ply1fermltl - Mathbox for Thierry Arnoux

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Theorem ply1fermltl 31384

Description: Fermat's little theorem for polynomials. If 𝑃 is prime, Then (𝑋 + 𝐴)↑𝑃 = ((𝑋↑𝑃) + 𝐴) modulo 𝑃. (Contributed by Thierry Arnoux, 24-Jul-2024.)

**Hypotheses**
Ref	Expression
ply1fermltl.z	⊢ 𝑍 = (ℤ/nℤ‘𝑃)
ply1fermltl.w	⊢ 𝑊 = (Poly₁‘𝑍)
ply1fermltl.x	⊢ 𝑋 = (var₁‘𝑍)
ply1fermltl.l	⊢ + = (+_g‘𝑊)
ply1fermltl.n	⊢ 𝑁 = (mulGrp‘𝑊)
ply1fermltl.t	⊢ ↑ = (._g‘𝑁)
ply1fermltl.c	⊢ 𝐶 = (algSc‘𝑊)
ply1fermltl.a	⊢ 𝐴 = (𝐶‘((ℤRHom‘𝑍)‘𝐸))
ply1fermltl.p	⊢ (𝜑 → 𝑃 ∈ ℙ)
ply1fermltl.1	⊢ (𝜑 → 𝐸 ∈ ℤ)

**Assertion**
Ref	Expression
ply1fermltl	⊢ (𝜑 → (𝑃 ↑ (𝑋 + 𝐴)) = ((𝑃 ↑ 𝑋) + 𝐴))

**Proof of Theorem ply1fermltl**
Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		ply1fermltl.l	. . 3 ⊢ + = (+_g‘𝑊)
3		ply1fermltl.t	. . . 4 ⊢ ↑ = (._g‘𝑁)
4		ply1fermltl.n	. . . . 5 ⊢ 𝑁 = (mulGrp‘𝑊)
5	4	fveq2i 6720	. . . 4 ⊢ (._g‘𝑁) = (._g‘(mulGrp‘𝑊))
6	3, 5	eqtri 2765	. . 3 ⊢ ↑ = (._g‘(mulGrp‘𝑊))
7		eqid 2737	. . 3 ⊢ (chr‘𝑊) = (chr‘𝑊)
8		ply1fermltl.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℙ)
9		prmnn 16231	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
10		nnnn0 12097	. . . . 5 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ₀)
11		ply1fermltl.z	. . . . . 6 ⊢ 𝑍 = (ℤ/nℤ‘𝑃)
12	11	zncrng 20509	. . . . 5 ⊢ (𝑃 ∈ ℕ₀ → 𝑍 ∈ CRing)
13	8, 9, 10, 12	4syl 19	. . . 4 ⊢ (𝜑 → 𝑍 ∈ CRing)
14		ply1fermltl.w	. . . . 5 ⊢ 𝑊 = (Poly₁‘𝑍)
15	14	ply1crng 21119	. . . 4 ⊢ (𝑍 ∈ CRing → 𝑊 ∈ CRing)
16	13, 15	syl 17	. . 3 ⊢ (𝜑 → 𝑊 ∈ CRing)
17	14	ply1chr 31383	. . . . . 6 ⊢ (𝑍 ∈ CRing → (chr‘𝑊) = (chr‘𝑍))
18	13, 17	syl 17	. . . . 5 ⊢ (𝜑 → (chr‘𝑊) = (chr‘𝑍))
19	11	znchr 20527	. . . . . 6 ⊢ (𝑃 ∈ ℕ₀ → (chr‘𝑍) = 𝑃)
20	8, 9, 10, 19	4syl 19	. . . . 5 ⊢ (𝜑 → (chr‘𝑍) = 𝑃)
21	18, 20	eqtrd 2777	. . . 4 ⊢ (𝜑 → (chr‘𝑊) = 𝑃)
22	21, 8	eqeltrd 2838	. . 3 ⊢ (𝜑 → (chr‘𝑊) ∈ ℙ)
23	13	crngringd 19575	. . . 4 ⊢ (𝜑 → 𝑍 ∈ Ring)
24		ply1fermltl.x	. . . . 5 ⊢ 𝑋 = (var₁‘𝑍)
25	24, 14, 1	vr1cl 21138	. . . 4 ⊢ (𝑍 ∈ Ring → 𝑋 ∈ (Base‘𝑊))
26	23, 25	syl 17	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊))
27		ply1fermltl.a	. . . 4 ⊢ 𝐴 = (𝐶‘((ℤRHom‘𝑍)‘𝐸))
28		eqid 2737	. . . . . . . 8 ⊢ (ℤRHom‘𝑍) = (ℤRHom‘𝑍)
29	28	zrhrhm 20478	. . . . . . 7 ⊢ (𝑍 ∈ Ring → (ℤRHom‘𝑍) ∈ (ℤ_ring RingHom 𝑍))
30		zringbas 20441	. . . . . . . 8 ⊢ ℤ = (Base‘ℤ_ring)
31		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝑍) = (Base‘𝑍)
32	30, 31	rhmf 19746	. . . . . . 7 ⊢ ((ℤRHom‘𝑍) ∈ (ℤ_ring RingHom 𝑍) → (ℤRHom‘𝑍):ℤ⟶(Base‘𝑍))
33	23, 29, 32	3syl 18	. . . . . 6 ⊢ (𝜑 → (ℤRHom‘𝑍):ℤ⟶(Base‘𝑍))
34		ply1fermltl.1	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℤ)
35	33, 34	ffvelrnd 6905	. . . . 5 ⊢ (𝜑 → ((ℤRHom‘𝑍)‘𝐸) ∈ (Base‘𝑍))
36		ply1fermltl.c	. . . . . 6 ⊢ 𝐶 = (algSc‘𝑊)
37	14, 36, 31, 1	ply1sclcl 21207	. . . . 5 ⊢ ((𝑍 ∈ Ring ∧ ((ℤRHom‘𝑍)‘𝐸) ∈ (Base‘𝑍)) → (𝐶‘((ℤRHom‘𝑍)‘𝐸)) ∈ (Base‘𝑊))
38	23, 35, 37	syl2anc 587	. . . 4 ⊢ (𝜑 → (𝐶‘((ℤRHom‘𝑍)‘𝐸)) ∈ (Base‘𝑊))
39	27, 38	eqeltrid 2842	. . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑊))
40	1, 2, 6, 7, 16, 22, 26, 39	freshmansdream 31203	. 2 ⊢ (𝜑 → ((chr‘𝑊) ↑ (𝑋 + 𝐴)) = (((chr‘𝑊) ↑ 𝑋) + ((chr‘𝑊) ↑ 𝐴)))
41	21	oveq1d 7228	. 2 ⊢ (𝜑 → ((chr‘𝑊) ↑ (𝑋 + 𝐴)) = (𝑃 ↑ (𝑋 + 𝐴)))
42	21	oveq1d 7228	. . 3 ⊢ (𝜑 → ((chr‘𝑊) ↑ 𝑋) = (𝑃 ↑ 𝑋))
43	21	oveq1d 7228	. . . 4 ⊢ (𝜑 → ((chr‘𝑊) ↑ 𝐴) = (𝑃 ↑ 𝐴))
44	14	ply1assa 21120	. . . . . . . . 9 ⊢ (𝑍 ∈ CRing → 𝑊 ∈ AssAlg)
45		eqid 2737	. . . . . . . . . 10 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
46	36, 45	asclrhm 20850	. . . . . . . . 9 ⊢ (𝑊 ∈ AssAlg → 𝐶 ∈ ((Scalar‘𝑊) RingHom 𝑊))
47	13, 44, 46	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ((Scalar‘𝑊) RingHom 𝑊))
48	13	crnggrpd 19576	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ Grp)
49	14	ply1sca 21174	. . . . . . . . . 10 ⊢ (𝑍 ∈ Grp → 𝑍 = (Scalar‘𝑊))
50	48, 49	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 = (Scalar‘𝑊))
51	50	oveq1d 7228	. . . . . . . 8 ⊢ (𝜑 → (𝑍 RingHom 𝑊) = ((Scalar‘𝑊) RingHom 𝑊))
52	47, 51	eleqtrrd 2841	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (𝑍 RingHom 𝑊))
53		eqid 2737	. . . . . . . 8 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
54	53, 4	rhmmhm 19742	. . . . . . 7 ⊢ (𝐶 ∈ (𝑍 RingHom 𝑊) → 𝐶 ∈ ((mulGrp‘𝑍) MndHom 𝑁))
55	52, 54	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ((mulGrp‘𝑍) MndHom 𝑁))
56	8, 9, 10	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
57	53, 31	mgpbas 19510	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
58		eqid 2737	. . . . . . 7 ⊢ (._g‘(mulGrp‘𝑍)) = (._g‘(mulGrp‘𝑍))
59	57, 58, 3	mhmmulg 18532	. . . . . 6 ⊢ ((𝐶 ∈ ((mulGrp‘𝑍) MndHom 𝑁) ∧ 𝑃 ∈ ℕ₀ ∧ ((ℤRHom‘𝑍)‘𝐸) ∈ (Base‘𝑍)) → (𝐶‘(𝑃(._g‘(mulGrp‘𝑍))((ℤRHom‘𝑍)‘𝐸))) = (𝑃 ↑ (𝐶‘((ℤRHom‘𝑍)‘𝐸))))
60	55, 56, 35, 59	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝐶‘(𝑃(._g‘(mulGrp‘𝑍))((ℤRHom‘𝑍)‘𝐸))) = (𝑃 ↑ (𝐶‘((ℤRHom‘𝑍)‘𝐸))))
61	27	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐴 = (𝐶‘((ℤRHom‘𝑍)‘𝐸)))
62	61	oveq2d 7229	. . . . 5 ⊢ (𝜑 → (𝑃 ↑ 𝐴) = (𝑃 ↑ (𝐶‘((ℤRHom‘𝑍)‘𝐸))))
63	60, 62	eqtr4d 2780	. . . 4 ⊢ (𝜑 → (𝐶‘(𝑃(._g‘(mulGrp‘𝑍))((ℤRHom‘𝑍)‘𝐸))) = (𝑃 ↑ 𝐴))
64	11, 31, 58	znfermltl 31276	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ ((ℤRHom‘𝑍)‘𝐸) ∈ (Base‘𝑍)) → (𝑃(._g‘(mulGrp‘𝑍))((ℤRHom‘𝑍)‘𝐸)) = ((ℤRHom‘𝑍)‘𝐸))
65	8, 35, 64	syl2anc 587	. . . . . 6 ⊢ (𝜑 → (𝑃(._g‘(mulGrp‘𝑍))((ℤRHom‘𝑍)‘𝐸)) = ((ℤRHom‘𝑍)‘𝐸))
66	65	fveq2d 6721	. . . . 5 ⊢ (𝜑 → (𝐶‘(𝑃(._g‘(mulGrp‘𝑍))((ℤRHom‘𝑍)‘𝐸))) = (𝐶‘((ℤRHom‘𝑍)‘𝐸)))
67	66, 27	eqtr4di 2796	. . . 4 ⊢ (𝜑 → (𝐶‘(𝑃(._g‘(mulGrp‘𝑍))((ℤRHom‘𝑍)‘𝐸))) = 𝐴)
68	43, 63, 67	3eqtr2d 2783	. . 3 ⊢ (𝜑 → ((chr‘𝑊) ↑ 𝐴) = 𝐴)
69	42, 68	oveq12d 7231	. 2 ⊢ (𝜑 → (((chr‘𝑊) ↑ 𝑋) + ((chr‘𝑊) ↑ 𝐴)) = ((𝑃 ↑ 𝑋) + 𝐴))
70	40, 41, 69	3eqtr3d 2785	1 ⊢ (𝜑 → (𝑃 ↑ (𝑋 + 𝐴)) = ((𝑃 ↑ 𝑋) + 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ℕcn 11830 ℕ₀cn0 12090 ℤcz 12176 ℙcprime 16228 Basecbs 16760 +_gcplusg 16802 Scalarcsca 16805 MndHom cmhm 18216 Grpcgrp 18365 ._gcmg 18488 mulGrpcmgp 19504 Ringcrg 19562 CRingccrg 19563 RingHom crh 19732 ℤ_ringzring 20435 ℤRHomczrh 20466 chrcchr 20468 ℤ/nℤczn 20469 AssAlgcasa 20812 algSccascl 20814 var₁cv1 21097 Poly₁cpl1 21098

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-ofr 7470 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-tpos 7968 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-oadd 8206 df-er 8391 df-ec 8393 df-qs 8397 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-sup 9058 df-inf 9059 df-oi 9126 df-dju 9517 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-xnn0 12163 df-z 12177 df-dec 12294 df-uz 12439 df-rp 12587 df-fz 13096 df-fzo 13239 df-fl 13367 df-mod 13443 df-seq 13575 df-exp 13636 df-fac 13840 df-bc 13869 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-dvds 15816 df-gcd 16054 df-prm 16229 df-phi 16319 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-0g 16946 df-gsum 16947 df-imas 17013 df-qus 17014 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-mhm 18218 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-mulg 18489 df-subg 18540 df-nsg 18541 df-eqg 18542 df-ghm 18620 df-cntz 18711 df-od 18920 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-srg 19521 df-ring 19564 df-cring 19565 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-invr 19690 df-dvr 19701 df-rnghom 19735 df-drng 19769 df-subrg 19798 df-lmod 19901 df-lss 19969 df-lsp 20009 df-sra 20209 df-rgmod 20210 df-lidl 20211 df-rsp 20212 df-2idl 20270 df-cnfld 20364 df-zring 20436 df-zrh 20470 df-chr 20472 df-zn 20473 df-assa 20815 df-ascl 20817 df-psr 20868 df-mvr 20869 df-mpl 20870 df-opsr 20872 df-psr1 21101 df-vr1 21102 df-ply1 21103 df-coe1 21104

This theorem is referenced by: (None)

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