·𝑠𝑀)𝑠) ∈ 𝐵)
105104fmpttd 7063 . . . . . . . . 9 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)):(𝑆 ∖ {𝑋})⟶𝐵)
10625adantr 481 . . . . . . . . . . . . . . 15 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → (𝑆 ∖ {𝑋}) ∈ V)
107 ssdifss 4095 . . . . . . . . . . . . . . . . . 18 (𝑆𝐵 → (𝑆 ∖ {𝑋}) ⊆ 𝐵)
10877, 107syl 17 . . . . . . . . . . . . . . . . 17 (𝑆 ∈ 𝒫 𝐵 → (𝑆 ∖ {𝑋}) ⊆ 𝐵)
109108adantr 481 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → (𝑆 ∖ {𝑋}) ⊆ 𝐵)
110109, 8sseqtrdi 3994 . . . . . . . . . . . . . . 15 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → (𝑆 ∖ {𝑋}) ⊆ (Base‘𝑀))
111106, 110elpwd 4566 . . . . . . . . . . . . . 14 ((𝑆 ∈ 𝒫 𝐵𝑋𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
1121113adant2 1131 . . . . . . . . . . . . 13 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
1131, 112jca 512 . . . . . . . . . . . 12 ((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) → (𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)))
114113adantr 481 . . . . . . . . . . 11 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)))
1158, 9, 10, 11, 12, 13, 14, 15, 16, 17lincresunit2 46549 . . . . . . . . . . . 12 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝐺 finSupp 0 )
116115, 12breqtrdi 5146 . . . . . . . . . . 11 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝐺 finSupp (0g𝑅))
1179, 10scmfsupp 46444 . . . . . . . . . . 11 (((𝑀 ∈ LMod ∧ (𝑆 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)) ∧ 𝐺 ∈ (𝐸m (𝑆 ∖ {𝑋})) ∧ 𝐺 finSupp (0g𝑅)) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)) finSupp (0g𝑀))
118114, 93, 116, 117syl3anc 1371 . . . . . . . . . 10 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)) finSupp (0g𝑀))
119118, 13breqtrrdi 5147 . . . . . . . . 9 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠)) finSupp 𝑍)
1208, 13, 89, 90, 105, 119gsumcl 19692 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))) ∈ 𝐵)
1218, 9, 53, 10lmodvscl 20339 . . . . . . . 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)
12272, 86, 120, 121syl3anc 1371 . . . . . . 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)
123 eqid 2736 . . . . . . . 8 (invg𝑀) = (invg𝑀)
1248, 54, 13, 123grpinvid2 18803 . . . . . . 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))
12571, 82, 122, 124syl3anc 1371 . . . . . 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))
1268, 9, 53, 123, 10, 14, 72, 80, 76lmodvsneg 20366 . . . . . . . 8 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → ((invg𝑀)‘((𝐹𝑋)( ·𝑠𝑀)𝑋)) = ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)𝑋))
127126eqeq1d 2738 . . . . . . 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))))))
128 simpr2 1195 . . . . . . . . 9 (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → (𝐹𝑋) ∈ 𝑈)
1298, 9, 10, 11, 14, 53lincresunit3lem3 46545 . . . . . . . . . 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)))))
130 eqcom 2743 . . . . . . . . . 10 (𝑋 = (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))) ↔ (𝑀 Σg (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑠)( ·𝑠𝑀)𝑠))) = 𝑋)
131129, 130bitrdi 286 . . . . . . . . 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))
13272, 80, 120, 128, 131syl31anc 1373 . . . . . . . 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))
133132biimpd 228 . . . . . . 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))
134127, 133sylbid 239 . . . . . 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))
135125, 134sylbird 259 . . . . 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))
13668, 135sylbid 239 . . . 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))
13757, 136sylbid 239 . . 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))
1381373impia 1117 . 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)
13945, 138eqtrd 2776 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3445  cdif 3907  wss 3910  𝒫 cpw 4560  {csn 4586   class class class wbr 5105  cmpt 5188  cres 5635  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765   finSupp cfsupp 9305  Basecbs 17083  +gcplusg 17133  .rcmulr 17134  Scalarcsca 17136   ·𝑠 cvsca 17137  0gc0g 17321   Σg cgsu 17322  Grpcgrp 18748  invgcminusg 18749  CMndccmn 19562  Unitcui 20068  invrcinvr 20100  LModclmod 20322   linC clinc 46475
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-0g 17323  df-gsum 17324  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-mulg 18873  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-lmod 20324  df-linc 46477
This theorem is referenced by:  lincreslvec3  46553
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