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Theorem cayhamlem1 20437
Description: Lemma 1 for cayleyhamilton 20461. (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 2609 . . 3 (+g𝑌) = (+g𝑌)
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11chfacfpmmulgsum2 20436 . 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 12170 . . . . . . . . . . . . . . 15 (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℤ)
1413zcnd 11317 . . . . . . . . . . . . . 14 (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℂ)
15 pncan1 10305 . . . . . . . . . . . . . 14 (𝑖 ∈ ℂ → ((𝑖 + 1) − 1) = 𝑖)
1614, 15syl 17 . . . . . . . . . . . . 13 (𝑖 ∈ (1...𝑠) → ((𝑖 + 1) − 1) = 𝑖)
1716eqcomd 2615 . . . . . . . . . . . 12 (𝑖 ∈ (1...𝑠) → 𝑖 = ((𝑖 + 1) − 1))
1817adantl 480 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑖 ∈ (1...𝑠)) → 𝑖 = ((𝑖 + 1) − 1))
1918fveq2d 6091 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑖 ∈ (1...𝑠)) → (𝑏𝑖) = (𝑏‘((𝑖 + 1) − 1)))
2019fveq2d 6091 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑖 ∈ (1...𝑠)) → (𝑇‘(𝑏𝑖)) = (𝑇‘(𝑏‘((𝑖 + 1) − 1))))
2120oveq2d 6542 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑖 ∈ (1...𝑠)) → (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑖))) = (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1)))))
2221oveq2d 6542 . . . . . . 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 4666 . . . . . 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 6542 . . . . 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 479 . . . 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 2609 . . . . 5 (Base‘𝑌) = (Base‘𝑌)
27 crngring 18329 . . . . . . . . . 10 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2827anim2i 590 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
29283adant3 1073 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
303, 4pmatring 20264 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
3129, 30syl 17 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Ring)
32 ringabl 18351 . . . . . . 7 (𝑌 ∈ Ring → 𝑌 ∈ Abel)
3331, 32syl 17 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Abel)
3433adantr 479 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑌 ∈ Abel)
35 elnnuz 11558 . . . . . . 7 (𝑠 ∈ ℕ ↔ 𝑠 ∈ (ℤ‘1))
3635biimpi 204 . . . . . 6 (𝑠 ∈ ℕ → 𝑠 ∈ (ℤ‘1))
3736ad2antrl 759 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑠 ∈ (ℤ‘1))
3831adantr 479 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑌 ∈ Ring)
3938adantr 479 . . . . . . 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 1073 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Ring)
42 eqid 2609 . . . . . . . . . . . . 13 (mulGrp‘𝑌) = (mulGrp‘𝑌)
4342ringmgp 18324 . . . . . . . . . . . 12 (𝑌 ∈ Ring → (mulGrp‘𝑌) ∈ Mnd)
4441, 43syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (mulGrp‘𝑌) ∈ Mnd)
4544adantr 479 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (mulGrp‘𝑌) ∈ Mnd)
4645adantr 479 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (mulGrp‘𝑌) ∈ Mnd)
47 mndmgm 17071 . . . . . . . . 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 12198 . . . . . . . . 9 (𝑘 ∈ (1...(𝑠 + 1)) → 𝑘 ∈ ℕ)
5049adantl 480 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑘 ∈ ℕ)
518, 1, 2, 3, 4mat2pmatbas 20297 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑇𝑀) ∈ (Base‘𝑌))
5227, 51syl3an2 1351 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑇𝑀) ∈ (Base‘𝑌))
5352adantr 479 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑇𝑀) ∈ (Base‘𝑌))
5453adantr 479 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (𝑇𝑀) ∈ (Base‘𝑌))
5542, 26mgpbas 18266 . . . . . . . . 9 (Base‘𝑌) = (Base‘(mulGrp‘𝑌))
5655, 10mulgnncl 17327 . . . . . . . 8 (((mulGrp‘𝑌) ∈ Mgm ∧ 𝑘 ∈ ℕ ∧ (𝑇𝑀) ∈ (Base‘𝑌)) → (𝑘 (𝑇𝑀)) ∈ (Base‘𝑌))
5748, 50, 54, 56syl3anc 1317 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (𝑘 (𝑇𝑀)) ∈ (Base‘𝑌))
58 simpl1 1056 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑁 ∈ Fin)
5958adantr 479 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑁 ∈ Fin)
60273ad2ant2 1075 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
6160adantr 479 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑅 ∈ Ring)
6261adantr 479 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑅 ∈ Ring)
63 elmapi 7742 . . . . . . . . . . . 12 (𝑏 ∈ (𝐵𝑚 (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
6463adantl 480 . . . . . . . . . . 11 ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → 𝑏:(0...𝑠)⟶𝐵)
6564adantl 480 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑏:(0...𝑠)⟶𝐵)
6665adantr 479 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → 𝑏:(0...𝑠)⟶𝐵)
67 nnz 11234 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
68 peano2nn 10881 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℕ)
6968nnzd 11315 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℤ)
70 elfzm1b 12244 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℤ ∧ (𝑠 + 1) ∈ ℤ) → (𝑘 ∈ (1...(𝑠 + 1)) ↔ (𝑘 − 1) ∈ (0...((𝑠 + 1) − 1))))
7167, 69, 70syl2an 492 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (𝑘 ∈ (1...(𝑠 + 1)) ↔ (𝑘 − 1) ∈ (0...((𝑠 + 1) − 1))))
72 nncn 10877 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ ℕ → 𝑠 ∈ ℂ)
73 pncan1 10305 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ ℂ → ((𝑠 + 1) − 1) = 𝑠)
7472, 73syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ∈ ℕ → ((𝑠 + 1) − 1) = 𝑠)
7574adantl 480 . . . . . . . . . . . . . . . . . . 19 ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → ((𝑠 + 1) − 1) = 𝑠)
7675oveq2d 6542 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (0...((𝑠 + 1) − 1)) = (0...𝑠))
7776eleq2d 2672 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → ((𝑘 − 1) ∈ (0...((𝑠 + 1) − 1)) ↔ (𝑘 − 1) ∈ (0...𝑠)))
7877biimpd 217 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → ((𝑘 − 1) ∈ (0...((𝑠 + 1) − 1)) → (𝑘 − 1) ∈ (0...𝑠)))
7971, 78sylbid 228 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℕ ∧ 𝑠 ∈ ℕ) → (𝑘 ∈ (1...(𝑠 + 1)) → (𝑘 − 1) ∈ (0...𝑠)))
8079expcom 449 . . . . . . . . . . . . . 14 (𝑠 ∈ ℕ → (𝑘 ∈ ℕ → (𝑘 ∈ (1...(𝑠 + 1)) → (𝑘 − 1) ∈ (0...𝑠))))
8180com13 85 . . . . . . . . . . . . 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 759 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑘 ∈ (1...(𝑠 + 1)) → (𝑘 − 1) ∈ (0...𝑠)))
8584imp 443 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (𝑘 − 1) ∈ (0...𝑠))
8666, 85ffvelrnd 6252 . . . . . . . 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 20297 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑘 − 1)) ∈ 𝐵) → (𝑇‘(𝑏‘(𝑘 − 1))) ∈ (Base‘𝑌))
8859, 62, 86, 87syl3anc 1317 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑘 ∈ (1...(𝑠 + 1))) → (𝑇‘(𝑏‘(𝑘 − 1))) ∈ (Base‘𝑌))
8926, 5ringcl 18332 . . . . . . 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 1317 . . . . . 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 2948 . . . . 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 6533 . . . . . 6 (𝑘 = 𝑖 → (𝑘 (𝑇𝑀)) = (𝑖 (𝑇𝑀)))
93 oveq1 6533 . . . . . . . 8 (𝑘 = 𝑖 → (𝑘 − 1) = (𝑖 − 1))
9493fveq2d 6091 . . . . . . 7 (𝑘 = 𝑖 → (𝑏‘(𝑘 − 1)) = (𝑏‘(𝑖 − 1)))
9594fveq2d 6091 . . . . . 6 (𝑘 = 𝑖 → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘(𝑖 − 1))))
9692, 95oveq12d 6544 . . . . 5 (𝑘 = 𝑖 → ((𝑘 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) = ((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))))
97 oveq1 6533 . . . . . 6 (𝑘 = (𝑖 + 1) → (𝑘 (𝑇𝑀)) = ((𝑖 + 1) (𝑇𝑀)))
98 oveq1 6533 . . . . . . . 8 (𝑘 = (𝑖 + 1) → (𝑘 − 1) = ((𝑖 + 1) − 1))
9998fveq2d 6091 . . . . . . 7 (𝑘 = (𝑖 + 1) → (𝑏‘(𝑘 − 1)) = (𝑏‘((𝑖 + 1) − 1)))
10099fveq2d 6091 . . . . . 6 (𝑘 = (𝑖 + 1) → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘((𝑖 + 1) − 1))))
10197, 100oveq12d 6544 . . . . 5 (𝑘 = (𝑖 + 1) → ((𝑘 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) = (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏‘((𝑖 + 1) − 1)))))
102 oveq1 6533 . . . . . 6 (𝑘 = 1 → (𝑘 (𝑇𝑀)) = (1 (𝑇𝑀)))
103 oveq1 6533 . . . . . . . 8 (𝑘 = 1 → (𝑘 − 1) = (1 − 1))
104103fveq2d 6091 . . . . . . 7 (𝑘 = 1 → (𝑏‘(𝑘 − 1)) = (𝑏‘(1 − 1)))
105104fveq2d 6091 . . . . . 6 (𝑘 = 1 → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘(1 − 1))))
106102, 105oveq12d 6544 . . . . 5 (𝑘 = 1 → ((𝑘 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) = ((1 (𝑇𝑀)) × (𝑇‘(𝑏‘(1 − 1)))))
107 oveq1 6533 . . . . . 6 (𝑘 = (𝑠 + 1) → (𝑘 (𝑇𝑀)) = ((𝑠 + 1) (𝑇𝑀)))
108 oveq1 6533 . . . . . . . 8 (𝑘 = (𝑠 + 1) → (𝑘 − 1) = ((𝑠 + 1) − 1))
109108fveq2d 6091 . . . . . . 7 (𝑘 = (𝑠 + 1) → (𝑏‘(𝑘 − 1)) = (𝑏‘((𝑠 + 1) − 1)))
110109fveq2d 6091 . . . . . 6 (𝑘 = (𝑠 + 1) → (𝑇‘(𝑏‘(𝑘 − 1))) = (𝑇‘(𝑏‘((𝑠 + 1) − 1))))
111107, 110oveq12d 6544 . . . . 5 (𝑘 = (𝑠 + 1) → ((𝑘 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑘 − 1)))) = (((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏‘((𝑠 + 1) − 1)))))
11226, 34, 6, 37, 91, 96, 101, 106, 111telgsumfz 18158 . . . 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))))))
11325, 112eqtrd 2643 . . 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))))))
114113oveq1d 6541 . 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))))))
11555, 10mulg1 17319 . . . . . . . 8 ((𝑇𝑀) ∈ (Base‘𝑌) → (1 (𝑇𝑀)) = (𝑇𝑀))
11652, 115syl 17 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (1 (𝑇𝑀)) = (𝑇𝑀))
117116adantr 479 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (1 (𝑇𝑀)) = (𝑇𝑀))
118 1cnd 9912 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 1 ∈ ℂ)
119118subidd 10231 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (1 − 1) = 0)
120119fveq2d 6091 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑏‘(1 − 1)) = (𝑏‘0))
121120fveq2d 6091 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑇‘(𝑏‘(1 − 1))) = (𝑇‘(𝑏‘0)))
122117, 121oveq12d 6544 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((1 (𝑇𝑀)) × (𝑇‘(𝑏‘(1 − 1)))) = ((𝑇𝑀) × (𝑇‘(𝑏‘0))))
12372ad2antrl 759 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑠 ∈ ℂ)
124123, 118pncand 10244 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((𝑠 + 1) − 1) = 𝑠)
125124fveq2d 6091 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑏‘((𝑠 + 1) − 1)) = (𝑏𝑠))
126125fveq2d 6091 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑇‘(𝑏‘((𝑠 + 1) − 1))) = (𝑇‘(𝑏𝑠)))
127126oveq2d 6542 . . . . 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))))
128122, 127oveq12d 6544 . . . 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)))))
129128oveq1d 6541 . . 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))))))
130 ringgrp 18323 . . . . . 6 (𝑌 ∈ Ring → 𝑌 ∈ Grp)
13131, 130syl 17 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Grp)
132131adantr 479 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑌 ∈ Grp)
133 nnnn0 11148 . . . . . . . . 9 (𝑠 ∈ ℕ → 𝑠 ∈ ℕ0)
134 0elfz 12262 . . . . . . . . 9 (𝑠 ∈ ℕ0 → 0 ∈ (0...𝑠))
135133, 134syl 17 . . . . . . . 8 (𝑠 ∈ ℕ → 0 ∈ (0...𝑠))
136135ad2antrl 759 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 0 ∈ (0...𝑠))
13765, 136ffvelrnd 6252 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑏‘0) ∈ 𝐵)
1388, 1, 2, 3, 4mat2pmatbas 20297 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
13958, 61, 137, 138syl3anc 1317 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
14026, 5ringcl 18332 . . . . 5 ((𝑌 ∈ Ring ∧ (𝑇𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌)) → ((𝑇𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
14138, 53, 139, 140syl3anc 1317 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((𝑇𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
14245, 47syl 17 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (mulGrp‘𝑌) ∈ Mgm)
143 simprl 789 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑠 ∈ ℕ)
144143peano2nnd 10886 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑠 + 1) ∈ ℕ)
14555, 10mulgnncl 17327 . . . . . 6 (((mulGrp‘𝑌) ∈ Mgm ∧ (𝑠 + 1) ∈ ℕ ∧ (𝑇𝑀) ∈ (Base‘𝑌)) → ((𝑠 + 1) (𝑇𝑀)) ∈ (Base‘𝑌))
146142, 144, 53, 145syl3anc 1317 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((𝑠 + 1) (𝑇𝑀)) ∈ (Base‘𝑌))
147 nn0fz0 12263 . . . . . . . . 9 (𝑠 ∈ ℕ0𝑠 ∈ (0...𝑠))
148133, 147sylib 206 . . . . . . . 8 (𝑠 ∈ ℕ → 𝑠 ∈ (0...𝑠))
149148ad2antrl 759 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑠 ∈ (0...𝑠))
15065, 149ffvelrnd 6252 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑏𝑠) ∈ 𝐵)
1518, 1, 2, 3, 4mat2pmatbas 20297 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏𝑠) ∈ 𝐵) → (𝑇‘(𝑏𝑠)) ∈ (Base‘𝑌))
15258, 61, 150, 151syl3anc 1317 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑇‘(𝑏𝑠)) ∈ (Base‘𝑌))
15326, 5ringcl 18332 . . . . 5 ((𝑌 ∈ Ring ∧ ((𝑠 + 1) (𝑇𝑀)) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏𝑠)) ∈ (Base‘𝑌)) → (((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ∈ (Base‘𝑌))
15438, 146, 152, 153syl3anc 1317 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ∈ (Base‘𝑌))
15526, 11, 6, 7grpnpncan0 17282 . . . 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 )
156132, 141, 154, 155syl12anc 1315 . . 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 )
157129, 156eqtrd 2643 . 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 )
15812, 114, 1573eqtrd 2647 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  ifcif 4035   class class class wbr 4577  cmpt 4637  wf 5785  cfv 5789  (class class class)co 6526  𝑚 cmap 7721  Fincfn 7818  cc 9790  0cc0 9792  1c1 9793   + caddc 9795   < clt 9930  cmin 10117  cn 10869  0cn0 11141  cz 11212  cuz 11521  ...cfz 12154  Basecbs 15643  +gcplusg 15716  .rcmulr 15717  0gc0g 15871   Σg cgsu 15872  Mgmcmgm 17011  Mndcmnd 17065  Grpcgrp 17193  -gcsg 17195  .gcmg 17311  Abelcabl 17965  mulGrpcmgp 18260  Ringcrg 18318  CRingccrg 18319  Poly1cpl1 19316   Mat cmat 19979   matToPolyMat cmat2pmat 20275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-inf2 8398  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-ot 4133  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-se 4987  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-isom 5798  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-of 6772  df-ofr 6773  df-om 6935  df-1st 7036  df-2nd 7037  df-supp 7160  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-ixp 7772  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-fsupp 8136  df-sup 8208  df-oi 8275  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10870  df-2 10928  df-3 10929  df-4 10930  df-5 10931  df-6 10932  df-7 10933  df-8 10934  df-9 10935  df-n0 11142  df-z 11213  df-dec 11328  df-uz 11522  df-rp 11667  df-fz 12155  df-fzo 12292  df-seq 12621  df-hash 12937  df-struct 15645  df-ndx 15646  df-slot 15647  df-base 15648  df-sets 15649  df-ress 15650  df-plusg 15729  df-mulr 15730  df-sca 15732  df-vsca 15733  df-ip 15734  df-tset 15735  df-ple 15736  df-ds 15739  df-hom 15741  df-cco 15742  df-0g 15873  df-gsum 15874  df-prds 15879  df-pws 15881  df-mre 16017  df-mrc 16018  df-acs 16020  df-mgm 17013  df-sgrp 17055  df-mnd 17066  df-mhm 17106  df-submnd 17107  df-grp 17196  df-minusg 17197  df-sbg 17198  df-mulg 17312  df-subg 17362  df-ghm 17429  df-cntz 17521  df-cmn 17966  df-abl 17967  df-mgp 18261  df-ur 18273  df-ring 18320  df-cring 18321  df-subrg 18549  df-lmod 18636  df-lss 18702  df-sra 18941  df-rgmod 18942  df-ascl 19083  df-psr 19125  df-mpl 19127  df-opsr 19129  df-psr1 19319  df-ply1 19321  df-dsmm 19842  df-frlm 19857  df-mamu 19956  df-mat 19980  df-mat2pmat 20278
This theorem is referenced by:  cayleyhamilton0  20460  cayleyhamiltonALT  20462
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