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Theorem chfacfisfcpmat 20479
Description: The "characteristic factor function" is a function from the nonnegative integers to constant polynomial matrices. (Contributed by AV, 19-Nov-2019.)
Hypotheses
Ref Expression
chfacfisf.a 𝐴 = (𝑁 Mat 𝑅)
chfacfisf.b 𝐵 = (Base‘𝐴)
chfacfisf.p 𝑃 = (Poly1𝑅)
chfacfisf.y 𝑌 = (𝑁 Mat 𝑃)
chfacfisf.r × = (.r𝑌)
chfacfisf.s = (-g𝑌)
chfacfisf.0 0 = (0g𝑌)
chfacfisf.t 𝑇 = (𝑁 matToPolyMat 𝑅)
chfacfisf.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))))))))
chfacfisfcpmat.s 𝑆 = (𝑁 ConstPolyMat 𝑅)
Assertion
Ref Expression
chfacfisfcpmat (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝐺:ℕ0𝑆)
Distinct variable groups:   𝐵,𝑛   𝑛,𝑀   𝑛,𝑁   𝑅,𝑛   𝑛,𝑌   𝑛,𝑏   𝑛,𝑠   𝑆,𝑛
Allowed substitution hints:   𝐴(𝑛,𝑠,𝑏)   𝐵(𝑠,𝑏)   𝑃(𝑛,𝑠,𝑏)   𝑅(𝑠,𝑏)   𝑆(𝑠,𝑏)   𝑇(𝑛,𝑠,𝑏)   × (𝑛,𝑠,𝑏)   𝐺(𝑛,𝑠,𝑏)   𝑀(𝑠,𝑏)   (𝑛,𝑠,𝑏)   𝑁(𝑠,𝑏)   𝑌(𝑠,𝑏)   0 (𝑛,𝑠,𝑏)

Proof of Theorem chfacfisfcpmat
StepHypRef Expression
1 chfacfisfcpmat.s . . . . . . . 8 𝑆 = (𝑁 ConstPolyMat 𝑅)
2 chfacfisf.p . . . . . . . 8 𝑃 = (Poly1𝑅)
3 chfacfisf.y . . . . . . . 8 𝑌 = (𝑁 Mat 𝑃)
41, 2, 3cpmatsubgpmat 20344 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝑌))
543adant3 1074 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝑆 ∈ (SubGrp‘𝑌))
65adantr 480 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑆 ∈ (SubGrp‘𝑌))
7 subgsubm 17439 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝑌) → 𝑆 ∈ (SubMnd‘𝑌))
8 chfacfisf.0 . . . . . . . 8 0 = (0g𝑌)
98subm0cl 17175 . . . . . . 7 (𝑆 ∈ (SubMnd‘𝑌) → 0𝑆)
105, 7, 93syl 18 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → 0𝑆)
1110adantr 480 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 0𝑆)
121, 2, 3cpmatsrgpmat 20345 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝑌))
13123adant3 1074 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝑆 ∈ (SubRing‘𝑌))
1413adantr 480 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑆 ∈ (SubRing‘𝑌))
15 chfacfisf.t . . . . . . . 8 𝑇 = (𝑁 matToPolyMat 𝑅)
16 chfacfisf.a . . . . . . . 8 𝐴 = (𝑁 Mat 𝑅)
17 chfacfisf.b . . . . . . . 8 𝐵 = (Base‘𝐴)
181, 15, 16, 17m2cpm 20365 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑇𝑀) ∈ 𝑆)
1918adantr 480 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑇𝑀) ∈ 𝑆)
20 3simpa 1051 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
21 elmapi 7765 . . . . . . . . . . 11 (𝑏 ∈ (𝐵𝑚 (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
2221adantl 481 . . . . . . . . . 10 ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → 𝑏:(0...𝑠)⟶𝐵)
23 nnnn0 11176 . . . . . . . . . . . . 13 (𝑠 ∈ ℕ → 𝑠 ∈ ℕ0)
24 nn0uz 11598 . . . . . . . . . . . . 13 0 = (ℤ‘0)
2523, 24syl6eleq 2698 . . . . . . . . . . . 12 (𝑠 ∈ ℕ → 𝑠 ∈ (ℤ‘0))
26 eluzfz1 12219 . . . . . . . . . . . 12 (𝑠 ∈ (ℤ‘0) → 0 ∈ (0...𝑠))
2725, 26syl 17 . . . . . . . . . . 11 (𝑠 ∈ ℕ → 0 ∈ (0...𝑠))
2827adantr 480 . . . . . . . . . 10 ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → 0 ∈ (0...𝑠))
2922, 28ffvelrnd 6268 . . . . . . . . 9 ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → (𝑏‘0) ∈ 𝐵)
3020, 29anim12i 588 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑏‘0) ∈ 𝐵))
31 df-3an 1033 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑏‘0) ∈ 𝐵))
3230, 31sylibr 223 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵))
331, 15, 16, 17m2cpm 20365 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵) → (𝑇‘(𝑏‘0)) ∈ 𝑆)
3432, 33syl 17 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑇‘(𝑏‘0)) ∈ 𝑆)
35 chfacfisf.r . . . . . . 7 × = (.r𝑌)
3635subrgmcl 18615 . . . . . 6 ((𝑆 ∈ (SubRing‘𝑌) ∧ (𝑇𝑀) ∈ 𝑆 ∧ (𝑇‘(𝑏‘0)) ∈ 𝑆) → ((𝑇𝑀) × (𝑇‘(𝑏‘0))) ∈ 𝑆)
3714, 19, 34, 36syl3anc 1318 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((𝑇𝑀) × (𝑇‘(𝑏‘0))) ∈ 𝑆)
38 chfacfisf.s . . . . . 6 = (-g𝑌)
3938subgsubcl 17428 . . . . 5 ((𝑆 ∈ (SubGrp‘𝑌) ∧ 0𝑆 ∧ ((𝑇𝑀) × (𝑇‘(𝑏‘0))) ∈ 𝑆) → ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))) ∈ 𝑆)
406, 11, 37, 39syl3anc 1318 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))) ∈ 𝑆)
4140ad2antrr 758 . . 3 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))) ∈ 𝑆)
42 simpl1 1057 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑁 ∈ Fin)
43 simpl2 1058 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑅 ∈ Ring)
4422adantl 481 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑏:(0...𝑠)⟶𝐵)
45 eluzfz2 12220 . . . . . . . . . 10 (𝑠 ∈ (ℤ‘0) → 𝑠 ∈ (0...𝑠))
4625, 45syl 17 . . . . . . . . 9 (𝑠 ∈ ℕ → 𝑠 ∈ (0...𝑠))
4746ad2antrl 760 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝑠 ∈ (0...𝑠))
4844, 47ffvelrnd 6268 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑏𝑠) ∈ 𝐵)
491, 15, 16, 17m2cpm 20365 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏𝑠) ∈ 𝐵) → (𝑇‘(𝑏𝑠)) ∈ 𝑆)
5042, 43, 48, 49syl3anc 1318 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑇‘(𝑏𝑠)) ∈ 𝑆)
5150adantr 480 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑏𝑠)) ∈ 𝑆)
5251ad2antrr 758 . . . 4 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 = (𝑠 + 1)) → (𝑇‘(𝑏𝑠)) ∈ 𝑆)
5311ad4antr 764 . . . . 5 (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ (𝑠 + 1) < 𝑛) → 0𝑆)
54 nn0re 11178 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ0𝑛 ∈ ℝ)
5554adantl 481 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℝ)
56 peano2nn 10909 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℕ)
5756nnred 10912 . . . . . . . . . . . . . . . 16 (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℝ)
5857adantr 480 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑠 + 1) ∈ ℝ)
5955, 58lenltd 10062 . . . . . . . . . . . . . 14 ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑛 ≤ (𝑠 + 1) ↔ ¬ (𝑠 + 1) < 𝑛))
60 nesym 2838 . . . . . . . . . . . . . . . 16 ((𝑠 + 1) ≠ 𝑛 ↔ ¬ 𝑛 = (𝑠 + 1))
61 ltlen 10017 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ) → (𝑛 < (𝑠 + 1) ↔ (𝑛 ≤ (𝑠 + 1) ∧ (𝑠 + 1) ≠ 𝑛)))
6254, 57, 61syl2anr 494 . . . . . . . . . . . . . . . . . 18 ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑛 < (𝑠 + 1) ↔ (𝑛 ≤ (𝑠 + 1) ∧ (𝑠 + 1) ≠ 𝑛)))
6362biimprd 237 . . . . . . . . . . . . . . . . 17 ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → ((𝑛 ≤ (𝑠 + 1) ∧ (𝑠 + 1) ≠ 𝑛) → 𝑛 < (𝑠 + 1)))
6463expcomd 453 . . . . . . . . . . . . . . . 16 ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → ((𝑠 + 1) ≠ 𝑛 → (𝑛 ≤ (𝑠 + 1) → 𝑛 < (𝑠 + 1))))
6560, 64syl5bir 232 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (¬ 𝑛 = (𝑠 + 1) → (𝑛 ≤ (𝑠 + 1) → 𝑛 < (𝑠 + 1))))
6665com23 84 . . . . . . . . . . . . . 14 ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑛 ≤ (𝑠 + 1) → (¬ 𝑛 = (𝑠 + 1) → 𝑛 < (𝑠 + 1))))
6759, 66sylbird 249 . . . . . . . . . . . . 13 ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (¬ (𝑠 + 1) < 𝑛 → (¬ 𝑛 = (𝑠 + 1) → 𝑛 < (𝑠 + 1))))
6867com23 84 . . . . . . . . . . . 12 ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (¬ 𝑛 = (𝑠 + 1) → (¬ (𝑠 + 1) < 𝑛𝑛 < (𝑠 + 1))))
6968impd 446 . . . . . . . . . . 11 ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → ((¬ 𝑛 = (𝑠 + 1) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 < (𝑠 + 1)))
7069ex 449 . . . . . . . . . 10 (𝑠 ∈ ℕ → (𝑛 ∈ ℕ0 → ((¬ 𝑛 = (𝑠 + 1) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 < (𝑠 + 1))))
7170ad2antrl 760 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑛 ∈ ℕ0 → ((¬ 𝑛 = (𝑠 + 1) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 < (𝑠 + 1))))
7271imp 444 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → ((¬ 𝑛 = (𝑠 + 1) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 < (𝑠 + 1)))
7372adantr 480 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → ((¬ 𝑛 = (𝑠 + 1) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 < (𝑠 + 1)))
745ad4antr 764 . . . . . . . . 9 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → 𝑆 ∈ (SubGrp‘𝑌))
7520ad4antr 764 . . . . . . . . . . 11 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
7622ad4antlr 765 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → 𝑏:(0...𝑠)⟶𝐵)
77 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 0 → 𝑛 = 0)
7877necon3bi 2808 . . . . . . . . . . . . . . . . . 18 𝑛 = 0 → 𝑛 ≠ 0)
7978anim2i 591 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → (𝑛 ∈ ℕ0𝑛 ≠ 0))
80 elnnne0 11183 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0𝑛 ≠ 0))
8179, 80sylibr 223 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ)
82 nnm1nn0 11211 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
8381, 82syl 17 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℕ0 ∧ ¬ 𝑛 = 0) → (𝑛 − 1) ∈ ℕ0)
8483adantll 746 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → (𝑛 − 1) ∈ ℕ0)
8584adantr 480 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑛 − 1) ∈ ℕ0)
8623adantr 480 . . . . . . . . . . . . . 14 ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → 𝑠 ∈ ℕ0)
8786ad4antlr 765 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → 𝑠 ∈ ℕ0)
8862simprbda 651 . . . . . . . . . . . . . . . . . . 19 (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑛 ≤ (𝑠 + 1))
8955adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑛 ∈ ℝ)
90 1red 9934 . . . . . . . . . . . . . . . . . . . 20 (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 1 ∈ ℝ)
91 nnre 10904 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ ℕ → 𝑠 ∈ ℝ)
9291ad2antrr 758 . . . . . . . . . . . . . . . . . . . 20 (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑠 ∈ ℝ)
9389, 90, 92lesubaddd 10503 . . . . . . . . . . . . . . . . . . 19 (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → ((𝑛 − 1) ≤ 𝑠𝑛 ≤ (𝑠 + 1)))
9488, 93mpbird 246 . . . . . . . . . . . . . . . . . 18 (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → (𝑛 − 1) ≤ 𝑠)
9594exp31 628 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ ℕ → (𝑛 ∈ ℕ0 → (𝑛 < (𝑠 + 1) → (𝑛 − 1) ≤ 𝑠)))
9695ad2antrl 760 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑛 ∈ ℕ0 → (𝑛 < (𝑠 + 1) → (𝑛 − 1) ≤ 𝑠)))
9796imp 444 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (𝑛 < (𝑠 + 1) → (𝑛 − 1) ≤ 𝑠))
9897adantr 480 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → (𝑛 < (𝑠 + 1) → (𝑛 − 1) ≤ 𝑠))
9998imp 444 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑛 − 1) ≤ 𝑠)
100 elfz2nn0 12300 . . . . . . . . . . . . 13 ((𝑛 − 1) ∈ (0...𝑠) ↔ ((𝑛 − 1) ∈ ℕ0𝑠 ∈ ℕ0 ∧ (𝑛 − 1) ≤ 𝑠))
10185, 87, 99, 100syl3anbrc 1239 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑛 − 1) ∈ (0...𝑠))
10276, 101ffvelrnd 6268 . . . . . . . . . . 11 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑏‘(𝑛 − 1)) ∈ 𝐵)
103 df-3an 1033 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑛 − 1)) ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑏‘(𝑛 − 1)) ∈ 𝐵))
10475, 102, 103sylanbrc 695 . . . . . . . . . 10 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑛 − 1)) ∈ 𝐵))
1051, 15, 16, 17m2cpm 20365 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘(𝑛 − 1)) ∈ 𝐵) → (𝑇‘(𝑏‘(𝑛 − 1))) ∈ 𝑆)
106104, 105syl 17 . . . . . . . . 9 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → (𝑇‘(𝑏‘(𝑛 − 1))) ∈ 𝑆)
10714ad2antrr 758 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑆 ∈ (SubRing‘𝑌))
10819ad2antrr 758 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → (𝑇𝑀) ∈ 𝑆)
10920, 86anim12i 588 . . . . . . . . . . . . . . 15 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑠 ∈ ℕ0))
110 df-3an 1033 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑠 ∈ ℕ0))
111109, 110sylibr 223 . . . . . . . . . . . . . 14 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0))
112111ad2antrr 758 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0))
113112simp1d 1066 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑁 ∈ Fin)
114112simp2d 1067 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑅 ∈ Ring)
11544ad2antrr 758 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑏:(0...𝑠)⟶𝐵)
116 simplr 788 . . . . . . . . . . . . . . . . 17 (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑛 ∈ ℕ0)
11723ad2antrr 758 . . . . . . . . . . . . . . . . 17 (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑠 ∈ ℕ0)
118 nn0z 11277 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ0𝑛 ∈ ℤ)
119 nnz 11276 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ ℕ → 𝑠 ∈ ℤ)
120 zleltp1 11305 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ ℤ ∧ 𝑠 ∈ ℤ) → (𝑛𝑠𝑛 < (𝑠 + 1)))
121118, 119, 120syl2anr 494 . . . . . . . . . . . . . . . . . 18 ((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (𝑛𝑠𝑛 < (𝑠 + 1)))
122121biimpar 501 . . . . . . . . . . . . . . . . 17 (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑛𝑠)
123 elfz2nn0 12300 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (0...𝑠) ↔ (𝑛 ∈ ℕ0𝑠 ∈ ℕ0𝑛𝑠))
124116, 117, 122, 123syl3anbrc 1239 . . . . . . . . . . . . . . . 16 (((𝑠 ∈ ℕ ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑛 ∈ (0...𝑠))
125124exp31 628 . . . . . . . . . . . . . . 15 (𝑠 ∈ ℕ → (𝑛 ∈ ℕ0 → (𝑛 < (𝑠 + 1) → 𝑛 ∈ (0...𝑠))))
126125ad2antrl 760 . . . . . . . . . . . . . 14 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑛 ∈ ℕ0 → (𝑛 < (𝑠 + 1) → 𝑛 ∈ (0...𝑠))))
127126imp31 447 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → 𝑛 ∈ (0...𝑠))
128115, 127ffvelrnd 6268 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → (𝑏𝑛) ∈ 𝐵)
1291, 15, 16, 17m2cpm 20365 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏𝑛) ∈ 𝐵) → (𝑇‘(𝑏𝑛)) ∈ 𝑆)
130113, 114, 128, 129syl3anc 1318 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → (𝑇‘(𝑏𝑛)) ∈ 𝑆)
13135subrgmcl 18615 . . . . . . . . . . 11 ((𝑆 ∈ (SubRing‘𝑌) ∧ (𝑇𝑀) ∈ 𝑆 ∧ (𝑇‘(𝑏𝑛)) ∈ 𝑆) → ((𝑇𝑀) × (𝑇‘(𝑏𝑛))) ∈ 𝑆)
132107, 108, 130, 131syl3anc 1318 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑠 + 1)) → ((𝑇𝑀) × (𝑇‘(𝑏𝑛))) ∈ 𝑆)
133132adantlr 747 . . . . . . . . 9 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → ((𝑇𝑀) × (𝑇‘(𝑏𝑛))) ∈ 𝑆)
13438subgsubcl 17428 . . . . . . . . 9 ((𝑆 ∈ (SubGrp‘𝑌) ∧ (𝑇‘(𝑏‘(𝑛 − 1))) ∈ 𝑆 ∧ ((𝑇𝑀) × (𝑇‘(𝑏𝑛))) ∈ 𝑆) → ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛)))) ∈ 𝑆)
13574, 106, 133, 134syl3anc 1318 . . . . . . . 8 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) → ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛)))) ∈ 𝑆)
136135ex 449 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → (𝑛 < (𝑠 + 1) → ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛)))) ∈ 𝑆))
13773, 136syld 46 . . . . . 6 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → ((¬ 𝑛 = (𝑠 + 1) ∧ ¬ (𝑠 + 1) < 𝑛) → ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛)))) ∈ 𝑆))
138137impl 648 . . . . 5 (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛)))) ∈ 𝑆)
13953, 138ifclda 4070 . . . 4 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) → if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))) ∈ 𝑆)
14052, 139ifclda 4070 . . 3 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛)))))) ∈ 𝑆)
14141, 140ifclda 4070 . 2 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 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))))))) ∈ 𝑆)
142 chfacfisf.g . 2 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))))))))
143141, 142fmptd 6292 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝐺:ℕ0𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  ifcif 4036   class class class wbr 4583  cmpt 4643  wf 5800  cfv 5804  (class class class)co 6549  𝑚 cmap 7744  Fincfn 7841  cr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953  cle 9954  cmin 10145  cn 10897  0cn0 11169  cz 11254  cuz 11563  ...cfz 12197  Basecbs 15695  .rcmulr 15769  0gc0g 15923  SubMndcsubmnd 17157  -gcsg 17247  SubGrpcsubg 17411  Ringcrg 18370  SubRingcsubrg 18599  Poly1cpl1 19368   Mat cmat 20032   ConstPolyMat ccpmat 20327   matToPolyMat cmat2pmat 20328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-hom 15793  df-cco 15794  df-0g 15925  df-gsum 15926  df-prds 15931  df-pws 15933  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-srg 18329  df-ring 18372  df-subrg 18601  df-lmod 18688  df-lss 18754  df-sra 18993  df-rgmod 18994  df-ascl 19135  df-psr 19177  df-mvr 19178  df-mpl 19179  df-opsr 19181  df-psr1 19371  df-vr1 19372  df-ply1 19373  df-coe1 19374  df-dsmm 19895  df-frlm 19910  df-mamu 20009  df-mat 20033  df-cpmat 20330  df-mat2pmat 20331
This theorem is referenced by:  cpmadumatpolylem1  20505  cpmadumatpolylem2  20506  cpmadumatpoly  20507  chcoeffeqlem  20509  cayhamlem4  20512
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