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Theorem chfacfisfcpmat 22348
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 𝐡 = (Baseβ€˜π΄)
chfacfisf.p 𝑃 = (Poly1β€˜π‘…)
chfacfisf.y π‘Œ = (𝑁 Mat 𝑃)
chfacfisf.r Γ— = (.rβ€˜π‘Œ)
chfacfisf.s βˆ’ = (-gβ€˜π‘Œ)
chfacfisf.0 0 = (0gβ€˜π‘Œ)
chfacfisf.t 𝑇 = (𝑁 matToPolyMat 𝑅)
chfacfisf.g 𝐺 = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))
chfacfisfcpmat.s 𝑆 = (𝑁 ConstPolyMat 𝑅)
Assertion
Ref Expression
chfacfisfcpmat (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ 𝐺:β„•0βŸΆπ‘†)
Distinct variable groups:   𝐡,𝑛   𝑛,𝑀   𝑛,𝑁   𝑅,𝑛   𝑛,π‘Œ   𝑛,𝑏   𝑛,𝑠   𝑆,𝑛
Allowed substitution hints:   𝐴(𝑛,𝑠,𝑏)   𝐡(𝑠,𝑏)   𝑃(𝑛,𝑠,𝑏)   𝑅(𝑠,𝑏)   𝑆(𝑠,𝑏)   𝑇(𝑛,𝑠,𝑏)   Γ— (𝑛,𝑠,𝑏)   𝐺(𝑛,𝑠,𝑏)   𝑀(𝑠,𝑏)   βˆ’ (𝑛,𝑠,𝑏)   𝑁(𝑠,𝑏)   π‘Œ(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 = (Poly1β€˜π‘…)
3 chfacfisf.y . . . . . . . 8 π‘Œ = (𝑁 Mat 𝑃)
41, 2, 3cpmatsubgpmat 22213 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ 𝑆 ∈ (SubGrpβ€˜π‘Œ))
543adant3 1132 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) β†’ 𝑆 ∈ (SubGrpβ€˜π‘Œ))
65adantr 481 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ 𝑆 ∈ (SubGrpβ€˜π‘Œ))
7 subgsubm 19022 . . . . . . 7 (𝑆 ∈ (SubGrpβ€˜π‘Œ) β†’ 𝑆 ∈ (SubMndβ€˜π‘Œ))
8 chfacfisf.0 . . . . . . . 8 0 = (0gβ€˜π‘Œ)
98subm0cl 18688 . . . . . . 7 (𝑆 ∈ (SubMndβ€˜π‘Œ) β†’ 0 ∈ 𝑆)
105, 7, 93syl 18 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) β†’ 0 ∈ 𝑆)
1110adantr 481 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ 0 ∈ 𝑆)
121, 2, 3cpmatsrgpmat 22214 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ 𝑆 ∈ (SubRingβ€˜π‘Œ))
13123adant3 1132 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) β†’ 𝑆 ∈ (SubRingβ€˜π‘Œ))
1413adantr 481 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ 𝑆 ∈ (SubRingβ€˜π‘Œ))
15 chfacfisf.t . . . . . . . 8 𝑇 = (𝑁 matToPolyMat 𝑅)
16 chfacfisf.a . . . . . . . 8 𝐴 = (𝑁 Mat 𝑅)
17 chfacfisf.b . . . . . . . 8 𝐡 = (Baseβ€˜π΄)
181, 15, 16, 17m2cpm 22234 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) β†’ (π‘‡β€˜π‘€) ∈ 𝑆)
1918adantr 481 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ (π‘‡β€˜π‘€) ∈ 𝑆)
20 3simpa 1148 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) β†’ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
21 elmapi 8839 . . . . . . . . . . 11 (𝑏 ∈ (𝐡 ↑m (0...𝑠)) β†’ 𝑏:(0...𝑠)⟢𝐡)
2221adantl 482 . . . . . . . . . 10 ((𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠))) β†’ 𝑏:(0...𝑠)⟢𝐡)
23 nnnn0 12475 . . . . . . . . . . . . 13 (𝑠 ∈ β„• β†’ 𝑠 ∈ β„•0)
24 nn0uz 12860 . . . . . . . . . . . . 13 β„•0 = (β„€β‰₯β€˜0)
2523, 24eleqtrdi 2843 . . . . . . . . . . . 12 (𝑠 ∈ β„• β†’ 𝑠 ∈ (β„€β‰₯β€˜0))
26 eluzfz1 13504 . . . . . . . . . . . 12 (𝑠 ∈ (β„€β‰₯β€˜0) β†’ 0 ∈ (0...𝑠))
2725, 26syl 17 . . . . . . . . . . 11 (𝑠 ∈ β„• β†’ 0 ∈ (0...𝑠))
2827adantr 481 . . . . . . . . . 10 ((𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠))) β†’ 0 ∈ (0...𝑠))
2922, 28ffvelcdmd 7084 . . . . . . . . 9 ((𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠))) β†’ (π‘β€˜0) ∈ 𝐡)
3020, 29anim12i 613 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘β€˜0) ∈ 𝐡))
31 df-3an 1089 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (π‘β€˜0) ∈ 𝐡) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘β€˜0) ∈ 𝐡))
3230, 31sylibr 233 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (π‘β€˜0) ∈ 𝐡))
331, 15, 16, 17m2cpm 22234 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (π‘β€˜0) ∈ 𝐡) β†’ (π‘‡β€˜(π‘β€˜0)) ∈ 𝑆)
3432, 33syl 17 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ (π‘‡β€˜(π‘β€˜0)) ∈ 𝑆)
35 chfacfisf.r . . . . . . 7 Γ— = (.rβ€˜π‘Œ)
3635subrgmcl 20367 . . . . . 6 ((𝑆 ∈ (SubRingβ€˜π‘Œ) ∧ (π‘‡β€˜π‘€) ∈ 𝑆 ∧ (π‘‡β€˜(π‘β€˜0)) ∈ 𝑆) β†’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0))) ∈ 𝑆)
3714, 19, 34, 36syl3anc 1371 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0))) ∈ 𝑆)
38 chfacfisf.s . . . . . 6 βˆ’ = (-gβ€˜π‘Œ)
3938subgsubcl 19011 . . . . 5 ((𝑆 ∈ (SubGrpβ€˜π‘Œ) ∧ 0 ∈ 𝑆 ∧ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0))) ∈ 𝑆) β†’ ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))) ∈ 𝑆)
406, 11, 37, 39syl3anc 1371 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))) ∈ 𝑆)
4140ad2antrr 724 . . 3 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ 𝑛 = 0) β†’ ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))) ∈ 𝑆)
42 simpl1 1191 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ 𝑁 ∈ Fin)
43 simpl2 1192 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ 𝑅 ∈ Ring)
4422adantl 482 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ 𝑏:(0...𝑠)⟢𝐡)
45 eluzfz2 13505 . . . . . . . . . 10 (𝑠 ∈ (β„€β‰₯β€˜0) β†’ 𝑠 ∈ (0...𝑠))
4625, 45syl 17 . . . . . . . . 9 (𝑠 ∈ β„• β†’ 𝑠 ∈ (0...𝑠))
4746ad2antrl 726 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ 𝑠 ∈ (0...𝑠))
4844, 47ffvelcdmd 7084 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ (π‘β€˜π‘ ) ∈ 𝐡)
491, 15, 16, 17m2cpm 22234 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (π‘β€˜π‘ ) ∈ 𝐡) β†’ (π‘‡β€˜(π‘β€˜π‘ )) ∈ 𝑆)
5042, 43, 48, 49syl3anc 1371 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ (π‘‡β€˜(π‘β€˜π‘ )) ∈ 𝑆)
5150adantr 481 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) β†’ (π‘‡β€˜(π‘β€˜π‘ )) ∈ 𝑆)
5251ad2antrr 724 . . . 4 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) ∧ 𝑛 = (𝑠 + 1)) β†’ (π‘‡β€˜(π‘β€˜π‘ )) ∈ 𝑆)
5311ad4antr 730 . . . . 5 (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) ∧ Β¬ 𝑛 = (𝑠 + 1)) ∧ (𝑠 + 1) < 𝑛) β†’ 0 ∈ 𝑆)
54 nn0re 12477 . . . . . . . . . . . . . . 15 (𝑛 ∈ β„•0 β†’ 𝑛 ∈ ℝ)
5554adantl 482 . . . . . . . . . . . . . 14 ((𝑠 ∈ β„• ∧ 𝑛 ∈ β„•0) β†’ 𝑛 ∈ ℝ)
56 peano2nn 12220 . . . . . . . . . . . . . . . 16 (𝑠 ∈ β„• β†’ (𝑠 + 1) ∈ β„•)
5756nnred 12223 . . . . . . . . . . . . . . 15 (𝑠 ∈ β„• β†’ (𝑠 + 1) ∈ ℝ)
5857adantr 481 . . . . . . . . . . . . . 14 ((𝑠 ∈ β„• ∧ 𝑛 ∈ β„•0) β†’ (𝑠 + 1) ∈ ℝ)
5955, 58lenltd 11356 . . . . . . . . . . . . 13 ((𝑠 ∈ β„• ∧ 𝑛 ∈ β„•0) β†’ (𝑛 ≀ (𝑠 + 1) ↔ Β¬ (𝑠 + 1) < 𝑛))
60 nesym 2997 . . . . . . . . . . . . . . 15 ((𝑠 + 1) β‰  𝑛 ↔ Β¬ 𝑛 = (𝑠 + 1))
61 ltlen 11311 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ) β†’ (𝑛 < (𝑠 + 1) ↔ (𝑛 ≀ (𝑠 + 1) ∧ (𝑠 + 1) β‰  𝑛)))
6254, 57, 61syl2anr 597 . . . . . . . . . . . . . . . . 17 ((𝑠 ∈ β„• ∧ 𝑛 ∈ β„•0) β†’ (𝑛 < (𝑠 + 1) ↔ (𝑛 ≀ (𝑠 + 1) ∧ (𝑠 + 1) β‰  𝑛)))
6362biimprd 247 . . . . . . . . . . . . . . . 16 ((𝑠 ∈ β„• ∧ 𝑛 ∈ β„•0) β†’ ((𝑛 ≀ (𝑠 + 1) ∧ (𝑠 + 1) β‰  𝑛) β†’ 𝑛 < (𝑠 + 1)))
6463expcomd 417 . . . . . . . . . . . . . . 15 ((𝑠 ∈ β„• ∧ 𝑛 ∈ β„•0) β†’ ((𝑠 + 1) β‰  𝑛 β†’ (𝑛 ≀ (𝑠 + 1) β†’ 𝑛 < (𝑠 + 1))))
6560, 64biimtrrid 242 . . . . . . . . . . . . . 14 ((𝑠 ∈ β„• ∧ 𝑛 ∈ β„•0) β†’ (Β¬ 𝑛 = (𝑠 + 1) β†’ (𝑛 ≀ (𝑠 + 1) β†’ 𝑛 < (𝑠 + 1))))
6665com23 86 . . . . . . . . . . . . 13 ((𝑠 ∈ β„• ∧ 𝑛 ∈ β„•0) β†’ (𝑛 ≀ (𝑠 + 1) β†’ (Β¬ 𝑛 = (𝑠 + 1) β†’ 𝑛 < (𝑠 + 1))))
6759, 66sylbird 259 . . . . . . . . . . . 12 ((𝑠 ∈ β„• ∧ 𝑛 ∈ β„•0) β†’ (Β¬ (𝑠 + 1) < 𝑛 β†’ (Β¬ 𝑛 = (𝑠 + 1) β†’ 𝑛 < (𝑠 + 1))))
6867impcomd 412 . . . . . . . . . . 11 ((𝑠 ∈ β„• ∧ 𝑛 ∈ β„•0) β†’ ((Β¬ 𝑛 = (𝑠 + 1) ∧ Β¬ (𝑠 + 1) < 𝑛) β†’ 𝑛 < (𝑠 + 1)))
6968ex 413 . . . . . . . . . 10 (𝑠 ∈ β„• β†’ (𝑛 ∈ β„•0 β†’ ((Β¬ 𝑛 = (𝑠 + 1) ∧ Β¬ (𝑠 + 1) < 𝑛) β†’ 𝑛 < (𝑠 + 1))))
7069ad2antrl 726 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ (𝑛 ∈ β„•0 β†’ ((Β¬ 𝑛 = (𝑠 + 1) ∧ Β¬ (𝑠 + 1) < 𝑛) β†’ 𝑛 < (𝑠 + 1))))
7170imp 407 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) β†’ ((Β¬ 𝑛 = (𝑠 + 1) ∧ Β¬ (𝑠 + 1) < 𝑛) β†’ 𝑛 < (𝑠 + 1)))
7271adantr 481 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ ((Β¬ 𝑛 = (𝑠 + 1) ∧ Β¬ (𝑠 + 1) < 𝑛) β†’ 𝑛 < (𝑠 + 1)))
735ad4antr 730 . . . . . . . . 9 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) β†’ 𝑆 ∈ (SubGrpβ€˜π‘Œ))
7420ad4antr 730 . . . . . . . . . . 11 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) β†’ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
7522ad4antlr 731 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) β†’ 𝑏:(0...𝑠)⟢𝐡)
76 neqne 2948 . . . . . . . . . . . . . . . . 17 (Β¬ 𝑛 = 0 β†’ 𝑛 β‰  0)
7776anim2i 617 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ β„•0 ∧ Β¬ 𝑛 = 0) β†’ (𝑛 ∈ β„•0 ∧ 𝑛 β‰  0))
78 elnnne0 12482 . . . . . . . . . . . . . . . 16 (𝑛 ∈ β„• ↔ (𝑛 ∈ β„•0 ∧ 𝑛 β‰  0))
7977, 78sylibr 233 . . . . . . . . . . . . . . 15 ((𝑛 ∈ β„•0 ∧ Β¬ 𝑛 = 0) β†’ 𝑛 ∈ β„•)
80 nnm1nn0 12509 . . . . . . . . . . . . . . 15 (𝑛 ∈ β„• β†’ (𝑛 βˆ’ 1) ∈ β„•0)
8179, 80syl 17 . . . . . . . . . . . . . 14 ((𝑛 ∈ β„•0 ∧ Β¬ 𝑛 = 0) β†’ (𝑛 βˆ’ 1) ∈ β„•0)
8281ad4ant23 751 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) β†’ (𝑛 βˆ’ 1) ∈ β„•0)
8323adantr 481 . . . . . . . . . . . . . 14 ((𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠))) β†’ 𝑠 ∈ β„•0)
8483ad4antlr 731 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) β†’ 𝑠 ∈ β„•0)
8562simprbda 499 . . . . . . . . . . . . . . . . . . 19 (((𝑠 ∈ β„• ∧ 𝑛 ∈ β„•0) ∧ 𝑛 < (𝑠 + 1)) β†’ 𝑛 ≀ (𝑠 + 1))
8655adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝑠 ∈ β„• ∧ 𝑛 ∈ β„•0) ∧ 𝑛 < (𝑠 + 1)) β†’ 𝑛 ∈ ℝ)
87 1red 11211 . . . . . . . . . . . . . . . . . . . 20 (((𝑠 ∈ β„• ∧ 𝑛 ∈ β„•0) ∧ 𝑛 < (𝑠 + 1)) β†’ 1 ∈ ℝ)
88 nnre 12215 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ β„• β†’ 𝑠 ∈ ℝ)
8988ad2antrr 724 . . . . . . . . . . . . . . . . . . . 20 (((𝑠 ∈ β„• ∧ 𝑛 ∈ β„•0) ∧ 𝑛 < (𝑠 + 1)) β†’ 𝑠 ∈ ℝ)
9086, 87, 89lesubaddd 11807 . . . . . . . . . . . . . . . . . . 19 (((𝑠 ∈ β„• ∧ 𝑛 ∈ β„•0) ∧ 𝑛 < (𝑠 + 1)) β†’ ((𝑛 βˆ’ 1) ≀ 𝑠 ↔ 𝑛 ≀ (𝑠 + 1)))
9185, 90mpbird 256 . . . . . . . . . . . . . . . . . 18 (((𝑠 ∈ β„• ∧ 𝑛 ∈ β„•0) ∧ 𝑛 < (𝑠 + 1)) β†’ (𝑛 βˆ’ 1) ≀ 𝑠)
9291exp31 420 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ β„• β†’ (𝑛 ∈ β„•0 β†’ (𝑛 < (𝑠 + 1) β†’ (𝑛 βˆ’ 1) ≀ 𝑠)))
9392ad2antrl 726 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ (𝑛 ∈ β„•0 β†’ (𝑛 < (𝑠 + 1) β†’ (𝑛 βˆ’ 1) ≀ 𝑠)))
9493imp 407 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) β†’ (𝑛 < (𝑠 + 1) β†’ (𝑛 βˆ’ 1) ≀ 𝑠))
9594adantr 481 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ (𝑛 < (𝑠 + 1) β†’ (𝑛 βˆ’ 1) ≀ 𝑠))
9695imp 407 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) β†’ (𝑛 βˆ’ 1) ≀ 𝑠)
97 elfz2nn0 13588 . . . . . . . . . . . . 13 ((𝑛 βˆ’ 1) ∈ (0...𝑠) ↔ ((𝑛 βˆ’ 1) ∈ β„•0 ∧ 𝑠 ∈ β„•0 ∧ (𝑛 βˆ’ 1) ≀ 𝑠))
9882, 84, 96, 97syl3anbrc 1343 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) β†’ (𝑛 βˆ’ 1) ∈ (0...𝑠))
9975, 98ffvelcdmd 7084 . . . . . . . . . . 11 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) β†’ (π‘β€˜(𝑛 βˆ’ 1)) ∈ 𝐡)
100 df-3an 1089 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (π‘β€˜(𝑛 βˆ’ 1)) ∈ 𝐡) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (π‘β€˜(𝑛 βˆ’ 1)) ∈ 𝐡))
10174, 99, 100sylanbrc 583 . . . . . . . . . 10 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) β†’ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (π‘β€˜(𝑛 βˆ’ 1)) ∈ 𝐡))
1021, 15, 16, 17m2cpm 22234 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (π‘β€˜(𝑛 βˆ’ 1)) ∈ 𝐡) β†’ (π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) ∈ 𝑆)
103101, 102syl 17 . . . . . . . . 9 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) β†’ (π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) ∈ 𝑆)
10414ad2antrr 724 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ 𝑛 < (𝑠 + 1)) β†’ 𝑆 ∈ (SubRingβ€˜π‘Œ))
10519ad2antrr 724 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ 𝑛 < (𝑠 + 1)) β†’ (π‘‡β€˜π‘€) ∈ 𝑆)
10620, 83anim12i 613 . . . . . . . . . . . . . . 15 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑠 ∈ β„•0))
107 df-3an 1089 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ β„•0) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑠 ∈ β„•0))
108106, 107sylibr 233 . . . . . . . . . . . . . 14 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ β„•0))
109108ad2antrr 724 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ 𝑛 < (𝑠 + 1)) β†’ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ β„•0))
110109simp1d 1142 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ 𝑛 < (𝑠 + 1)) β†’ 𝑁 ∈ Fin)
111109simp2d 1143 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ 𝑛 < (𝑠 + 1)) β†’ 𝑅 ∈ Ring)
11244ad2antrr 724 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ 𝑛 < (𝑠 + 1)) β†’ 𝑏:(0...𝑠)⟢𝐡)
113 simplr 767 . . . . . . . . . . . . . . . . 17 (((𝑠 ∈ β„• ∧ 𝑛 ∈ β„•0) ∧ 𝑛 < (𝑠 + 1)) β†’ 𝑛 ∈ β„•0)
11423ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((𝑠 ∈ β„• ∧ 𝑛 ∈ β„•0) ∧ 𝑛 < (𝑠 + 1)) β†’ 𝑠 ∈ β„•0)
115 nn0z 12579 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ β„•0 β†’ 𝑛 ∈ β„€)
116 nnz 12575 . . . . . . . . . . . . . . . . . . 19 (𝑠 ∈ β„• β†’ 𝑠 ∈ β„€)
117 zleltp1 12609 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ β„€ ∧ 𝑠 ∈ β„€) β†’ (𝑛 ≀ 𝑠 ↔ 𝑛 < (𝑠 + 1)))
118115, 116, 117syl2anr 597 . . . . . . . . . . . . . . . . . 18 ((𝑠 ∈ β„• ∧ 𝑛 ∈ β„•0) β†’ (𝑛 ≀ 𝑠 ↔ 𝑛 < (𝑠 + 1)))
119118biimpar 478 . . . . . . . . . . . . . . . . 17 (((𝑠 ∈ β„• ∧ 𝑛 ∈ β„•0) ∧ 𝑛 < (𝑠 + 1)) β†’ 𝑛 ≀ 𝑠)
120 elfz2nn0 13588 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (0...𝑠) ↔ (𝑛 ∈ β„•0 ∧ 𝑠 ∈ β„•0 ∧ 𝑛 ≀ 𝑠))
121113, 114, 119, 120syl3anbrc 1343 . . . . . . . . . . . . . . . 16 (((𝑠 ∈ β„• ∧ 𝑛 ∈ β„•0) ∧ 𝑛 < (𝑠 + 1)) β†’ 𝑛 ∈ (0...𝑠))
122121exp31 420 . . . . . . . . . . . . . . 15 (𝑠 ∈ β„• β†’ (𝑛 ∈ β„•0 β†’ (𝑛 < (𝑠 + 1) β†’ 𝑛 ∈ (0...𝑠))))
123122ad2antrl 726 . . . . . . . . . . . . . 14 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ (𝑛 ∈ β„•0 β†’ (𝑛 < (𝑠 + 1) β†’ 𝑛 ∈ (0...𝑠))))
124123imp31 418 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ 𝑛 < (𝑠 + 1)) β†’ 𝑛 ∈ (0...𝑠))
125112, 124ffvelcdmd 7084 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ 𝑛 < (𝑠 + 1)) β†’ (π‘β€˜π‘›) ∈ 𝐡)
1261, 15, 16, 17m2cpm 22234 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (π‘β€˜π‘›) ∈ 𝐡) β†’ (π‘‡β€˜(π‘β€˜π‘›)) ∈ 𝑆)
127110, 111, 125, 126syl3anc 1371 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ 𝑛 < (𝑠 + 1)) β†’ (π‘‡β€˜(π‘β€˜π‘›)) ∈ 𝑆)
12835subrgmcl 20367 . . . . . . . . . . 11 ((𝑆 ∈ (SubRingβ€˜π‘Œ) ∧ (π‘‡β€˜π‘€) ∈ 𝑆 ∧ (π‘‡β€˜(π‘β€˜π‘›)) ∈ 𝑆) β†’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))) ∈ 𝑆)
129104, 105, 127, 128syl3anc 1371 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ 𝑛 < (𝑠 + 1)) β†’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))) ∈ 𝑆)
130129adantlr 713 . . . . . . . . 9 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) β†’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))) ∈ 𝑆)
13138subgsubcl 19011 . . . . . . . . 9 ((𝑆 ∈ (SubGrpβ€˜π‘Œ) ∧ (π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) ∈ 𝑆 ∧ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))) ∈ 𝑆) β†’ ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›)))) ∈ 𝑆)
13273, 103, 130, 131syl3anc 1371 . . . . . . . 8 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) ∧ 𝑛 < (𝑠 + 1)) β†’ ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›)))) ∈ 𝑆)
133132ex 413 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ (𝑛 < (𝑠 + 1) β†’ ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›)))) ∈ 𝑆))
13472, 133syld 47 . . . . . 6 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ ((Β¬ 𝑛 = (𝑠 + 1) ∧ Β¬ (𝑠 + 1) < 𝑛) β†’ ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›)))) ∈ 𝑆))
135134impl 456 . . . . 5 (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) ∧ Β¬ 𝑛 = (𝑠 + 1)) ∧ Β¬ (𝑠 + 1) < 𝑛) β†’ ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›)))) ∈ 𝑆)
13653, 135ifclda 4562 . . . 4 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) ∧ Β¬ 𝑛 = (𝑠 + 1)) β†’ if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))) ∈ 𝑆)
13752, 136ifclda 4562 . . 3 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) ∧ Β¬ 𝑛 = 0) β†’ if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›)))))) ∈ 𝑆)
13841, 137ifclda 4562 . 2 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) ∧ 𝑛 ∈ β„•0) β†’ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))) ∈ 𝑆)
139 chfacfisf.g . 2 𝐺 = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))
140138, 139fmptd 7110 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ 𝐺:β„•0βŸΆπ‘†)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  ifcif 4527   class class class wbr 5147   ↦ cmpt 5230  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405   ↑m cmap 8816  Fincfn 8935  β„cr 11105  0cc0 11106  1c1 11107   + caddc 11109   < clt 11244   ≀ cle 11245   βˆ’ cmin 11440  β„•cn 12208  β„•0cn0 12468  β„€cz 12554  β„€β‰₯cuz 12818  ...cfz 13480  Basecbs 17140  .rcmulr 17194  0gc0g 17381  SubMndcsubmnd 18666  -gcsg 18817  SubGrpcsubg 18994  Ringcrg 20049  SubRingcsubrg 20351  Poly1cpl1 21692   Mat cmat 21898   ConstPolyMat ccpmat 22196   matToPolyMat cmat2pmat 22197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-ot 4636  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-ofr 7667  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-pm 8819  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-sup 9433  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-fz 13481  df-fzo 13624  df-seq 13963  df-hash 14287  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-sca 17209  df-vsca 17210  df-ip 17211  df-tset 17212  df-ple 17213  df-ds 17215  df-hom 17217  df-cco 17218  df-0g 17383  df-gsum 17384  df-prds 17389  df-pws 17391  df-mre 17526  df-mrc 17527  df-acs 17529  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-mhm 18667  df-submnd 18668  df-grp 18818  df-minusg 18819  df-sbg 18820  df-mulg 18945  df-subg 18997  df-ghm 19084  df-cntz 19175  df-cmn 19644  df-abl 19645  df-mgp 19982  df-ur 19999  df-srg 20003  df-ring 20051  df-subrg 20353  df-lmod 20465  df-lss 20535  df-sra 20777  df-rgmod 20778  df-dsmm 21278  df-frlm 21293  df-ascl 21401  df-psr 21453  df-mvr 21454  df-mpl 21455  df-opsr 21457  df-psr1 21695  df-vr1 21696  df-ply1 21697  df-coe1 21698  df-mamu 21877  df-mat 21899  df-cpmat 22199  df-mat2pmat 22200
This theorem is referenced by:  cpmadumatpolylem1  22374  cpmadumatpolylem2  22375  cpmadumatpoly  22376  chcoeffeqlem  22378  cayhamlem4  22381
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