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Theorem chpscmat 22864
Description: The characteristic polynomial of a (nonempty!) scalar matrix. (Contributed by AV, 21-Aug-2019.)
Hypotheses
Ref Expression
chp0mat.c 𝐶 = (𝑁 CharPlyMat 𝑅)
chp0mat.p 𝑃 = (Poly1𝑅)
chp0mat.a 𝐴 = (𝑁 Mat 𝑅)
chp0mat.x 𝑋 = (var1𝑅)
chp0mat.g 𝐺 = (mulGrp‘𝑃)
chp0mat.m = (.g𝐺)
chpscmat.d 𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖𝑁𝑗𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅))}
chpscmat.s 𝑆 = (algSc‘𝑃)
chpscmat.m = (-g𝑃)
Assertion
Ref Expression
chpscmat (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝐶𝑀) = ((♯‘𝑁) (𝑋 (𝑆𝐸))))
Distinct variable groups:   𝑖,𝑗,𝐴   𝑖,𝑁,𝑗   𝑃,𝑖,𝑗   𝑅,𝑖,𝑗   𝑖,𝑋,𝑗   𝐴,𝑐,𝑚   𝐷,𝑛   𝑛,𝐸   𝑛,𝐼   𝑀,𝑐,𝑖,𝑗,𝑚,𝑛   𝑁,𝑐,𝑚,𝑛   𝑃,𝑛   𝑅,𝑐,𝑚,𝑛   𝑆,𝑛
Allowed substitution hints:   𝐴(𝑛)   𝐶(𝑖,𝑗,𝑚,𝑛,𝑐)   𝐷(𝑖,𝑗,𝑚,𝑐)   𝑃(𝑚,𝑐)   𝑆(𝑖,𝑗,𝑚,𝑐)   𝐸(𝑖,𝑗,𝑚,𝑐)   (𝑖,𝑗,𝑚,𝑛,𝑐)   𝐺(𝑖,𝑗,𝑚,𝑛,𝑐)   𝐼(𝑖,𝑗,𝑚,𝑐)   (𝑖,𝑗,𝑚,𝑛,𝑐)   𝑋(𝑚,𝑛,𝑐)

Proof of Theorem chpscmat
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 simpll 767 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑁 ∈ Fin)
2 simplr 769 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑅 ∈ CRing)
3 elrabi 3690 . . . . . 6 (𝑀 ∈ {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖𝑁𝑗𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅))} → 𝑀 ∈ (Base‘𝐴))
4 chpscmat.d . . . . . 6 𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖𝑁𝑗𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅))}
53, 4eleq2s 2857 . . . . 5 (𝑀𝐷𝑀 ∈ (Base‘𝐴))
653ad2ant1 1132 . . . 4 ((𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸) → 𝑀 ∈ (Base‘𝐴))
76adantl 481 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑀 ∈ (Base‘𝐴))
8 oveq 7437 . . . . . . . . . . 11 (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
98eqeq1d 2737 . . . . . . . . . 10 (𝑚 = 𝑀 → ((𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅)) ↔ (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅))))
1092ralbidv 3219 . . . . . . . . 9 (𝑚 = 𝑀 → (∀𝑖𝑁𝑗𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅)) ↔ ∀𝑖𝑁𝑗𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅))))
1110rexbidv 3177 . . . . . . . 8 (𝑚 = 𝑀 → (∃𝑐 ∈ (Base‘𝑅)∀𝑖𝑁𝑗𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅)) ↔ ∃𝑐 ∈ (Base‘𝑅)∀𝑖𝑁𝑗𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅))))
1211elrab 3695 . . . . . . 7 (𝑀 ∈ {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖𝑁𝑗𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅))} ↔ (𝑀 ∈ (Base‘𝐴) ∧ ∃𝑐 ∈ (Base‘𝑅)∀𝑖𝑁𝑗𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅))))
13 ifnefalse 4543 . . . . . . . . . . . . . . . 16 (𝑖𝑗 → if(𝑖 = 𝑗, 𝑐, (0g𝑅)) = (0g𝑅))
1413eqeq2d 2746 . . . . . . . . . . . . . . 15 (𝑖𝑗 → ((𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅)) ↔ (𝑖𝑀𝑗) = (0g𝑅)))
1514biimpcd 249 . . . . . . . . . . . . . 14 ((𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅)) → (𝑖𝑗 → (𝑖𝑀𝑗) = (0g𝑅)))
1615a1i 11 . . . . . . . . . . . . 13 (((((𝑀 ∈ (Base‘𝐴) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → ((𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅)) → (𝑖𝑗 → (𝑖𝑀𝑗) = (0g𝑅))))
1716ralimdva 3165 . . . . . . . . . . . 12 ((((𝑀 ∈ (Base‘𝐴) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝑖𝑁) → (∀𝑗𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅)) → ∀𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = (0g𝑅))))
1817ralimdva 3165 . . . . . . . . . . 11 (((𝑀 ∈ (Base‘𝐴) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → (∀𝑖𝑁𝑗𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅)) → ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = (0g𝑅))))
1918ex 412 . . . . . . . . . 10 ((𝑀 ∈ (Base‘𝐴) ∧ 𝑐 ∈ (Base‘𝑅)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (∀𝑖𝑁𝑗𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅)) → ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = (0g𝑅)))))
2019com23 86 . . . . . . . . 9 ((𝑀 ∈ (Base‘𝐴) ∧ 𝑐 ∈ (Base‘𝑅)) → (∀𝑖𝑁𝑗𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = (0g𝑅)))))
2120rexlimdva 3153 . . . . . . . 8 (𝑀 ∈ (Base‘𝐴) → (∃𝑐 ∈ (Base‘𝑅)∀𝑖𝑁𝑗𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = (0g𝑅)))))
2221imp 406 . . . . . . 7 ((𝑀 ∈ (Base‘𝐴) ∧ ∃𝑐 ∈ (Base‘𝑅)∀𝑖𝑁𝑗𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅))) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = (0g𝑅))))
2312, 22sylbi 217 . . . . . 6 (𝑀 ∈ {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖𝑁𝑗𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅))} → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = (0g𝑅))))
2423, 4eleq2s 2857 . . . . 5 (𝑀𝐷 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = (0g𝑅))))
25243ad2ant1 1132 . . . 4 ((𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = (0g𝑅))))
2625impcom 407 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸)) → ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = (0g𝑅)))
27 chp0mat.c . . . 4 𝐶 = (𝑁 CharPlyMat 𝑅)
28 chp0mat.p . . . 4 𝑃 = (Poly1𝑅)
29 chp0mat.a . . . 4 𝐴 = (𝑁 Mat 𝑅)
30 chpscmat.s . . . 4 𝑆 = (algSc‘𝑃)
31 eqid 2735 . . . 4 (Base‘𝐴) = (Base‘𝐴)
32 chp0mat.x . . . 4 𝑋 = (var1𝑅)
33 eqid 2735 . . . 4 (0g𝑅) = (0g𝑅)
34 chp0mat.g . . . 4 𝐺 = (mulGrp‘𝑃)
35 chpscmat.m . . . 4 = (-g𝑃)
3627, 28, 29, 30, 31, 32, 33, 34, 35chpdmat 22863 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘𝐴)) ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = (0g𝑅))) → (𝐶𝑀) = (𝐺 Σg (𝑘𝑁 ↦ (𝑋 (𝑆‘(𝑘𝑀𝑘))))))
371, 2, 7, 26, 36syl31anc 1372 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝐶𝑀) = (𝐺 Σg (𝑘𝑁 ↦ (𝑋 (𝑆‘(𝑘𝑀𝑘))))))
38 id 22 . . . . . . . . . . . 12 (𝑛 = 𝑘𝑛 = 𝑘)
3938, 38oveq12d 7449 . . . . . . . . . . 11 (𝑛 = 𝑘 → (𝑛𝑀𝑛) = (𝑘𝑀𝑘))
4039eqeq1d 2737 . . . . . . . . . 10 (𝑛 = 𝑘 → ((𝑛𝑀𝑛) = 𝐸 ↔ (𝑘𝑀𝑘) = 𝐸))
4140rspccv 3619 . . . . . . . . 9 (∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝑘𝑁 → (𝑘𝑀𝑘) = 𝐸))
42413ad2ant3 1134 . . . . . . . 8 ((𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸) → (𝑘𝑁 → (𝑘𝑀𝑘) = 𝐸))
4342adantl 481 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑘𝑁 → (𝑘𝑀𝑘) = 𝐸))
4443imp 406 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸)) ∧ 𝑘𝑁) → (𝑘𝑀𝑘) = 𝐸)
4544fveq2d 6911 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸)) ∧ 𝑘𝑁) → (𝑆‘(𝑘𝑀𝑘)) = (𝑆𝐸))
4645oveq2d 7447 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸)) ∧ 𝑘𝑁) → (𝑋 (𝑆‘(𝑘𝑀𝑘))) = (𝑋 (𝑆𝐸)))
4746mpteq2dva 5248 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑘𝑁 ↦ (𝑋 (𝑆‘(𝑘𝑀𝑘)))) = (𝑘𝑁 ↦ (𝑋 (𝑆𝐸))))
4847oveq2d 7447 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝐺 Σg (𝑘𝑁 ↦ (𝑋 (𝑆‘(𝑘𝑀𝑘))))) = (𝐺 Σg (𝑘𝑁 ↦ (𝑋 (𝑆𝐸)))))
4928ply1crng 22216 . . . . 5 (𝑅 ∈ CRing → 𝑃 ∈ CRing)
5034crngmgp 20259 . . . . 5 (𝑃 ∈ CRing → 𝐺 ∈ CMnd)
51 cmnmnd 19830 . . . . 5 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
5249, 50, 513syl 18 . . . 4 (𝑅 ∈ CRing → 𝐺 ∈ Mnd)
5352ad2antlr 727 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝐺 ∈ Mnd)
54 crngring 20263 . . . . . . . 8 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5528ply1ring 22265 . . . . . . . 8 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
5654, 55syl 17 . . . . . . 7 (𝑅 ∈ CRing → 𝑃 ∈ Ring)
57 ringgrp 20256 . . . . . . 7 (𝑃 ∈ Ring → 𝑃 ∈ Grp)
5856, 57syl 17 . . . . . 6 (𝑅 ∈ CRing → 𝑃 ∈ Grp)
5958ad2antlr 727 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑃 ∈ Grp)
60 eqid 2735 . . . . . . . 8 (Base‘𝑃) = (Base‘𝑃)
6132, 28, 60vr1cl 22235 . . . . . . 7 (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
6254, 61syl 17 . . . . . 6 (𝑅 ∈ CRing → 𝑋 ∈ (Base‘𝑃))
6362ad2antlr 727 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸)) → 𝑋 ∈ (Base‘𝑃))
64 simpr 484 . . . . . . . . . . . 12 (((𝑀𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼𝑁) → 𝐼𝑁)
65 eqid 2735 . . . . . . . . . . . . . . . . 17 (Scalar‘𝑃) = (Scalar‘𝑃)
6656ad2antll 729 . . . . . . . . . . . . . . . . . 18 ((𝑀𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → 𝑃 ∈ Ring)
6766adantr 480 . . . . . . . . . . . . . . . . 17 (((𝑀𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼𝑁) → 𝑃 ∈ Ring)
6828ply1lmod 22269 . . . . . . . . . . . . . . . . . . . 20 (𝑅 ∈ Ring → 𝑃 ∈ LMod)
6954, 68syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑅 ∈ CRing → 𝑃 ∈ LMod)
7069ad2antll 729 . . . . . . . . . . . . . . . . . 18 ((𝑀𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → 𝑃 ∈ LMod)
7170adantr 480 . . . . . . . . . . . . . . . . 17 (((𝑀𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼𝑁) → 𝑃 ∈ LMod)
72 eqid 2735 . . . . . . . . . . . . . . . . 17 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
7330, 65, 67, 71, 72, 60asclf 21920 . . . . . . . . . . . . . . . 16 (((𝑀𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼𝑁) → 𝑆:(Base‘(Scalar‘𝑃))⟶(Base‘𝑃))
745adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑀𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → 𝑀 ∈ (Base‘𝐴))
7574adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝑀𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼𝑁) → 𝑀 ∈ (Base‘𝐴))
76 eqid 2735 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑅) = (Base‘𝑅)
7729, 76matecl 22447 . . . . . . . . . . . . . . . . . 18 ((𝐼𝑁𝐼𝑁𝑀 ∈ (Base‘𝐴)) → (𝐼𝑀𝐼) ∈ (Base‘𝑅))
7864, 64, 75, 77syl3anc 1370 . . . . . . . . . . . . . . . . 17 (((𝑀𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼𝑁) → (𝐼𝑀𝐼) ∈ (Base‘𝑅))
7928ply1sca 22270 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
8079ad2antll 729 . . . . . . . . . . . . . . . . . . . 20 ((𝑀𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → 𝑅 = (Scalar‘𝑃))
8180adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝑀𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼𝑁) → 𝑅 = (Scalar‘𝑃))
8281eqcomd 2741 . . . . . . . . . . . . . . . . . 18 (((𝑀𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼𝑁) → (Scalar‘𝑃) = 𝑅)
8382fveq2d 6911 . . . . . . . . . . . . . . . . 17 (((𝑀𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼𝑁) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
8478, 83eleqtrrd 2842 . . . . . . . . . . . . . . . 16 (((𝑀𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼𝑁) → (𝐼𝑀𝐼) ∈ (Base‘(Scalar‘𝑃)))
8573, 84ffvelcdmd 7105 . . . . . . . . . . . . . . 15 (((𝑀𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼𝑁) → (𝑆‘(𝐼𝑀𝐼)) ∈ (Base‘𝑃))
86 fveq2 6907 . . . . . . . . . . . . . . . . 17 (𝐸 = (𝐼𝑀𝐼) → (𝑆𝐸) = (𝑆‘(𝐼𝑀𝐼)))
8786eqcoms 2743 . . . . . . . . . . . . . . . 16 ((𝐼𝑀𝐼) = 𝐸 → (𝑆𝐸) = (𝑆‘(𝐼𝑀𝐼)))
8887eleq1d 2824 . . . . . . . . . . . . . . 15 ((𝐼𝑀𝐼) = 𝐸 → ((𝑆𝐸) ∈ (Base‘𝑃) ↔ (𝑆‘(𝐼𝑀𝐼)) ∈ (Base‘𝑃)))
8985, 88syl5ibrcom 247 . . . . . . . . . . . . . 14 (((𝑀𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼𝑁) → ((𝐼𝑀𝐼) = 𝐸 → (𝑆𝐸) ∈ (Base‘𝑃)))
9089adantr 480 . . . . . . . . . . . . 13 ((((𝑀𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼𝑁) ∧ 𝑛 = 𝐼) → ((𝐼𝑀𝐼) = 𝐸 → (𝑆𝐸) ∈ (Base‘𝑃)))
91 id 22 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝐼𝑛 = 𝐼)
9291, 91oveq12d 7449 . . . . . . . . . . . . . . . 16 (𝑛 = 𝐼 → (𝑛𝑀𝑛) = (𝐼𝑀𝐼))
9392eqeq1d 2737 . . . . . . . . . . . . . . 15 (𝑛 = 𝐼 → ((𝑛𝑀𝑛) = 𝐸 ↔ (𝐼𝑀𝐼) = 𝐸))
9493imbi1d 341 . . . . . . . . . . . . . 14 (𝑛 = 𝐼 → (((𝑛𝑀𝑛) = 𝐸 → (𝑆𝐸) ∈ (Base‘𝑃)) ↔ ((𝐼𝑀𝐼) = 𝐸 → (𝑆𝐸) ∈ (Base‘𝑃))))
9594adantl 481 . . . . . . . . . . . . 13 ((((𝑀𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼𝑁) ∧ 𝑛 = 𝐼) → (((𝑛𝑀𝑛) = 𝐸 → (𝑆𝐸) ∈ (Base‘𝑃)) ↔ ((𝐼𝑀𝐼) = 𝐸 → (𝑆𝐸) ∈ (Base‘𝑃))))
9690, 95mpbird 257 . . . . . . . . . . . 12 ((((𝑀𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼𝑁) ∧ 𝑛 = 𝐼) → ((𝑛𝑀𝑛) = 𝐸 → (𝑆𝐸) ∈ (Base‘𝑃)))
9764, 96rspcimdv 3612 . . . . . . . . . . 11 (((𝑀𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) ∧ 𝐼𝑁) → (∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝑆𝐸) ∈ (Base‘𝑃)))
9897ex 412 . . . . . . . . . 10 ((𝑀𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → (𝐼𝑁 → (∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝑆𝐸) ∈ (Base‘𝑃))))
9998com23 86 . . . . . . . . 9 ((𝑀𝐷 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing)) → (∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝐼𝑁 → (𝑆𝐸) ∈ (Base‘𝑃))))
10099ex 412 . . . . . . . 8 (𝑀𝐷 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸 → (𝐼𝑁 → (𝑆𝐸) ∈ (Base‘𝑃)))))
101100com24 95 . . . . . . 7 (𝑀𝐷 → (𝐼𝑁 → (∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑆𝐸) ∈ (Base‘𝑃)))))
1021013imp 1110 . . . . . 6 ((𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑆𝐸) ∈ (Base‘𝑃)))
103102impcom 407 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑆𝐸) ∈ (Base‘𝑃))
10460, 35grpsubcl 19051 . . . . 5 ((𝑃 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑃) ∧ (𝑆𝐸) ∈ (Base‘𝑃)) → (𝑋 (𝑆𝐸)) ∈ (Base‘𝑃))
10559, 63, 103, 104syl3anc 1370 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑋 (𝑆𝐸)) ∈ (Base‘𝑃))
10634, 60mgpbas 20158 . . . 4 (Base‘𝑃) = (Base‘𝐺)
107105, 106eleqtrdi 2849 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝑋 (𝑆𝐸)) ∈ (Base‘𝐺))
108 eqid 2735 . . . 4 (Base‘𝐺) = (Base‘𝐺)
109 chp0mat.m . . . 4 = (.g𝐺)
110108, 109gsumconst 19967 . . 3 ((𝐺 ∈ Mnd ∧ 𝑁 ∈ Fin ∧ (𝑋 (𝑆𝐸)) ∈ (Base‘𝐺)) → (𝐺 Σg (𝑘𝑁 ↦ (𝑋 (𝑆𝐸)))) = ((♯‘𝑁) (𝑋 (𝑆𝐸))))
11153, 1, 107, 110syl3anc 1370 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝐺 Σg (𝑘𝑁 ↦ (𝑋 (𝑆𝐸)))) = ((♯‘𝑁) (𝑋 (𝑆𝐸))))
11237, 48, 1113eqtrd 2779 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝐶𝑀) = ((♯‘𝑁) (𝑋 (𝑆𝐸))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  ifcif 4531  cmpt 5231  cfv 6563  (class class class)co 7431  Fincfn 8984  chash 14366  Basecbs 17245  Scalarcsca 17301  0gc0g 17486   Σg cgsu 17487  Mndcmnd 18760  Grpcgrp 18964  -gcsg 18966  .gcmg 19098  CMndccmn 19813  mulGrpcmgp 20152  Ringcrg 20251  CRingccrg 20252  LModclmod 20875  algSccascl 21890  var1cv1 22193  Poly1cpl1 22194   Mat cmat 22427   CharPlyMat cchpmat 22848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-addf 11232  ax-mulf 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1509  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-xnn0 12598  df-z 12612  df-dec 12732  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-hash 14367  df-word 14550  df-lsw 14598  df-concat 14606  df-s1 14631  df-substr 14676  df-pfx 14706  df-splice 14785  df-reverse 14794  df-s2 14884  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-efmnd 18895  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-gim 19290  df-cntz 19348  df-oppg 19377  df-symg 19402  df-pmtr 19475  df-psgn 19524  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-cring 20254  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-dvr 20418  df-rhm 20489  df-subrng 20563  df-subrg 20587  df-drng 20748  df-lmod 20877  df-lss 20948  df-sra 21190  df-rgmod 21191  df-cnfld 21383  df-zring 21476  df-zrh 21532  df-dsmm 21770  df-frlm 21785  df-ascl 21893  df-psr 21947  df-mvr 21948  df-mpl 21949  df-opsr 21951  df-psr1 22197  df-vr1 22198  df-ply1 22199  df-mamu 22411  df-mat 22428  df-mdet 22607  df-mat2pmat 22729  df-chpmat 22849
This theorem is referenced by:  chpscmat0  22865
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