ply1fermltl - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > ply1fermltl		Structured version Visualization version GIF version

Theorem ply1fermltl 31572

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 2738	. . 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 6759	. . . 4 ⊢ (._g‘𝑁) = (._g‘(mulGrp‘𝑊))
6	3, 5	eqtri 2766	. . 3 ⊢ ↑ = (._g‘(mulGrp‘𝑊))
7		eqid 2738	. . 3 ⊢ (chr‘𝑊) = (chr‘𝑊)
8		ply1fermltl.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℙ)
9		prmnn 16307	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
10		nnnn0 12170	. . . . 5 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ₀)
11		ply1fermltl.z	. . . . . 6 ⊢ 𝑍 = (ℤ/nℤ‘𝑃)
12	11	zncrng 20664	. . . . 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 21279	. . . 4 ⊢ (𝑍 ∈ CRing → 𝑊 ∈ CRing)
16	13, 15	syl 17	. . 3 ⊢ (𝜑 → 𝑊 ∈ CRing)
17	14	ply1chr 31571	. . . . . 6 ⊢ (𝑍 ∈ CRing → (chr‘𝑊) = (chr‘𝑍))
18	13, 17	syl 17	. . . . 5 ⊢ (𝜑 → (chr‘𝑊) = (chr‘𝑍))
19	11	znchr 20682	. . . . . 6 ⊢ (𝑃 ∈ ℕ₀ → (chr‘𝑍) = 𝑃)
20	8, 9, 10, 19	4syl 19	. . . . 5 ⊢ (𝜑 → (chr‘𝑍) = 𝑃)
21	18, 20	eqtrd 2778	. . . 4 ⊢ (𝜑 → (chr‘𝑊) = 𝑃)
22	21, 8	eqeltrd 2839	. . 3 ⊢ (𝜑 → (chr‘𝑊) ∈ ℙ)
23	13	crngringd 19711	. . . 4 ⊢ (𝜑 → 𝑍 ∈ Ring)
24		ply1fermltl.x	. . . . 5 ⊢ 𝑋 = (var₁‘𝑍)
25	24, 14, 1	vr1cl 21298	. . . 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 2738	. . . . . . . 8 ⊢ (ℤRHom‘𝑍) = (ℤRHom‘𝑍)
29	28	zrhrhm 20625	. . . . . . 7 ⊢ (𝑍 ∈ Ring → (ℤRHom‘𝑍) ∈ (ℤ_ring RingHom 𝑍))
30		zringbas 20588	. . . . . . . 8 ⊢ ℤ = (Base‘ℤ_ring)
31		eqid 2738	. . . . . . . 8 ⊢ (Base‘𝑍) = (Base‘𝑍)
32	30, 31	rhmf 19885	. . . . . . 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 6944	. . . . 5 ⊢ (𝜑 → ((ℤRHom‘𝑍)‘𝐸) ∈ (Base‘𝑍))
36		ply1fermltl.c	. . . . . 6 ⊢ 𝐶 = (algSc‘𝑊)
37	14, 36, 31, 1	ply1sclcl 21367	. . . . 5 ⊢ ((𝑍 ∈ Ring ∧ ((ℤRHom‘𝑍)‘𝐸) ∈ (Base‘𝑍)) → (𝐶‘((ℤRHom‘𝑍)‘𝐸)) ∈ (Base‘𝑊))
38	23, 35, 37	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝐶‘((ℤRHom‘𝑍)‘𝐸)) ∈ (Base‘𝑊))
39	27, 38	eqeltrid 2843	. . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑊))
40	1, 2, 6, 7, 16, 22, 26, 39	freshmansdream 31386	. 2 ⊢ (𝜑 → ((chr‘𝑊) ↑ (𝑋 + 𝐴)) = (((chr‘𝑊) ↑ 𝑋) + ((chr‘𝑊) ↑ 𝐴)))
41	21	oveq1d 7270	. 2 ⊢ (𝜑 → ((chr‘𝑊) ↑ (𝑋 + 𝐴)) = (𝑃 ↑ (𝑋 + 𝐴)))
42	21	oveq1d 7270	. . 3 ⊢ (𝜑 → ((chr‘𝑊) ↑ 𝑋) = (𝑃 ↑ 𝑋))
43	21	oveq1d 7270	. . . 4 ⊢ (𝜑 → ((chr‘𝑊) ↑ 𝐴) = (𝑃 ↑ 𝐴))
44	14	ply1assa 21280	. . . . . . . . 9 ⊢ (𝑍 ∈ CRing → 𝑊 ∈ AssAlg)
45		eqid 2738	. . . . . . . . . 10 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
46	36, 45	asclrhm 21004	. . . . . . . . 9 ⊢ (𝑊 ∈ AssAlg → 𝐶 ∈ ((Scalar‘𝑊) RingHom 𝑊))
47	13, 44, 46	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ((Scalar‘𝑊) RingHom 𝑊))
48	13	crnggrpd 19712	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ Grp)
49	14	ply1sca 21334	. . . . . . . . . 10 ⊢ (𝑍 ∈ Grp → 𝑍 = (Scalar‘𝑊))
50	48, 49	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 = (Scalar‘𝑊))
51	50	oveq1d 7270	. . . . . . . 8 ⊢ (𝜑 → (𝑍 RingHom 𝑊) = ((Scalar‘𝑊) RingHom 𝑊))
52	47, 51	eleqtrrd 2842	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (𝑍 RingHom 𝑊))
53		eqid 2738	. . . . . . . 8 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
54	53, 4	rhmmhm 19881	. . . . . . 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 19641	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
58		eqid 2738	. . . . . . 7 ⊢ (._g‘(mulGrp‘𝑍)) = (._g‘(mulGrp‘𝑍))
59	57, 58, 3	mhmmulg 18659	. . . . . 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 1369	. . . . 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 7271	. . . . 5 ⊢ (𝜑 → (𝑃 ↑ 𝐴) = (𝑃 ↑ (𝐶‘((ℤRHom‘𝑍)‘𝐸))))
63	60, 62	eqtr4d 2781	. . . 4 ⊢ (𝜑 → (𝐶‘(𝑃(._g‘(mulGrp‘𝑍))((ℤRHom‘𝑍)‘𝐸))) = (𝑃 ↑ 𝐴))
64	11, 31, 58	znfermltl 31464	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ ((ℤRHom‘𝑍)‘𝐸) ∈ (Base‘𝑍)) → (𝑃(._g‘(mulGrp‘𝑍))((ℤRHom‘𝑍)‘𝐸)) = ((ℤRHom‘𝑍)‘𝐸))
65	8, 35, 64	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝑃(._g‘(mulGrp‘𝑍))((ℤRHom‘𝑍)‘𝐸)) = ((ℤRHom‘𝑍)‘𝐸))
66	65	fveq2d 6760	. . . . 5 ⊢ (𝜑 → (𝐶‘(𝑃(._g‘(mulGrp‘𝑍))((ℤRHom‘𝑍)‘𝐸))) = (𝐶‘((ℤRHom‘𝑍)‘𝐸)))
67	66, 27	eqtr4di 2797	. . . 4 ⊢ (𝜑 → (𝐶‘(𝑃(._g‘(mulGrp‘𝑍))((ℤRHom‘𝑍)‘𝐸))) = 𝐴)
68	43, 63, 67	3eqtr2d 2784	. . 3 ⊢ (𝜑 → ((chr‘𝑊) ↑ 𝐴) = 𝐴)
69	42, 68	oveq12d 7273	. 2 ⊢ (𝜑 → (((chr‘𝑊) ↑ 𝑋) + ((chr‘𝑊) ↑ 𝐴)) = ((𝑃 ↑ 𝑋) + 𝐴))
70	40, 41, 69	3eqtr3d 2786	1 ⊢ (𝜑 → (𝑃 ↑ (𝑋 + 𝐴)) = ((𝑃 ↑ 𝑋) + 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2108 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ℕcn 11903 ℕ₀cn0 12163 ℤcz 12249 ℙcprime 16304 Basecbs 16840 +_gcplusg 16888 Scalarcsca 16891 MndHom cmhm 18343 Grpcgrp 18492 ._gcmg 18615 mulGrpcmgp 19635 Ringcrg 19698 CRingccrg 19699 RingHom crh 19871 ℤ_ringzring 20582 ℤRHomczrh 20613 chrcchr 20615 ℤ/nℤczn 20616 AssAlgcasa 20967 algSccascl 20969 var₁cv1 21257 Poly₁cpl1 21258

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-ofr 7512 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-er 8456 df-ec 8458 df-qs 8462 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-xnn0 12236 df-z 12250 df-dec 12367 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-dvds 15892 df-gcd 16130 df-prm 16305 df-phi 16395 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-0g 17069 df-gsum 17070 df-imas 17136 df-qus 17137 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-nsg 18668 df-eqg 18669 df-ghm 18747 df-cntz 18838 df-od 19051 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-srg 19657 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-rnghom 19874 df-drng 19908 df-subrg 19937 df-lmod 20040 df-lss 20109 df-lsp 20149 df-sra 20349 df-rgmod 20350 df-lidl 20351 df-rsp 20352 df-2idl 20416 df-cnfld 20511 df-zring 20583 df-zrh 20617 df-chr 20619 df-zn 20620 df-assa 20970 df-ascl 20972 df-psr 21022 df-mvr 21023 df-mpl 21024 df-opsr 21026 df-psr1 21261 df-vr1 21262 df-ply1 21263 df-coe1 21264

This theorem is referenced by: (None)

W3C validator