MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cayhamlem1 Structured version   Visualization version   GIF version

Theorem cayhamlem1 22888
Description: Lemma 1 for cayleyhamilton 22912. (Contributed by AV, 11-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))
Assertion
Ref Expression
cayhamlem1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) = 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,𝑖
Allowed substitution hints:   𝐴(𝑖,𝑛,𝑠,𝑏)   𝐵(𝑠,𝑏)   𝑃(𝑖,𝑛,𝑠,𝑏)   𝑅(𝑠,𝑏)   𝑇(𝑠,𝑏)   × (𝑠,𝑏)   (𝑛,𝑠,𝑏)   𝐺(𝑛,𝑠,𝑏)   𝑀(𝑠,𝑏)   (𝑠,𝑏)   𝑁(𝑠,𝑏)   𝑌(𝑠,𝑏)   0 (𝑖,𝑠,𝑏)

Proof of Theorem cayhamlem1
Dummy variable 𝑘 is distinct from all other variables.
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 eqid 2735 . . 3 (+g𝑌) = (+g𝑌)
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11chfacfpmmulgsum2 22887 . 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)))) (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑖))))))(+g𝑌)((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
13 elfzelz 13561 . . . . . . . . . . . . . . 15 (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℤ)
1413zcnd 12721 . . . . . . . . . . . . . 14 (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℂ)
15 pncan1 11685 . . . . . . . . . . . . . 14 (𝑖 ∈ ℂ → ((𝑖 + 1) − 1) = 𝑖)
1614, 15syl 17 . . . . . . . . . . . . 13 (𝑖 ∈ (1...𝑠) → ((𝑖 + 1) − 1) = 𝑖)
1716eqcomd 2741 . . . . . . . . . . . 12 (𝑖 ∈ (1...𝑠) → 𝑖 = ((𝑖 + 1) − 1))
1817adantl 481 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑖 ∈ (1...𝑠)) → 𝑖 = ((𝑖 + 1) − 1))
1918fveq2d 6911 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑖 ∈ (1...𝑠)) → (𝑏𝑖) = (𝑏‘((𝑖 + 1) − 1)))
2019fveq2d 6911 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑖 ∈ (1...𝑠)) → (𝑇‘(𝑏𝑖)) = (𝑇‘(𝑏‘((𝑖 + 1) − 1))))
2120oveq2d 7447 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑖 ∈ (1...𝑠)) → (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑖))) = (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1)))))
2221oveq2d 7447 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑖 ∈ (1...𝑠)) → (((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑖)))) = (((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1))))))
2322mpteq2dva 5248 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑖 ∈ (1...𝑠) ↦ (((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑖))))) = (𝑖 ∈ (1...𝑠) ↦ (((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1)))))))
2423oveq2d 7447 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑖)))))) = (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1))))))))
2524adantr 480 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑖)))))) = (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1))))))))
26 eqid 2735 . . . . 5 (Base‘𝑌) = (Base‘𝑌)
27 crngring 20263 . . . . . . . . . 10 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2827anim2i 617 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
29283adant3 1131 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
303, 4pmatring 22714 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
3129, 30syl 17 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Ring)
32 ringabl 20295 . . . . . . 7 (𝑌 ∈ Ring → 𝑌 ∈ Abel)
3331, 32syl 17 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Abel)
3433adantr 480 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑌 ∈ Abel)
35 elnnuz 12920 . . . . . . 7 (𝑠 ∈ ℕ ↔ 𝑠 ∈ (ℤ‘1))
3635biimpi 216 . . . . . 6 (𝑠 ∈ ℕ → 𝑠 ∈ (ℤ‘1))
3736ad2antrl 728 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑠 ∈ (ℤ‘1))
3831adantr 480 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑌 ∈ Ring)
3938adantr 480 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑌 ∈ Ring)
4028, 30syl 17 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
41403adant3 1131 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Ring)
42 eqid 2735 . . . . . . . . . . . . 13 (mulGrp‘𝑌) = (mulGrp‘𝑌)
4342ringmgp 20257 . . . . . . . . . . . 12 (𝑌 ∈ Ring → (mulGrp‘𝑌) ∈ Mnd)
4441, 43syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (mulGrp‘𝑌) ∈ Mnd)
4544adantr 480 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (mulGrp‘𝑌) ∈ Mnd)
4645adantr 480 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (mulGrp‘𝑌) ∈ Mnd)
47 mndmgm 18767 . . . . . . . . 9 ((mulGrp‘𝑌) ∈ Mnd → (mulGrp‘𝑌) ∈ Mgm)
4846, 47syl 17 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (mulGrp‘𝑌) ∈ Mgm)
49 elfznn 13590 . . . . . . . . 9 (𝑘 ∈ (1...(𝑠 + 1)) → 𝑘 ∈ ℕ)
5049adantl 481 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑘 ∈ ℕ)
518, 1, 2, 3, 4mat2pmatbas 22748 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑇𝑀) ∈ (Base‘𝑌))
5227, 51syl3an2 1163 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑇𝑀) ∈ (Base‘𝑌))
5352adantr 480 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑇𝑀) ∈ (Base‘𝑌))
5453adantr 480 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (𝑇𝑀) ∈ (Base‘𝑌))
5542, 26mgpbas 20158 . . . . . . . . 9 (Base‘𝑌) = (Base‘(mulGrp‘𝑌))
5655, 10mulgnncl 19120 . . . . . . . 8 (((mulGrp‘𝑌) ∈ Mgm ∧ 𝑘 ∈ ℕ ∧ (𝑇𝑀) ∈ (Base‘𝑌)) → (𝑘 (𝑇𝑀)) ∈ (Base‘𝑌))
5748, 50, 54, 56syl3anc 1370 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (𝑘 (𝑇𝑀)) ∈ (Base‘𝑌))
58 simpl1 1190 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑁 ∈ Fin)
5958adantr 480 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑁 ∈ Fin)
60273ad2ant2 1133 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
6160adantr 480 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑅 ∈ Ring)
6261adantr 480 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑅 ∈ Ring)
63 elmapi 8888 . . . . . . . . . . . 12 (𝑏 ∈ (𝐵m (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
6463adantl 481 . . . . . . . . . . 11 ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑏:(0...𝑠)⟶𝐵)
6564adantl 481 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑏:(0...𝑠)⟶𝐵)
6665adantr 480 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑏:(0...𝑠)⟶𝐵)
67 nnz 12632 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
68 peano2nn 12276 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℕ)
6968nnzd 12638 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℤ)
70 elfzm1b 13639 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℤ ∧ (𝑠 + 1) ∈ ℤ) → (𝑘 ∈ (1...(𝑠 + 1)) ↔ (𝑘 − 1) ∈ (0...((𝑠 + 1) − 1))))
7167, 69, 70syl2an 596 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (𝑘 ∈ (1...(𝑠 + 1)) ↔ (𝑘 − 1) ∈ (0...((𝑠 + 1) − 1))))
72 nncn 12272 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ ℕ → 𝑠 ∈ ℂ)
73 pncan1 11685 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ ℂ → ((𝑠 + 1) − 1) = 𝑠)
7472, 73syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ∈ ℕ → ((𝑠 + 1) − 1) = 𝑠)
7574adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → ((𝑠 + 1) − 1) = 𝑠)
7675oveq2d 7447 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (0...((𝑠 + 1) − 1)) = (0...𝑠))
7776eleq2d 2825 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → ((𝑘 − 1) ∈ (0...((𝑠 + 1) − 1)) ↔ (𝑘 − 1) ∈ (0...𝑠)))
7877biimpd 229 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → ((𝑘 − 1) ∈ (0...((𝑠 + 1) − 1)) → (𝑘 − 1) ∈ (0...𝑠)))
7971, 78sylbid 240 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (𝑘 ∈ (1...(𝑠 + 1)) → (𝑘 − 1) ∈ (0...𝑠)))
8079expcom 413 . . . . . . . . . . . . . 14 (𝑠 ∈ ℕ → (𝑘 ∈ ℕ → (𝑘 ∈ (1...(𝑠 + 1)) → (𝑘 − 1) ∈ (0...𝑠))))
8180com13 88 . . . . . . . . . . . . 13 (𝑘 ∈ (1...(𝑠 + 1)) → (𝑘 ∈ ℕ → (𝑠 ∈ ℕ → (𝑘 − 1) ∈ (0...𝑠))))
8249, 81mpd 15 . . . . . . . . . . . 12 (𝑘 ∈ (1...(𝑠 + 1)) → (𝑠 ∈ ℕ → (𝑘 − 1) ∈ (0...𝑠)))
8382com12 32 . . . . . . . . . . 11 (𝑠 ∈ ℕ → (𝑘 ∈ (1...(𝑠 + 1)) → (𝑘 − 1) ∈ (0...𝑠)))
8483ad2antrl 728 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑘 ∈ (1...(𝑠 + 1)) → (𝑘 − 1) ∈ (0...𝑠)))
8584imp 406 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (𝑘 − 1) ∈ (0...𝑠))
8666, 85ffvelcdmd 7105 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (𝑏‘(𝑘 − 1)) ∈ 𝐵)
878, 1, 2, 3, 4mat2pmatbas 22748 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑘 − 1)) ∈ 𝐵) → (𝑇‘(𝑏‘(𝑘 − 1))) ∈ (Base‘𝑌))
8859, 62, 86, 87syl3anc 1370 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (𝑇‘(𝑏‘(𝑘 − 1))) ∈ (Base‘𝑌))
8926, 5ringcl 20268 . . . . . . 7 ((𝑌 ∈ Ring ∧ (𝑘 (𝑇𝑀)) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘(𝑘 − 1))) ∈ (Base‘𝑌)) → ((𝑘 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) ∈ (Base‘𝑌))
9039, 57, 88, 89syl3anc 1370 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → ((𝑘 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) ∈ (Base‘𝑌))
9190ralrimiva 3144 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ∀𝑘 ∈ (1...(𝑠 + 1))((𝑘 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) ∈ (Base‘𝑌))
92 oveq1 7438 . . . . . 6 (𝑘 = 𝑖 → (𝑘 (𝑇𝑀)) = (𝑖 (𝑇𝑀)))
93 fvoveq1 7454 . . . . . . 7 (𝑘 = 𝑖 → (𝑏‘(𝑘 − 1)) = (𝑏‘(𝑖 − 1)))
9493fveq2d 6911 . . . . . 6 (𝑘 = 𝑖 → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘(𝑖 − 1))))
9592, 94oveq12d 7449 . . . . 5 (𝑘 = 𝑖 → ((𝑘 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) = ((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))))
96 oveq1 7438 . . . . . 6 (𝑘 = (𝑖 + 1) → (𝑘 (𝑇𝑀)) = ((𝑖 + 1) (𝑇𝑀)))
97 fvoveq1 7454 . . . . . . 7 (𝑘 = (𝑖 + 1) → (𝑏‘(𝑘 − 1)) = (𝑏‘((𝑖 + 1) − 1)))
9897fveq2d 6911 . . . . . 6 (𝑘 = (𝑖 + 1) → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘((𝑖 + 1) − 1))))
9996, 98oveq12d 7449 . . . . 5 (𝑘 = (𝑖 + 1) → ((𝑘 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) = (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1)))))
100 oveq1 7438 . . . . . 6 (𝑘 = 1 → (𝑘 (𝑇𝑀)) = (1 (𝑇𝑀)))
101 fvoveq1 7454 . . . . . . 7 (𝑘 = 1 → (𝑏‘(𝑘 − 1)) = (𝑏‘(1 − 1)))
102101fveq2d 6911 . . . . . 6 (𝑘 = 1 → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘(1 − 1))))
103100, 102oveq12d 7449 . . . . 5 (𝑘 = 1 → ((𝑘 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) = ((1 (𝑇𝑀)) × (𝑇‘(𝑏‘(1 − 1)))))
104 oveq1 7438 . . . . . 6 (𝑘 = (𝑠 + 1) → (𝑘 (𝑇𝑀)) = ((𝑠 + 1) (𝑇𝑀)))
105 fvoveq1 7454 . . . . . . 7 (𝑘 = (𝑠 + 1) → (𝑏‘(𝑘 − 1)) = (𝑏‘((𝑠 + 1) − 1)))
106105fveq2d 6911 . . . . . 6 (𝑘 = (𝑠 + 1) → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘((𝑠 + 1) − 1))))
107104, 106oveq12d 7449 . . . . 5 (𝑘 = (𝑠 + 1) → ((𝑘 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) = (((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) − 1)))))
10826, 34, 6, 37, 91, 95, 99, 103, 107telgsumfz 20023 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1))))))) = (((1 (𝑇𝑀)) × (𝑇‘(𝑏‘(1 − 1)))) (((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) − 1))))))
10925, 108eqtrd 2775 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑖)))))) = (((1 (𝑇𝑀)) × (𝑇‘(𝑏‘(1 − 1)))) (((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) − 1))))))
110109oveq1d 7446 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑖))))))(+g𝑌)((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))) = ((((1 (𝑇𝑀)) × (𝑇‘(𝑏‘(1 − 1)))) (((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) − 1)))))(+g𝑌)((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
11155, 10mulg1 19112 . . . . . . . 8 ((𝑇𝑀) ∈ (Base‘𝑌) → (1 (𝑇𝑀)) = (𝑇𝑀))
11252, 111syl 17 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (1 (𝑇𝑀)) = (𝑇𝑀))
113112adantr 480 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (1 (𝑇𝑀)) = (𝑇𝑀))
114 1cnd 11254 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 1 ∈ ℂ)
115114subidd 11606 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (1 − 1) = 0)
116115fveq2d 6911 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑏‘(1 − 1)) = (𝑏‘0))
117116fveq2d 6911 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑇‘(𝑏‘(1 − 1))) = (𝑇‘(𝑏‘0)))
118113, 117oveq12d 7449 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((1 (𝑇𝑀)) × (𝑇‘(𝑏‘(1 − 1)))) = ((𝑇𝑀) × (𝑇‘(𝑏‘0))))
11972ad2antrl 728 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑠 ∈ ℂ)
120119, 114pncand 11619 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑠 + 1) − 1) = 𝑠)
121120fveq2d 6911 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑏‘((𝑠 + 1) − 1)) = (𝑏𝑠))
122121fveq2d 6911 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑇‘(𝑏‘((𝑠 + 1) − 1))) = (𝑇‘(𝑏𝑠)))
123122oveq2d 7447 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) − 1)))) = (((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))))
124118, 123oveq12d 7449 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (((1 (𝑇𝑀)) × (𝑇‘(𝑏‘(1 − 1)))) (((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) − 1))))) = (((𝑇𝑀) × (𝑇‘(𝑏‘0))) (((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠)))))
125124oveq1d 7446 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((((1 (𝑇𝑀)) × (𝑇‘(𝑏‘(1 − 1)))) (((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) − 1)))))(+g𝑌)((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))) = ((((𝑇𝑀) × (𝑇‘(𝑏‘0))) (((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))))(+g𝑌)((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
126 ringgrp 20256 . . . . . 6 (𝑌 ∈ Ring → 𝑌 ∈ Grp)
12731, 126syl 17 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Grp)
128127adantr 480 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑌 ∈ Grp)
129 nnnn0 12531 . . . . . . . . 9 (𝑠 ∈ ℕ → 𝑠 ∈ ℕ0)
130 0elfz 13661 . . . . . . . . 9 (𝑠 ∈ ℕ0 → 0 ∈ (0...𝑠))
131129, 130syl 17 . . . . . . . 8 (𝑠 ∈ ℕ → 0 ∈ (0...𝑠))
132131ad2antrl 728 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 0 ∈ (0...𝑠))
13365, 132ffvelcdmd 7105 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑏‘0) ∈ 𝐵)
1348, 1, 2, 3, 4mat2pmatbas 22748 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
13558, 61, 133, 134syl3anc 1370 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
13626, 5ringcl 20268 . . . . 5 ((𝑌 ∈ Ring ∧ (𝑇𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌)) → ((𝑇𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
13738, 53, 135, 136syl3anc 1370 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑇𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
13845, 47syl 17 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (mulGrp‘𝑌) ∈ Mgm)
139 simprl 771 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑠 ∈ ℕ)
140139peano2nnd 12281 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑠 + 1) ∈ ℕ)
14155, 10mulgnncl 19120 . . . . . 6 (((mulGrp‘𝑌) ∈ Mgm ∧ (𝑠 + 1) ∈ ℕ ∧ (𝑇𝑀) ∈ (Base‘𝑌)) → ((𝑠 + 1) (𝑇𝑀)) ∈ (Base‘𝑌))
142138, 140, 53, 141syl3anc 1370 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑠 + 1) (𝑇𝑀)) ∈ (Base‘𝑌))
143 nn0fz0 13662 . . . . . . . . 9 (𝑠 ∈ ℕ0𝑠 ∈ (0...𝑠))
144129, 143sylib 218 . . . . . . . 8 (𝑠 ∈ ℕ → 𝑠 ∈ (0...𝑠))
145144ad2antrl 728 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑠 ∈ (0...𝑠))
14665, 145ffvelcdmd 7105 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑏𝑠) ∈ 𝐵)
1478, 1, 2, 3, 4mat2pmatbas 22748 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏𝑠) ∈ 𝐵) → (𝑇‘(𝑏𝑠)) ∈ (Base‘𝑌))
14858, 61, 146, 147syl3anc 1370 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑇‘(𝑏𝑠)) ∈ (Base‘𝑌))
14926, 5ringcl 20268 . . . . 5 ((𝑌 ∈ Ring ∧ ((𝑠 + 1) (𝑇𝑀)) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏𝑠)) ∈ (Base‘𝑌)) → (((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ∈ (Base‘𝑌))
15038, 142, 148, 149syl3anc 1370 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ∈ (Base‘𝑌))
15126, 11, 6, 7grpnpncan0 19067 . . . 4 ((𝑌 ∈ Grp ∧ (((𝑇𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌) ∧ (((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ∈ (Base‘𝑌))) → ((((𝑇𝑀) × (𝑇‘(𝑏‘0))) (((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))))(+g𝑌)((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))) = 0 )
152128, 137, 150, 151syl12anc 837 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((((𝑇𝑀) × (𝑇‘(𝑏‘0))) (((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))))(+g𝑌)((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))) = 0 )
153125, 152eqtrd 2775 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((((1 (𝑇𝑀)) × (𝑇‘(𝑏‘(1 − 1)))) (((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) − 1)))))(+g𝑌)((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))) = 0 )
15412, 110, 1533eqtrd 2779 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  ifcif 4531   class class class wbr 5148  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865  Fincfn 8984  cc 11151  0cc0 11153  1c1 11154   + caddc 11156   < clt 11293  cmin 11490  cn 12264  0cn0 12524  cz 12611  cuz 12876  ...cfz 13544  Basecbs 17245  +gcplusg 17298  .rcmulr 17299  0gc0g 17486   Σg cgsu 17487  Mgmcmgm 18664  Mndcmnd 18760  Grpcgrp 18964  -gcsg 18966  .gcmg 19098  Abelcabl 19814  mulGrpcmgp 20152  Ringcrg 20251  CRingccrg 20252  Poly1cpl1 22194   Mat cmat 22427   matToPolyMat cmat2pmat 22726
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
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  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-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-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-z 12612  df-dec 12732  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  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-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  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-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-cntz 19348  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-cring 20254  df-subrng 20563  df-subrg 20587  df-lmod 20877  df-lss 20948  df-sra 21190  df-rgmod 21191  df-dsmm 21770  df-frlm 21785  df-ascl 21893  df-psr 21947  df-mpl 21949  df-opsr 21951  df-psr1 22197  df-ply1 22199  df-mamu 22411  df-mat 22428  df-mat2pmat 22729
This theorem is referenced by:  cayleyhamilton0  22911  cayleyhamiltonALT  22913
  Copyright terms: Public domain W3C validator