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Theorem chfacfpmmulgsum 20948
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
chfacfpmmulgsum (((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))))))
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 chfacfpmmulgsum
StepHypRef Expression
1 eqid 2765 . . 3 (Base‘𝑌) = (Base‘𝑌)
2 cayhamlem1.0 . . 3 0 = (0g𝑌)
3 chfacfpmmulgsum.p . . 3 + = (+g𝑌)
4 crngring 18825 . . . . . . . 8 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
54anim2i 610 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
653adant3 1162 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
7 cayhamlem1.p . . . . . . 7 𝑃 = (Poly1𝑅)
8 cayhamlem1.y . . . . . . 7 𝑌 = (𝑁 Mat 𝑃)
97, 8pmatring 20777 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
106, 9syl 17 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Ring)
11 ringcmn 18848 . . . . 5 (𝑌 ∈ Ring → 𝑌 ∈ CMnd)
1210, 11syl 17 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ CMnd)
1312adantr 472 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑌 ∈ CMnd)
14 nn0ex 11545 . . . 4 0 ∈ V
1514a1i 11 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ℕ0 ∈ V)
16 simpll 783 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵))
17 simplr 785 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ ℕ0) → (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))))
18 simpr 477 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℕ0)
1916, 17, 183jca 1158 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ ℕ0) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑖 ∈ ℕ0))
20 cayhamlem1.a . . . . 5 𝐴 = (𝑁 Mat 𝑅)
21 cayhamlem1.b . . . . 5 𝐵 = (Base‘𝐴)
22 cayhamlem1.r . . . . 5 × = (.r𝑌)
23 cayhamlem1.s . . . . 5 = (-g𝑌)
24 cayhamlem1.t . . . . 5 𝑇 = (𝑁 matToPolyMat 𝑅)
25 cayhamlem1.g . . . . 5 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))))))))
26 cayhamlem1.e . . . . 5 = (.g‘(mulGrp‘𝑌))
2720, 21, 7, 8, 22, 23, 2, 24, 25, 26chfacfpmmulcl 20945 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑖 ∈ ℕ0) → ((𝑖 (𝑇𝑀)) × (𝐺𝑖)) ∈ (Base‘𝑌))
2819, 27syl 17 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ ℕ0) → ((𝑖 (𝑇𝑀)) × (𝐺𝑖)) ∈ (Base‘𝑌))
2920, 21, 7, 8, 22, 23, 2, 24, 25, 26chfacfpmmulfsupp 20947 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑖 ∈ ℕ0 ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖))) finSupp 0 )
30 nn0disj 12663 . . . 4 ((0...(𝑠 + 1)) ∩ (ℤ‘((𝑠 + 1) + 1))) = ∅
3130a1i 11 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((0...(𝑠 + 1)) ∩ (ℤ‘((𝑠 + 1) + 1))) = ∅)
32 nnnn0 11546 . . . . . 6 (𝑠 ∈ ℕ → 𝑠 ∈ ℕ0)
33 peano2nn0 11580 . . . . . 6 (𝑠 ∈ ℕ0 → (𝑠 + 1) ∈ ℕ0)
3432, 33syl 17 . . . . 5 (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℕ0)
35 nn0split 12662 . . . . 5 ((𝑠 + 1) ∈ ℕ0 → ℕ0 = ((0...(𝑠 + 1)) ∪ (ℤ‘((𝑠 + 1) + 1))))
3634, 35syl 17 . . . 4 (𝑠 ∈ ℕ → ℕ0 = ((0...(𝑠 + 1)) ∪ (ℤ‘((𝑠 + 1) + 1))))
3736ad2antrl 719 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ℕ0 = ((0...(𝑠 + 1)) ∪ (ℤ‘((𝑠 + 1) + 1))))
381, 2, 3, 13, 15, 28, 29, 31, 37gsumsplit2 18595 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) + (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖))))))
39 simpll 783 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (ℤ‘((𝑠 + 1) + 1))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵))
40 simplr 785 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (ℤ‘((𝑠 + 1) + 1))) → (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))))
41 nncn 11283 . . . . . . . . . . . . 13 (𝑠 ∈ ℕ → 𝑠 ∈ ℂ)
42 add1p1 11529 . . . . . . . . . . . . 13 (𝑠 ∈ ℂ → ((𝑠 + 1) + 1) = (𝑠 + 2))
4341, 42syl 17 . . . . . . . . . . . 12 (𝑠 ∈ ℕ → ((𝑠 + 1) + 1) = (𝑠 + 2))
4443ad2antrl 719 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((𝑠 + 1) + 1) = (𝑠 + 2))
4544fveq2d 6379 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (ℤ‘((𝑠 + 1) + 1)) = (ℤ‘(𝑠 + 2)))
4645eleq2d 2830 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↔ 𝑖 ∈ (ℤ‘(𝑠 + 2))))
4746biimpa 468 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (ℤ‘((𝑠 + 1) + 1))) → 𝑖 ∈ (ℤ‘(𝑠 + 2)))
4820, 21, 7, 8, 22, 23, 2, 24, 25, 26chfacfpmmul0 20946 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑖 ∈ (ℤ‘(𝑠 + 2))) → ((𝑖 (𝑇𝑀)) × (𝐺𝑖)) = 0 )
4939, 40, 47, 48syl3anc 1490 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (ℤ‘((𝑠 + 1) + 1))) → ((𝑖 (𝑇𝑀)) × (𝐺𝑖)) = 0 )
5049mpteq2dva 4903 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖))) = (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ 0 ))
5150oveq2d 6858 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) = (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ 0 )))
524, 9sylan2 586 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
53 ringmnd 18823 . . . . . . . . . 10 (𝑌 ∈ Ring → 𝑌 ∈ Mnd)
5452, 53syl 17 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Mnd)
55543adant3 1162 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Mnd)
56 fvex 6388 . . . . . . . 8 (ℤ‘((𝑠 + 1) + 1)) ∈ V
5755, 56jctir 516 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑌 ∈ Mnd ∧ (ℤ‘((𝑠 + 1) + 1)) ∈ V))
5857adantr 472 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 ∈ Mnd ∧ (ℤ‘((𝑠 + 1) + 1)) ∈ V))
592gsumz 17640 . . . . . 6 ((𝑌 ∈ Mnd ∧ (ℤ‘((𝑠 + 1) + 1)) ∈ V) → (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ 0 )) = 0 )
6058, 59syl 17 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ 0 )) = 0 )
6151, 60eqtrd 2799 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) = 0 )
6261oveq2d 6858 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) + (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖))))) = ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) + 0 ))
6355adantr 472 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑌 ∈ Mnd)
64 fzfid 12980 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (0...(𝑠 + 1)) ∈ Fin)
65 elfznn0 12640 . . . . . . . 8 (𝑖 ∈ (0...(𝑠 + 1)) → 𝑖 ∈ ℕ0)
6665, 19sylan2 586 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...(𝑠 + 1))) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝑖 ∈ ℕ0))
6766, 27syl 17 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...(𝑠 + 1))) → ((𝑖 (𝑇𝑀)) × (𝐺𝑖)) ∈ (Base‘𝑌))
6867ralrimiva 3113 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ∀𝑖 ∈ (0...(𝑠 + 1))((𝑖 (𝑇𝑀)) × (𝐺𝑖)) ∈ (Base‘𝑌))
691, 13, 64, 68gsummptcl 18632 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) ∈ (Base‘𝑌))
701, 3, 2mndrid 17578 . . . 4 ((𝑌 ∈ Mnd ∧ (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) ∈ (Base‘𝑌)) → ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) + 0 ) = (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))))
7163, 69, 70syl2anc 579 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) + 0 ) = (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))))
7262, 71eqtrd 2799 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) + (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖))))) = (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))))
7332ad2antrl 719 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑠 ∈ ℕ0)
741, 3, 13, 73, 67gsummptfzsplit 18598 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) + (𝑌 Σg (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖))))))
75 elfznn0 12640 . . . . . . 7 (𝑖 ∈ (0...𝑠) → 𝑖 ∈ ℕ0)
7675, 28sylan2 586 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 (𝑇𝑀)) × (𝐺𝑖)) ∈ (Base‘𝑌))
771, 3, 13, 73, 76gsummptfzsplitl 18599 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) + (𝑌 Σg (𝑖 ∈ {0} ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖))))))
78 0nn0 11555 . . . . . . . 8 0 ∈ ℕ0
7978a1i 11 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 0 ∈ ℕ0)
8020, 21, 7, 8, 22, 23, 2, 24, 25, 26chfacfpmmulcl 20945 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 0 ∈ ℕ0) → ((0 (𝑇𝑀)) × (𝐺‘0)) ∈ (Base‘𝑌))
8179, 80mpd3an3 1586 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((0 (𝑇𝑀)) × (𝐺‘0)) ∈ (Base‘𝑌))
82 oveq1 6849 . . . . . . . . 9 (𝑖 = 0 → (𝑖 (𝑇𝑀)) = (0 (𝑇𝑀)))
83 fveq2 6375 . . . . . . . . 9 (𝑖 = 0 → (𝐺𝑖) = (𝐺‘0))
8482, 83oveq12d 6860 . . . . . . . 8 (𝑖 = 0 → ((𝑖 (𝑇𝑀)) × (𝐺𝑖)) = ((0 (𝑇𝑀)) × (𝐺‘0)))
851, 84gsumsn 18620 . . . . . . 7 ((𝑌 ∈ Mnd ∧ 0 ∈ ℕ0 ∧ ((0 (𝑇𝑀)) × (𝐺‘0)) ∈ (Base‘𝑌)) → (𝑌 Σg (𝑖 ∈ {0} ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) = ((0 (𝑇𝑀)) × (𝐺‘0)))
8663, 79, 81, 85syl3anc 1490 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ {0} ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) = ((0 (𝑇𝑀)) × (𝐺‘0)))
8786oveq2d 6858 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) + (𝑌 Σg (𝑖 ∈ {0} ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖))))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) + ((0 (𝑇𝑀)) × (𝐺‘0))))
8877, 87eqtrd 2799 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) + ((0 (𝑇𝑀)) × (𝐺‘0))))
89 ovexd 6876 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑠 + 1) ∈ V)
90 1nn0 11556 . . . . . . . 8 1 ∈ ℕ0
9190a1i 11 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 1 ∈ ℕ0)
9273, 91nn0addcld 11602 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑠 + 1) ∈ ℕ0)
9320, 21, 7, 8, 22, 23, 2, 24, 25, 26chfacfpmmulcl 20945 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ (𝑠 + 1) ∈ ℕ0) → (((𝑠 + 1) (𝑇𝑀)) × (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌))
9492, 93mpd3an3 1586 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (((𝑠 + 1) (𝑇𝑀)) × (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌))
95 oveq1 6849 . . . . . . 7 (𝑖 = (𝑠 + 1) → (𝑖 (𝑇𝑀)) = ((𝑠 + 1) (𝑇𝑀)))
96 fveq2 6375 . . . . . . 7 (𝑖 = (𝑠 + 1) → (𝐺𝑖) = (𝐺‘(𝑠 + 1)))
9795, 96oveq12d 6860 . . . . . 6 (𝑖 = (𝑠 + 1) → ((𝑖 (𝑇𝑀)) × (𝐺𝑖)) = (((𝑠 + 1) (𝑇𝑀)) × (𝐺‘(𝑠 + 1))))
981, 97gsumsn 18620 . . . . 5 ((𝑌 ∈ Mnd ∧ (𝑠 + 1) ∈ V ∧ (((𝑠 + 1) (𝑇𝑀)) × (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌)) → (𝑌 Σg (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) = (((𝑠 + 1) (𝑇𝑀)) × (𝐺‘(𝑠 + 1))))
9963, 89, 94, 98syl3anc 1490 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) = (((𝑠 + 1) (𝑇𝑀)) × (𝐺‘(𝑠 + 1))))
10088, 99oveq12d 6860 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) + (𝑌 Σg (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖))))) = (((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) + ((0 (𝑇𝑀)) × (𝐺‘0))) + (((𝑠 + 1) (𝑇𝑀)) × (𝐺‘(𝑠 + 1)))))
101 fzfid 12980 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (1...𝑠) ∈ Fin)
102 simpll 783 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵))
103 simplr 785 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))))
104 elfznn 12577 . . . . . . . . . 10 (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℕ)
105104nnnn0d 11598 . . . . . . . . 9 (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℕ0)
106105adantl 473 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑖 ∈ ℕ0)
107102, 103, 106, 27syl3anc 1490 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 (𝑇𝑀)) × (𝐺𝑖)) ∈ (Base‘𝑌))
108107ralrimiva 3113 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ∀𝑖 ∈ (1...𝑠)((𝑖 (𝑇𝑀)) × (𝐺𝑖)) ∈ (Base‘𝑌))
1091, 13, 101, 108gsummptcl 18632 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) ∈ (Base‘𝑌))
1101, 3mndass 17568 . . . . 5 ((𝑌 ∈ Mnd ∧ ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) ∈ (Base‘𝑌) ∧ ((0 (𝑇𝑀)) × (𝐺‘0)) ∈ (Base‘𝑌) ∧ (((𝑠 + 1) (𝑇𝑀)) × (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌))) → (((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) + ((0 (𝑇𝑀)) × (𝐺‘0))) + (((𝑠 + 1) (𝑇𝑀)) × (𝐺‘(𝑠 + 1)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) + (((0 (𝑇𝑀)) × (𝐺‘0)) + (((𝑠 + 1) (𝑇𝑀)) × (𝐺‘(𝑠 + 1))))))
11163, 109, 81, 94, 110syl13anc 1491 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) + ((0 (𝑇𝑀)) × (𝐺‘0))) + (((𝑠 + 1) (𝑇𝑀)) × (𝐺‘(𝑠 + 1)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) + (((0 (𝑇𝑀)) × (𝐺‘0)) + (((𝑠 + 1) (𝑇𝑀)) × (𝐺‘(𝑠 + 1))))))
11225a1i 11 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 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)))))))))
113104nnne0d 11322 . . . . . . . . . . . . . 14 (𝑖 ∈ (1...𝑠) → 𝑖 ≠ 0)
114113ad2antlr 718 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑖 ≠ 0)
115 neeq1 2999 . . . . . . . . . . . . . 14 (𝑛 = 𝑖 → (𝑛 ≠ 0 ↔ 𝑖 ≠ 0))
116115adantl 473 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑛 ≠ 0 ↔ 𝑖 ≠ 0))
117114, 116mpbird 248 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 ≠ 0)
118 eqneqall 2948 . . . . . . . . . . . 12 (𝑛 = 0 → (𝑛 ≠ 0 → 0 = (𝑇‘(𝑏‘(𝑖 − 1)))))
119117, 118mpan9 502 . . . . . . . . . . 11 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → 0 = (𝑇‘(𝑏‘(𝑖 − 1))))
120 simplr 785 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → 𝑛 = 𝑖)
121 eqeq1 2769 . . . . . . . . . . . . . . . . 17 (0 = 𝑛 → (0 = 𝑖𝑛 = 𝑖))
122121eqcoms 2773 . . . . . . . . . . . . . . . 16 (𝑛 = 0 → (0 = 𝑖𝑛 = 𝑖))
123122adantl 473 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → (0 = 𝑖𝑛 = 𝑖))
124120, 123mpbird 248 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → 0 = 𝑖)
125124fveq2d 6379 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → (𝑏‘0) = (𝑏𝑖))
126125fveq2d 6379 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → (𝑇‘(𝑏‘0)) = (𝑇‘(𝑏𝑖)))
127126oveq2d 6858 . . . . . . . . . . 11 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → ((𝑇𝑀) × (𝑇‘(𝑏‘0))) = ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))
128119, 127oveq12d 6860 . . . . . . . . . 10 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
129 elfz2 12540 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (1...𝑠) ↔ ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖𝑖𝑠)))
130 zleltp1 11675 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ ℤ ∧ 𝑠 ∈ ℤ) → (𝑖𝑠𝑖 < (𝑠 + 1)))
131130ancoms 450 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖𝑠𝑖 < (𝑠 + 1)))
1321313adant1 1160 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖𝑠𝑖 < (𝑠 + 1)))
133132biimpcd 240 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖𝑠 → ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → 𝑖 < (𝑠 + 1)))
134133adantl 473 . . . . . . . . . . . . . . . . . . . . 21 ((1 ≤ 𝑖𝑖𝑠) → ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → 𝑖 < (𝑠 + 1)))
135134impcom 396 . . . . . . . . . . . . . . . . . . . 20 (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖𝑖𝑠)) → 𝑖 < (𝑠 + 1))
136135orcd 899 . . . . . . . . . . . . . . . . . . 19 (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖𝑖𝑠)) → (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖))
137 zre 11628 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑠 ∈ ℤ → 𝑠 ∈ ℝ)
138 1red 10294 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑠 ∈ ℤ → 1 ∈ ℝ)
139137, 138readdcld 10323 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠 ∈ ℤ → (𝑠 + 1) ∈ ℝ)
140 zre 11628 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
141139, 140anim12ci 607 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ))
1421413adant1 1160 . . . . . . . . . . . . . . . . . . . . 21 ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ))
143 lttri2 10374 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ) → (𝑖 ≠ (𝑠 + 1) ↔ (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖)))
144142, 143syl 17 . . . . . . . . . . . . . . . . . . . 20 ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ≠ (𝑠 + 1) ↔ (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖)))
145144adantr 472 . . . . . . . . . . . . . . . . . . 19 (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖𝑖𝑠)) → (𝑖 ≠ (𝑠 + 1) ↔ (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖)))
146136, 145mpbird 248 . . . . . . . . . . . . . . . . . 18 (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖𝑖𝑠)) → 𝑖 ≠ (𝑠 + 1))
147129, 146sylbi 208 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...𝑠) → 𝑖 ≠ (𝑠 + 1))
148147ad2antlr 718 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑖 ≠ (𝑠 + 1))
149 neeq1 2999 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑖 → (𝑛 ≠ (𝑠 + 1) ↔ 𝑖 ≠ (𝑠 + 1)))
150149adantl 473 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑛 ≠ (𝑠 + 1) ↔ 𝑖 ≠ (𝑠 + 1)))
151148, 150mpbird 248 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 ≠ (𝑠 + 1))
152151adantr 472 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → 𝑛 ≠ (𝑠 + 1))
153152neneqd 2942 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → ¬ 𝑛 = (𝑠 + 1))
154153pm2.21d 119 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → (𝑛 = (𝑠 + 1) → (𝑇‘(𝑏𝑠)) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))
155154imp 395 . . . . . . . . . . 11 (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ 𝑛 = (𝑠 + 1)) → (𝑇‘(𝑏𝑠)) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
156104nnred 11291 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℝ)
157 eleq1w 2827 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑖 → (𝑛 ∈ ℝ ↔ 𝑖 ∈ ℝ))
158156, 157syl5ibrcom 238 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (1...𝑠) → (𝑛 = 𝑖𝑛 ∈ ℝ))
159158adantl 473 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑛 = 𝑖𝑛 ∈ ℝ))
160159imp 395 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 ∈ ℝ)
16173nn0red 11599 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑠 ∈ ℝ)
162161ad2antrr 717 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑠 ∈ ℝ)
163 1red 10294 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 1 ∈ ℝ)
164162, 163readdcld 10323 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑠 + 1) ∈ ℝ)
165129, 135sylbi 208 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (1...𝑠) → 𝑖 < (𝑠 + 1))
166165ad2antlr 718 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑖 < (𝑠 + 1))
167 breq1 4812 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑖 → (𝑛 < (𝑠 + 1) ↔ 𝑖 < (𝑠 + 1)))
168167adantl 473 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑛 < (𝑠 + 1) ↔ 𝑖 < (𝑠 + 1)))
169166, 168mpbird 248 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 < (𝑠 + 1))
170160, 164, 169ltnsymd 10440 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → ¬ (𝑠 + 1) < 𝑛)
171170pm2.21d 119 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → ((𝑠 + 1) < 𝑛0 = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))
172171ad2antrr 717 . . . . . . . . . . . . 13 (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) → ((𝑠 + 1) < 𝑛0 = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))
173172imp 395 . . . . . . . . . . . 12 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ (𝑠 + 1) < 𝑛) → 0 = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
174 simp-4r 803 . . . . . . . . . . . . . . 15 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 = 𝑖)
175174fvoveq1d 6864 . . . . . . . . . . . . . 14 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑏‘(𝑛 − 1)) = (𝑏‘(𝑖 − 1)))
176175fveq2d 6379 . . . . . . . . . . . . 13 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑇‘(𝑏‘(𝑛 − 1))) = (𝑇‘(𝑏‘(𝑖 − 1))))
177174fveq2d 6379 . . . . . . . . . . . . . . 15 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑏𝑛) = (𝑏𝑖))
178177fveq2d 6379 . . . . . . . . . . . . . 14 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑇‘(𝑏𝑛)) = (𝑇‘(𝑏𝑖)))
179178oveq2d 6858 . . . . . . . . . . . . 13 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → ((𝑇𝑀) × (𝑇‘(𝑏𝑛))) = ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))
180176, 179oveq12d 6860 . . . . . . . . . . . 12 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛)))) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
181173, 180ifeqda 4278 . . . . . . . . . . 11 (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) → if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
182155, 181ifeqda 4278 . . . . . . . . . 10 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛)))))) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
183128, 182ifeqda 4278 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 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))))))) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
184 ovexd 6876 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))) ∈ V)
185112, 183, 106, 184fvmptd 6477 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝐺𝑖) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
186185oveq2d 6858 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 (𝑇𝑀)) × (𝐺𝑖)) = ((𝑖 (𝑇𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))
187186mpteq2dva 4903 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖))) = (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))))
188187oveq2d 6858 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) = (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))))
18925a1i 11 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 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)))))))))
190 nn0p1gt0 11569 . . . . . . . . . . . . . . 15 (𝑠 ∈ ℕ0 → 0 < (𝑠 + 1))
191 0red 10297 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ ℕ0 → 0 ∈ ℝ)
192 ltne 10388 . . . . . . . . . . . . . . . . 17 ((0 ∈ ℝ ∧ 0 < (𝑠 + 1)) → (𝑠 + 1) ≠ 0)
193191, 192sylan 575 . . . . . . . . . . . . . . . 16 ((𝑠 ∈ ℕ0 ∧ 0 < (𝑠 + 1)) → (𝑠 + 1) ≠ 0)
194 neeq1 2999 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑠 + 1) → (𝑛 ≠ 0 ↔ (𝑠 + 1) ≠ 0))
195193, 194syl5ibrcom 238 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℕ0 ∧ 0 < (𝑠 + 1)) → (𝑛 = (𝑠 + 1) → 𝑛 ≠ 0))
196190, 195mpdan 678 . . . . . . . . . . . . . 14 (𝑠 ∈ ℕ0 → (𝑛 = (𝑠 + 1) → 𝑛 ≠ 0))
19732, 196syl 17 . . . . . . . . . . . . 13 (𝑠 ∈ ℕ → (𝑛 = (𝑠 + 1) → 𝑛 ≠ 0))
198197ad2antrl 719 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑛 = (𝑠 + 1) → 𝑛 ≠ 0))
199198imp 395 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) → 𝑛 ≠ 0)
200 eqneqall 2948 . . . . . . . . . . 11 (𝑛 = 0 → (𝑛 ≠ 0 → ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))) = (𝑇‘(𝑏𝑠))))
201199, 200mpan9 502 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) ∧ 𝑛 = 0) → ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))) = (𝑇‘(𝑏𝑠)))
202 iftrue 4249 . . . . . . . . . . 11 (𝑛 = (𝑠 + 1) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛)))))) = (𝑇‘(𝑏𝑠)))
203202ad2antlr 718 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) ∧ ¬ 𝑛 = 0) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛)))))) = (𝑇‘(𝑏𝑠)))
204201, 203ifeqda 4278 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) → 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))))))) = (𝑇‘(𝑏𝑠)))
20573, 33syl 17 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑠 + 1) ∈ ℕ0)
206 fvexd 6390 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑇‘(𝑏𝑠)) ∈ V)
207189, 204, 205, 206fvmptd 6477 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝐺‘(𝑠 + 1)) = (𝑇‘(𝑏𝑠)))
208207oveq2d 6858 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (((𝑠 + 1) (𝑇𝑀)) × (𝐺‘(𝑠 + 1))) = (((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))))
20924, 20, 21, 7, 8mat2pmatbas 20810 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑇𝑀) ∈ (Base‘𝑌))
2104, 209syl3an2 1203 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑇𝑀) ∈ (Base‘𝑌))
211 eqid 2765 . . . . . . . . . . . . . 14 (mulGrp‘𝑌) = (mulGrp‘𝑌)
212211, 1mgpbas 18762 . . . . . . . . . . . . 13 (Base‘𝑌) = (Base‘(mulGrp‘𝑌))
213 eqid 2765 . . . . . . . . . . . . 13 (0g‘(mulGrp‘𝑌)) = (0g‘(mulGrp‘𝑌))
214212, 213, 26mulg0 17813 . . . . . . . . . . . 12 ((𝑇𝑀) ∈ (Base‘𝑌) → (0 (𝑇𝑀)) = (0g‘(mulGrp‘𝑌)))
215210, 214syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (0 (𝑇𝑀)) = (0g‘(mulGrp‘𝑌)))
216 eqid 2765 . . . . . . . . . . . 12 (1r𝑌) = (1r𝑌)
217211, 216ringidval 18770 . . . . . . . . . . 11 (1r𝑌) = (0g‘(mulGrp‘𝑌))
218215, 217syl6eqr 2817 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (0 (𝑇𝑀)) = (1r𝑌))
219218adantr 472 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (0 (𝑇𝑀)) = (1r𝑌))
220219oveq1d 6857 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((0 (𝑇𝑀)) × (𝐺‘0)) = ((1r𝑌) × (𝐺‘0)))
221523adant3 1162 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Ring)
222221adantr 472 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑌 ∈ Ring)
22320, 21, 7, 8, 22, 23, 2, 24, 25chfacfisf 20938 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝐺:ℕ0⟶(Base‘𝑌))
2244, 223syl3anl2 1534 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝐺:ℕ0⟶(Base‘𝑌))
225224, 79ffvelrnd 6550 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝐺‘0) ∈ (Base‘𝑌))
2261, 22, 216ringlidm 18838 . . . . . . . . 9 ((𝑌 ∈ Ring ∧ (𝐺‘0) ∈ (Base‘𝑌)) → ((1r𝑌) × (𝐺‘0)) = (𝐺‘0))
227222, 225, 226syl2anc 579 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((1r𝑌) × (𝐺‘0)) = (𝐺‘0))
228 iftrue 4249 . . . . . . . . . 10 (𝑛 = 0 → 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))))))) = ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))))
229228adantl 473 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 = 0) → 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))))))) = ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))))
230 ovexd 6876 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))) ∈ V)
231189, 229, 79, 230fvmptd 6477 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝐺‘0) = ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))))
232220, 227, 2313eqtrd 2803 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((0 (𝑇𝑀)) × (𝐺‘0)) = ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))))
233208, 232oveq12d 6860 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((((𝑠 + 1) (𝑇𝑀)) × (𝐺‘(𝑠 + 1))) + ((0 (𝑇𝑀)) × (𝐺‘0))) = ((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) + ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
2341, 3cmncom 18475 . . . . . . 7 ((𝑌 ∈ CMnd ∧ ((0 (𝑇𝑀)) × (𝐺‘0)) ∈ (Base‘𝑌) ∧ (((𝑠 + 1) (𝑇𝑀)) × (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌)) → (((0 (𝑇𝑀)) × (𝐺‘0)) + (((𝑠 + 1) (𝑇𝑀)) × (𝐺‘(𝑠 + 1)))) = ((((𝑠 + 1) (𝑇𝑀)) × (𝐺‘(𝑠 + 1))) + ((0 (𝑇𝑀)) × (𝐺‘0))))
23513, 81, 94, 234syl3anc 1490 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (((0 (𝑇𝑀)) × (𝐺‘0)) + (((𝑠 + 1) (𝑇𝑀)) × (𝐺‘(𝑠 + 1)))) = ((((𝑠 + 1) (𝑇𝑀)) × (𝐺‘(𝑠 + 1))) + ((0 (𝑇𝑀)) × (𝐺‘0))))
236 ringgrp 18819 . . . . . . . . 9 (𝑌 ∈ Ring → 𝑌 ∈ Grp)
23710, 236syl 17 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Grp)
238237adantr 472 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑌 ∈ Grp)
239208, 94eqeltrrd 2845 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ∈ (Base‘𝑌))
24010adantr 472 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑌 ∈ Ring)
241210adantr 472 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑇𝑀) ∈ (Base‘𝑌))
242 simpl1 1242 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑁 ∈ Fin)
24343ad2ant2 1164 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
244243adantr 472 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑅 ∈ Ring)
245 elmapi 8082 . . . . . . . . . . . 12 (𝑏 ∈ (𝐵𝑚 (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
246245adantl 473 . . . . . . . . . . 11 ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → 𝑏:(0...𝑠)⟶𝐵)
247246adantl 473 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑏:(0...𝑠)⟶𝐵)
248 0elfz 12644 . . . . . . . . . . . 12 (𝑠 ∈ ℕ0 → 0 ∈ (0...𝑠))
24932, 248syl 17 . . . . . . . . . . 11 (𝑠 ∈ ℕ → 0 ∈ (0...𝑠))
250249ad2antrl 719 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 0 ∈ (0...𝑠))
251247, 250ffvelrnd 6550 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑏‘0) ∈ 𝐵)
25224, 20, 21, 7, 8mat2pmatbas 20810 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
253242, 244, 251, 252syl3anc 1490 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
2541, 22ringcl 18828 . . . . . . . 8 ((𝑌 ∈ Ring ∧ (𝑇𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌)) → ((𝑇𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
255240, 241, 253, 254syl3anc 1490 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((𝑇𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
2561, 2, 23, 3grpsubadd0sub 17769 . . . . . . 7 ((𝑌 ∈ Grp ∧ (((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ∈ (Base‘𝑌) ∧ ((𝑇𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌)) → ((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0)))) = ((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) + ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
257238, 239, 255, 256syl3anc 1490 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0)))) = ((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) + ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
258233, 235, 2573eqtr4d 2809 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (((0 (𝑇𝑀)) × (𝐺‘0)) + (((𝑠 + 1) (𝑇𝑀)) × (𝐺‘(𝑠 + 1)))) = ((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0)))))
259188, 258oveq12d 6860 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) + (((0 (𝑇𝑀)) × (𝐺‘0)) + (((𝑠 + 1) (𝑇𝑀)) × (𝐺‘(𝑠 + 1))))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) + ((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
260111, 259eqtrd 2799 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) + ((0 (𝑇𝑀)) × (𝐺‘0))) + (((𝑠 + 1) (𝑇𝑀)) × (𝐺‘(𝑠 + 1)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) + ((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
26174, 100, 2603eqtrd 2803 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((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))))))
26238, 72, 2613eqtrd 2803 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))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) + ((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wne 2937  Vcvv 3350  cun 3730  cin 3731  c0 4079  ifcif 4243  {csn 4334   class class class wbr 4809  cmpt 4888  wf 6064  cfv 6068  (class class class)co 6842  𝑚 cmap 8060  Fincfn 8160  cc 10187  cr 10188  0cc0 10189  1c1 10190   + caddc 10192   < clt 10328  cle 10329  cmin 10520  cn 11274  2c2 11327  0cn0 11538  cz 11624  cuz 11886  ...cfz 12533  Basecbs 16130  +gcplusg 16214  .rcmulr 16215  0gc0g 16366   Σg cgsu 16367  Mndcmnd 17560  Grpcgrp 17689  -gcsg 17691  .gcmg 17807  CMndccmn 18459  mulGrpcmgp 18756  1rcur 18768  Ringcrg 18814  CRingccrg 18815  Poly1cpl1 19820   Mat cmat 20489   matToPolyMat cmat2pmat 20788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-ot 4343  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-ofr 7096  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-sup 8555  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-dec 11741  df-uz 11887  df-rp 12029  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-struct 16132  df-ndx 16133  df-slot 16134  df-base 16136  df-sets 16137  df-ress 16138  df-plusg 16227  df-mulr 16228  df-sca 16230  df-vsca 16231  df-ip 16232  df-tset 16233  df-ple 16234  df-ds 16236  df-hom 16238  df-cco 16239  df-0g 16368  df-gsum 16369  df-prds 16374  df-pws 16376  df-mre 16512  df-mrc 16513  df-acs 16515  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-mhm 17601  df-submnd 17602  df-grp 17692  df-minusg 17693  df-sbg 17694  df-mulg 17808  df-subg 17855  df-ghm 17922  df-cntz 18013  df-cmn 18461  df-abl 18462  df-mgp 18757  df-ur 18769  df-ring 18816  df-cring 18817  df-subrg 19047  df-lmod 19134  df-lss 19202  df-sra 19446  df-rgmod 19447  df-ascl 19588  df-psr 19630  df-mpl 19632  df-opsr 19634  df-psr1 19823  df-ply1 19825  df-dsmm 20352  df-frlm 20367  df-mamu 20466  df-mat 20490  df-mat2pmat 20791
This theorem is referenced by:  chfacfpmmulgsum2  20949
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