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Theorem chfacfpmmulgsum2 22214
Description: Breaking up a sum of values of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)
Hypotheses
Ref Expression
cayhamlem1.a 𝐴 = (𝑁 Mat 𝑅)
cayhamlem1.b 𝐵 = (Base‘𝐴)
cayhamlem1.p 𝑃 = (Poly1𝑅)
cayhamlem1.y 𝑌 = (𝑁 Mat 𝑃)
cayhamlem1.r × = (.r𝑌)
cayhamlem1.s = (-g𝑌)
cayhamlem1.0 0 = (0g𝑌)
cayhamlem1.t 𝑇 = (𝑁 matToPolyMat 𝑅)
cayhamlem1.g 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))
cayhamlem1.e = (.g‘(mulGrp‘𝑌))
chfacfpmmulgsum.p + = (+g𝑌)
Assertion
Ref Expression
chfacfpmmulgsum2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑖)))))) + ((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
Distinct variable groups:   𝐵,𝑛   𝑛,𝑀   𝑛,𝑁   𝑅,𝑛   𝑛,𝑌   𝑛,𝑏   𝑛,𝑠   0 ,𝑛   𝐵,𝑖   𝑖,𝐺   𝑖,𝑀   𝑖,𝑁   𝑅,𝑖   𝑇,𝑖   × ,𝑖   ,𝑖   𝑖,𝑠   𝑖,𝑏   𝑇,𝑛,𝑖   𝑖,𝑌   × ,𝑛   ,𝑛
Allowed substitution hints:   𝐴(𝑖,𝑛,𝑠,𝑏)   𝐵(𝑠,𝑏)   𝑃(𝑖,𝑛,𝑠,𝑏)   + (𝑖,𝑛,𝑠,𝑏)   𝑅(𝑠,𝑏)   𝑇(𝑠,𝑏)   × (𝑠,𝑏)   (𝑛,𝑠,𝑏)   𝐺(𝑛,𝑠,𝑏)   𝑀(𝑠,𝑏)   (𝑖,𝑠,𝑏)   𝑁(𝑠,𝑏)   𝑌(𝑠,𝑏)   0 (𝑖,𝑠,𝑏)

Proof of Theorem chfacfpmmulgsum2
StepHypRef Expression
1 cayhamlem1.a . . 3 𝐴 = (𝑁 Mat 𝑅)
2 cayhamlem1.b . . 3 𝐵 = (Base‘𝐴)
3 cayhamlem1.p . . 3 𝑃 = (Poly1𝑅)
4 cayhamlem1.y . . 3 𝑌 = (𝑁 Mat 𝑃)
5 cayhamlem1.r . . 3 × = (.r𝑌)
6 cayhamlem1.s . . 3 = (-g𝑌)
7 cayhamlem1.0 . . 3 0 = (0g𝑌)
8 cayhamlem1.t . . 3 𝑇 = (𝑁 matToPolyMat 𝑅)
9 cayhamlem1.g . . 3 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))
10 cayhamlem1.e . . 3 = (.g‘(mulGrp‘𝑌))
11 chfacfpmmulgsum.p . . 3 + = (+g𝑌)
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11chfacfpmmulgsum 22213 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) + ((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
13 eqid 2736 . . . . . . 7 (Base‘𝑌) = (Base‘𝑌)
14 crngring 19976 . . . . . . . . . . 11 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
1514anim2i 617 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
163, 4pmatring 22041 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
1715, 16syl 17 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
18173adant3 1132 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Ring)
1918ad2antrr 724 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑌 ∈ Ring)
20 eqid 2736 . . . . . . . . . . . 12 (mulGrp‘𝑌) = (mulGrp‘𝑌)
2120ringmgp 19970 . . . . . . . . . . 11 (𝑌 ∈ Ring → (mulGrp‘𝑌) ∈ Mnd)
22 mndmgm 18563 . . . . . . . . . . 11 ((mulGrp‘𝑌) ∈ Mnd → (mulGrp‘𝑌) ∈ Mgm)
2321, 22syl 17 . . . . . . . . . 10 (𝑌 ∈ Ring → (mulGrp‘𝑌) ∈ Mgm)
2418, 23syl 17 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (mulGrp‘𝑌) ∈ Mgm)
2524ad2antrr 724 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (mulGrp‘𝑌) ∈ Mgm)
26 elfznn 13470 . . . . . . . . 9 (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℕ)
2726adantl 482 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑖 ∈ ℕ)
288, 1, 2, 3, 4mat2pmatbas 22075 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑇𝑀) ∈ (Base‘𝑌))
2914, 28syl3an2 1164 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑇𝑀) ∈ (Base‘𝑌))
3029ad2antrr 724 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑇𝑀) ∈ (Base‘𝑌))
3120, 13mgpbas 19902 . . . . . . . . 9 (Base‘𝑌) = (Base‘(mulGrp‘𝑌))
3231, 10mulgnncl 18891 . . . . . . . 8 (((mulGrp‘𝑌) ∈ Mgm ∧ 𝑖 ∈ ℕ ∧ (𝑇𝑀) ∈ (Base‘𝑌)) → (𝑖 (𝑇𝑀)) ∈ (Base‘𝑌))
3325, 27, 30, 32syl3anc 1371 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑖 (𝑇𝑀)) ∈ (Base‘𝑌))
34153adant3 1132 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
3534ad2antrr 724 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
36 elmapi 8787 . . . . . . . . . . . . 13 (𝑏 ∈ (𝐵m (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
3736adantl 482 . . . . . . . . . . . 12 ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑏:(0...𝑠)⟶𝐵)
3837adantl 482 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑏:(0...𝑠)⟶𝐵)
3938adantr 481 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
40 1nn0 12429 . . . . . . . . . . . . . . . 16 1 ∈ ℕ0
4140a1i 11 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → 1 ∈ ℕ0)
42 nnnn0 12420 . . . . . . . . . . . . . . . 16 (𝑠 ∈ ℕ → 𝑠 ∈ ℕ0)
4342adantr 481 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → 𝑠 ∈ ℕ0)
44 nnge1 12181 . . . . . . . . . . . . . . . 16 (𝑠 ∈ ℕ → 1 ≤ 𝑠)
4544adantr 481 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → 1 ≤ 𝑠)
46 elfz2nn0 13532 . . . . . . . . . . . . . . 15 (1 ∈ (0...𝑠) ↔ (1 ∈ ℕ0𝑠 ∈ ℕ0 ∧ 1 ≤ 𝑠))
4741, 43, 45, 46syl3anbrc 1343 . . . . . . . . . . . . . 14 ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → 1 ∈ (0...𝑠))
48 simpr 485 . . . . . . . . . . . . . 14 ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → 𝑖 ∈ (1...𝑠))
49 fz0fzdiffz0 13550 . . . . . . . . . . . . . 14 ((1 ∈ (0...𝑠) ∧ 𝑖 ∈ (1...𝑠)) → (𝑖 − 1) ∈ (0...𝑠))
5047, 48, 49syl2anc 584 . . . . . . . . . . . . 13 ((𝑠 ∈ ℕ ∧ 𝑖 ∈ (1...𝑠)) → (𝑖 − 1) ∈ (0...𝑠))
5150ex 413 . . . . . . . . . . . 12 (𝑠 ∈ ℕ → (𝑖 ∈ (1...𝑠) → (𝑖 − 1) ∈ (0...𝑠)))
5251ad2antrl 726 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑖 ∈ (1...𝑠) → (𝑖 − 1) ∈ (0...𝑠)))
5352imp 407 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑖 − 1) ∈ (0...𝑠))
5439, 53ffvelcdmd 7036 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑏‘(𝑖 − 1)) ∈ 𝐵)
55 df-3an 1089 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑖 − 1)) ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑏‘(𝑖 − 1)) ∈ 𝐵))
5635, 54, 55sylanbrc 583 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑖 − 1)) ∈ 𝐵))
578, 1, 2, 3, 4mat2pmatbas 22075 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑖 − 1)) ∈ 𝐵) → (𝑇‘(𝑏‘(𝑖 − 1))) ∈ (Base‘𝑌))
5856, 57syl 17 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑇‘(𝑏‘(𝑖 − 1))) ∈ (Base‘𝑌))
5934, 16syl 17 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Ring)
6059ad2antrr 724 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑌 ∈ Ring)
61 simpl1 1191 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑁 ∈ Fin)
62143ad2ant2 1134 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
6362adantr 481 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑅 ∈ Ring)
6442ad2antrl 726 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑠 ∈ ℕ0)
6561, 63, 643jca 1128 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0))
6665adantr 481 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0))
67 simpr 485 . . . . . . . . . . 11 ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑏 ∈ (𝐵m (0...𝑠)))
6867adantl 482 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑏 ∈ (𝐵m (0...𝑠)))
69 fz1ssfz0 13537 . . . . . . . . . . 11 (1...𝑠) ⊆ (0...𝑠)
7069sseli 3940 . . . . . . . . . 10 (𝑖 ∈ (1...𝑠) → 𝑖 ∈ (0...𝑠))
7168, 70anim12i 613 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑏 ∈ (𝐵m (0...𝑠)) ∧ 𝑖 ∈ (0...𝑠)))
721, 2, 3, 4, 8m2pmfzmap 22096 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ∧ (𝑏 ∈ (𝐵m (0...𝑠)) ∧ 𝑖 ∈ (0...𝑠))) → (𝑇‘(𝑏𝑖)) ∈ (Base‘𝑌))
7366, 71, 72syl2anc 584 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑇‘(𝑏𝑖)) ∈ (Base‘𝑌))
7413, 5ringcl 19981 . . . . . . . 8 ((𝑌 ∈ Ring ∧ (𝑇𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏𝑖)) ∈ (Base‘𝑌)) → ((𝑇𝑀) × (𝑇‘(𝑏𝑖))) ∈ (Base‘𝑌))
7560, 30, 73, 74syl3anc 1371 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑇𝑀) × (𝑇‘(𝑏𝑖))) ∈ (Base‘𝑌))
7613, 5, 6, 19, 33, 58, 75ringsubdi 20023 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 (𝑇𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))) = (((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) ((𝑖 (𝑇𝑀)) × ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))
7713, 5ringass 19984 . . . . . . . . . 10 ((𝑌 ∈ Ring ∧ ((𝑖 (𝑇𝑀)) ∈ (Base‘𝑌) ∧ (𝑇𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏𝑖)) ∈ (Base‘𝑌))) → (((𝑖 (𝑇𝑀)) × (𝑇𝑀)) × (𝑇‘(𝑏𝑖))) = ((𝑖 (𝑇𝑀)) × ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
7860, 33, 30, 73, 77syl13anc 1372 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (((𝑖 (𝑇𝑀)) × (𝑇𝑀)) × (𝑇‘(𝑏𝑖))) = ((𝑖 (𝑇𝑀)) × ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
7978eqcomd 2742 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 (𝑇𝑀)) × ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))) = (((𝑖 (𝑇𝑀)) × (𝑇𝑀)) × (𝑇‘(𝑏𝑖))))
8029, 31eleqtrdi 2848 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑇𝑀) ∈ (Base‘(mulGrp‘𝑌)))
8180adantr 481 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑇𝑀) ∈ (Base‘(mulGrp‘𝑌)))
82 eqid 2736 . . . . . . . . . . . . 13 (Base‘(mulGrp‘𝑌)) = (Base‘(mulGrp‘𝑌))
83 eqid 2736 . . . . . . . . . . . . 13 (+g‘(mulGrp‘𝑌)) = (+g‘(mulGrp‘𝑌))
8482, 10, 83mulgnnp1 18884 . . . . . . . . . . . 12 ((𝑖 ∈ ℕ ∧ (𝑇𝑀) ∈ (Base‘(mulGrp‘𝑌))) → ((𝑖 + 1) (𝑇𝑀)) = ((𝑖 (𝑇𝑀))(+g‘(mulGrp‘𝑌))(𝑇𝑀)))
8526, 81, 84syl2anr 597 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 + 1) (𝑇𝑀)) = ((𝑖 (𝑇𝑀))(+g‘(mulGrp‘𝑌))(𝑇𝑀)))
8620, 5mgpplusg 19900 . . . . . . . . . . . . . 14 × = (+g‘(mulGrp‘𝑌))
8786eqcomi 2745 . . . . . . . . . . . . 13 (+g‘(mulGrp‘𝑌)) = ×
8887a1i 11 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (+g‘(mulGrp‘𝑌)) = × )
8988oveqd 7374 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 (𝑇𝑀))(+g‘(mulGrp‘𝑌))(𝑇𝑀)) = ((𝑖 (𝑇𝑀)) × (𝑇𝑀)))
9085, 89eqtrd 2776 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 + 1) (𝑇𝑀)) = ((𝑖 (𝑇𝑀)) × (𝑇𝑀)))
9190eqcomd 2742 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 (𝑇𝑀)) × (𝑇𝑀)) = ((𝑖 + 1) (𝑇𝑀)))
9291oveq1d 7372 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (((𝑖 (𝑇𝑀)) × (𝑇𝑀)) × (𝑇‘(𝑏𝑖))) = (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑖))))
9379, 92eqtrd 2776 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 (𝑇𝑀)) × ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))) = (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑖))))
9493oveq2d 7373 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) ((𝑖 (𝑇𝑀)) × ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))) = (((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑖)))))
9576, 94eqtrd 2776 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 (𝑇𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))) = (((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑖)))))
9695mpteq2dva 5205 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))) = (𝑖 ∈ (1...𝑠) ↦ (((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑖))))))
9796oveq2d 7373 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) = (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑖)))))))
9897oveq1d 7372 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) + ((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑖)))))) + ((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
9912, 98eqtrd 2776 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑖)))))) + ((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  ifcif 4486   class class class wbr 5105  cmpt 5188  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765  Fincfn 8883  0cc0 11051  1c1 11052   + caddc 11054   < clt 11189  cle 11190  cmin 11385  cn 12153  0cn0 12413  ...cfz 13424  Basecbs 17083  +gcplusg 17133  .rcmulr 17134  0gc0g 17321   Σg cgsu 17322  Mgmcmgm 18495  Mndcmnd 18556  -gcsg 18750  .gcmg 18872  mulGrpcmgp 19896  Ringcrg 19964  CRingccrg 19965  Poly1cpl1 21548   Mat cmat 21754   matToPolyMat cmat2pmat 22053
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-tp 4591  df-op 4593  df-ot 4595  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-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  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-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  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-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-subrg 20220  df-lmod 20324  df-lss 20393  df-sra 20633  df-rgmod 20634  df-dsmm 21138  df-frlm 21153  df-ascl 21261  df-psr 21311  df-mpl 21313  df-opsr 21315  df-psr1 21551  df-ply1 21553  df-mamu 21733  df-mat 21755  df-mat2pmat 22056
This theorem is referenced by:  cayhamlem1  22215
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