ply1fermltl - Mathbox for Thierry Arnoux

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Theorem ply1fermltl 31670

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 6777	. . . 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 16379	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
10		nnnn0 12240	. . . . 5 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ₀)
11		ply1fermltl.z	. . . . . 6 ⊢ 𝑍 = (ℤ/nℤ‘𝑃)
12	11	zncrng 20752	. . . . 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 21369	. . . 4 ⊢ (𝑍 ∈ CRing → 𝑊 ∈ CRing)
16	13, 15	syl 17	. . 3 ⊢ (𝜑 → 𝑊 ∈ CRing)
17	14	ply1chr 31669	. . . . . 6 ⊢ (𝑍 ∈ CRing → (chr‘𝑊) = (chr‘𝑍))
18	13, 17	syl 17	. . . . 5 ⊢ (𝜑 → (chr‘𝑊) = (chr‘𝑍))
19	11	znchr 20770	. . . . . 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 19796	. . . 4 ⊢ (𝜑 → 𝑍 ∈ Ring)
24		ply1fermltl.x	. . . . 5 ⊢ 𝑋 = (var₁‘𝑍)
25	24, 14, 1	vr1cl 21388	. . . 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 20713	. . . . . . 7 ⊢ (𝑍 ∈ Ring → (ℤRHom‘𝑍) ∈ (ℤ_ring RingHom 𝑍))
30		zringbas 20676	. . . . . . . 8 ⊢ ℤ = (Base‘ℤ_ring)
31		eqid 2738	. . . . . . . 8 ⊢ (Base‘𝑍) = (Base‘𝑍)
32	30, 31	rhmf 19970	. . . . . . 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 6962	. . . . 5 ⊢ (𝜑 → ((ℤRHom‘𝑍)‘𝐸) ∈ (Base‘𝑍))
36		ply1fermltl.c	. . . . . 6 ⊢ 𝐶 = (algSc‘𝑊)
37	14, 36, 31, 1	ply1sclcl 21457	. . . . 5 ⊢ ((𝑍 ∈ Ring ∧ ((ℤRHom‘𝑍)‘𝐸) ∈ (Base‘𝑍)) → (𝐶‘((ℤRHom‘𝑍)‘𝐸)) ∈ (Base‘𝑊))
38	23, 35, 37	syl2anc 584	. . . 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 31484	. 2 ⊢ (𝜑 → ((chr‘𝑊) ↑ (𝑋 + 𝐴)) = (((chr‘𝑊) ↑ 𝑋) + ((chr‘𝑊) ↑ 𝐴)))
41	21	oveq1d 7290	. 2 ⊢ (𝜑 → ((chr‘𝑊) ↑ (𝑋 + 𝐴)) = (𝑃 ↑ (𝑋 + 𝐴)))
42	21	oveq1d 7290	. . 3 ⊢ (𝜑 → ((chr‘𝑊) ↑ 𝑋) = (𝑃 ↑ 𝑋))
43	21	oveq1d 7290	. . . 4 ⊢ (𝜑 → ((chr‘𝑊) ↑ 𝐴) = (𝑃 ↑ 𝐴))
44	14	ply1assa 21370	. . . . . . . . 9 ⊢ (𝑍 ∈ CRing → 𝑊 ∈ AssAlg)
45		eqid 2738	. . . . . . . . . 10 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
46	36, 45	asclrhm 21094	. . . . . . . . 9 ⊢ (𝑊 ∈ AssAlg → 𝐶 ∈ ((Scalar‘𝑊) RingHom 𝑊))
47	13, 44, 46	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ((Scalar‘𝑊) RingHom 𝑊))
48	13	crnggrpd 19797	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ Grp)
49	14	ply1sca 21424	. . . . . . . . . 10 ⊢ (𝑍 ∈ Grp → 𝑍 = (Scalar‘𝑊))
50	48, 49	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 = (Scalar‘𝑊))
51	50	oveq1d 7290	. . . . . . . 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 19966	. . . . . . 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 19726	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
58		eqid 2738	. . . . . . 7 ⊢ (._g‘(mulGrp‘𝑍)) = (._g‘(mulGrp‘𝑍))
59	57, 58, 3	mhmmulg 18744	. . . . . 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 1370	. . . . 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 7291	. . . . 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 31562	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ ((ℤRHom‘𝑍)‘𝐸) ∈ (Base‘𝑍)) → (𝑃(._g‘(mulGrp‘𝑍))((ℤRHom‘𝑍)‘𝐸)) = ((ℤRHom‘𝑍)‘𝐸))
65	8, 35, 64	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑃(._g‘(mulGrp‘𝑍))((ℤRHom‘𝑍)‘𝐸)) = ((ℤRHom‘𝑍)‘𝐸))
66	65	fveq2d 6778	. . . . 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 2784	. . 3 ⊢ (𝜑 → ((chr‘𝑊) ↑ 𝐴) = 𝐴)
69	42, 68	oveq12d 7293	. 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 2106 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℕcn 11973 ℕ₀cn0 12233 ℤcz 12319 ℙcprime 16376 Basecbs 16912 +_gcplusg 16962 Scalarcsca 16965 MndHom cmhm 18428 Grpcgrp 18577 ._gcmg 18700 mulGrpcmgp 19720 Ringcrg 19783 CRingccrg 19784 RingHom crh 19956 ℤ_ringczring 20670 ℤRHomczrh 20701 chrcchr 20703 ℤ/nℤczn 20704 AssAlgcasa 21057 algSccascl 21059 var₁cv1 21347 Poly₁cpl1 21348

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-ofr 7534 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-er 8498 df-ec 8500 df-qs 8504 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-xnn0 12306 df-z 12320 df-dec 12438 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-dvds 15964 df-gcd 16202 df-prm 16377 df-phi 16467 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-0g 17152 df-gsum 17153 df-imas 17219 df-qus 17220 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-nsg 18753 df-eqg 18754 df-ghm 18832 df-cntz 18923 df-od 19136 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-srg 19742 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-rnghom 19959 df-drng 19993 df-subrg 20022 df-lmod 20125 df-lss 20194 df-lsp 20234 df-sra 20434 df-rgmod 20435 df-lidl 20436 df-rsp 20437 df-2idl 20503 df-cnfld 20598 df-zring 20671 df-zrh 20705 df-chr 20707 df-zn 20708 df-assa 21060 df-ascl 21062 df-psr 21112 df-mvr 21113 df-mpl 21114 df-opsr 21116 df-psr1 21351 df-vr1 21352 df-ply1 21353 df-coe1 21354

This theorem is referenced by: (None)

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