ply1fermltl - Mathbox for Thierry Arnoux

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Theorem ply1fermltl 31195

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 2758	. . 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 6665	. . . 4 ⊢ (._g‘𝑁) = (._g‘(mulGrp‘𝑊))
6	3, 5	eqtri 2781	. . 3 ⊢ ↑ = (._g‘(mulGrp‘𝑊))
7		eqid 2758	. . 3 ⊢ (chr‘𝑊) = (chr‘𝑊)
8		ply1fermltl.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℙ)
9		prmnn 16075	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
10		nnnn0 11946	. . . . 5 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ₀)
11		ply1fermltl.z	. . . . . 6 ⊢ 𝑍 = (ℤ/nℤ‘𝑃)
12	11	zncrng 20317	. . . . 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 20927	. . . 4 ⊢ (𝑍 ∈ CRing → 𝑊 ∈ CRing)
16	13, 15	syl 17	. . 3 ⊢ (𝜑 → 𝑊 ∈ CRing)
17	14	ply1chr 31194	. . . . . 6 ⊢ (𝑍 ∈ CRing → (chr‘𝑊) = (chr‘𝑍))
18	13, 17	syl 17	. . . . 5 ⊢ (𝜑 → (chr‘𝑊) = (chr‘𝑍))
19	11	znchr 20335	. . . . . 6 ⊢ (𝑃 ∈ ℕ₀ → (chr‘𝑍) = 𝑃)
20	8, 9, 10, 19	4syl 19	. . . . 5 ⊢ (𝜑 → (chr‘𝑍) = 𝑃)
21	18, 20	eqtrd 2793	. . . 4 ⊢ (𝜑 → (chr‘𝑊) = 𝑃)
22	21, 8	eqeltrd 2852	. . 3 ⊢ (𝜑 → (chr‘𝑊) ∈ ℙ)
23	13	crngringd 19383	. . . 4 ⊢ (𝜑 → 𝑍 ∈ Ring)
24		ply1fermltl.x	. . . . 5 ⊢ 𝑋 = (var₁‘𝑍)
25	24, 14, 1	vr1cl 20946	. . . 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 2758	. . . . . . . 8 ⊢ (ℤRHom‘𝑍) = (ℤRHom‘𝑍)
29	28	zrhrhm 20286	. . . . . . 7 ⊢ (𝑍 ∈ Ring → (ℤRHom‘𝑍) ∈ (ℤ_ring RingHom 𝑍))
30		zringbas 20249	. . . . . . . 8 ⊢ ℤ = (Base‘ℤ_ring)
31		eqid 2758	. . . . . . . 8 ⊢ (Base‘𝑍) = (Base‘𝑍)
32	30, 31	rhmf 19554	. . . . . . 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 6848	. . . . 5 ⊢ (𝜑 → ((ℤRHom‘𝑍)‘𝐸) ∈ (Base‘𝑍))
36		ply1fermltl.c	. . . . . 6 ⊢ 𝐶 = (algSc‘𝑊)
37	14, 36, 31, 1	ply1sclcl 21015	. . . . 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 2856	. . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑊))
40	1, 2, 6, 7, 16, 22, 26, 39	freshmansdream 31014	. 2 ⊢ (𝜑 → ((chr‘𝑊) ↑ (𝑋 + 𝐴)) = (((chr‘𝑊) ↑ 𝑋) + ((chr‘𝑊) ↑ 𝐴)))
41	21	oveq1d 7170	. 2 ⊢ (𝜑 → ((chr‘𝑊) ↑ (𝑋 + 𝐴)) = (𝑃 ↑ (𝑋 + 𝐴)))
42	21	oveq1d 7170	. . 3 ⊢ (𝜑 → ((chr‘𝑊) ↑ 𝑋) = (𝑃 ↑ 𝑋))
43	21	oveq1d 7170	. . . 4 ⊢ (𝜑 → ((chr‘𝑊) ↑ 𝐴) = (𝑃 ↑ 𝐴))
44	14	ply1assa 20928	. . . . . . . . 9 ⊢ (𝑍 ∈ CRing → 𝑊 ∈ AssAlg)
45		eqid 2758	. . . . . . . . . 10 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
46	36, 45	asclrhm 20658	. . . . . . . . 9 ⊢ (𝑊 ∈ AssAlg → 𝐶 ∈ ((Scalar‘𝑊) RingHom 𝑊))
47	13, 44, 46	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ((Scalar‘𝑊) RingHom 𝑊))
48	13	crnggrpd 19384	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ Grp)
49	14	ply1sca 20982	. . . . . . . . . 10 ⊢ (𝑍 ∈ Grp → 𝑍 = (Scalar‘𝑊))
50	48, 49	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 = (Scalar‘𝑊))
51	50	oveq1d 7170	. . . . . . . 8 ⊢ (𝜑 → (𝑍 RingHom 𝑊) = ((Scalar‘𝑊) RingHom 𝑊))
52	47, 51	eleqtrrd 2855	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (𝑍 RingHom 𝑊))
53		eqid 2758	. . . . . . . 8 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
54	53, 4	rhmmhm 19550	. . . . . . 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 19318	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
58		eqid 2758	. . . . . . 7 ⊢ (._g‘(mulGrp‘𝑍)) = (._g‘(mulGrp‘𝑍))
59	57, 58, 3	mhmmulg 18340	. . . . . 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 1368	. . . . 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 7171	. . . . 5 ⊢ (𝜑 → (𝑃 ↑ 𝐴) = (𝑃 ↑ (𝐶‘((ℤRHom‘𝑍)‘𝐸))))
63	60, 62	eqtr4d 2796	. . . 4 ⊢ (𝜑 → (𝐶‘(𝑃(._g‘(mulGrp‘𝑍))((ℤRHom‘𝑍)‘𝐸))) = (𝑃 ↑ 𝐴))
64	11, 31, 58	znfermltl 31087	. . . . . . 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 6666	. . . . 5 ⊢ (𝜑 → (𝐶‘(𝑃(._g‘(mulGrp‘𝑍))((ℤRHom‘𝑍)‘𝐸))) = (𝐶‘((ℤRHom‘𝑍)‘𝐸)))
67	66, 27	eqtr4di 2811	. . . 4 ⊢ (𝜑 → (𝐶‘(𝑃(._g‘(mulGrp‘𝑍))((ℤRHom‘𝑍)‘𝐸))) = 𝐴)
68	43, 63, 67	3eqtr2d 2799	. . 3 ⊢ (𝜑 → ((chr‘𝑊) ↑ 𝐴) = 𝐴)
69	42, 68	oveq12d 7173	. 2 ⊢ (𝜑 → (((chr‘𝑊) ↑ 𝑋) + ((chr‘𝑊) ↑ 𝐴)) = ((𝑃 ↑ 𝑋) + 𝐴))
70	40, 41, 69	3eqtr3d 2801	1 ⊢ (𝜑 → (𝑃 ↑ (𝑋 + 𝐴)) = ((𝑃 ↑ 𝑋) + 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ⟶wf 6335 ‘cfv 6339 (class class class)co 7155 ℕcn 11679 ℕ₀cn0 11939 ℤcz 12025 ℙcprime 16072 Basecbs 16546 +_gcplusg 16628 Scalarcsca 16631 MndHom cmhm 18025 Grpcgrp 18174 ._gcmg 18296 mulGrpcmgp 19312 Ringcrg 19370 CRingccrg 19371 RingHom crh 19540 ℤ_ringzring 20243 ℤRHomczrh 20274 chrcchr 20276 ℤ/nℤczn 20277 AssAlgcasa 20620 algSccascl 20622 var₁cv1 20905 Poly₁cpl1 20906

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5159 ax-sep 5172 ax-nul 5179 ax-pow 5237 ax-pr 5301 ax-un 7464 ax-cnex 10636 ax-resscn 10637 ax-1cn 10638 ax-icn 10639 ax-addcl 10640 ax-addrcl 10641 ax-mulcl 10642 ax-mulrcl 10643 ax-mulcom 10644 ax-addass 10645 ax-mulass 10646 ax-distr 10647 ax-i2m1 10648 ax-1ne0 10649 ax-1rid 10650 ax-rnegex 10651 ax-rrecex 10652 ax-cnre 10653 ax-pre-lttri 10654 ax-pre-lttrn 10655 ax-pre-ltadd 10656 ax-pre-mulgt0 10657 ax-pre-sup 10658 ax-addf 10659 ax-mulf 10660

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-iin 4889 df-br 5036 df-opab 5098 df-mpt 5116 df-tr 5142 df-id 5433 df-eprel 5438 df-po 5446 df-so 5447 df-fr 5486 df-se 5487 df-we 5488 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-res 5539 df-ima 5540 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7410 df-ofr 7411 df-om 7585 df-1st 7698 df-2nd 7699 df-supp 7841 df-tpos 7907 df-wrecs 7962 df-recs 8023 df-rdg 8061 df-1o 8117 df-2o 8118 df-oadd 8121 df-er 8304 df-ec 8306 df-qs 8310 df-map 8423 df-pm 8424 df-ixp 8485 df-en 8533 df-dom 8534 df-sdom 8535 df-fin 8536 df-fsupp 8872 df-sup 8944 df-inf 8945 df-oi 9012 df-dju 9368 df-card 9406 df-pnf 10720 df-mnf 10721 df-xr 10722 df-ltxr 10723 df-le 10724 df-sub 10915 df-neg 10916 df-div 11341 df-nn 11680 df-2 11742 df-3 11743 df-4 11744 df-5 11745 df-6 11746 df-7 11747 df-8 11748 df-9 11749 df-n0 11940 df-xnn0 12012 df-z 12026 df-dec 12143 df-uz 12288 df-rp 12436 df-fz 12945 df-fzo 13088 df-fl 13216 df-mod 13292 df-seq 13424 df-exp 13485 df-fac 13689 df-bc 13718 df-hash 13746 df-cj 14511 df-re 14512 df-im 14513 df-sqrt 14647 df-abs 14648 df-dvds 15661 df-gcd 15899 df-prm 16073 df-phi 16163 df-struct 16548 df-ndx 16549 df-slot 16550 df-base 16552 df-sets 16553 df-ress 16554 df-plusg 16641 df-mulr 16642 df-starv 16643 df-sca 16644 df-vsca 16645 df-ip 16646 df-tset 16647 df-ple 16648 df-ds 16650 df-unif 16651 df-0g 16778 df-gsum 16779 df-imas 16844 df-qus 16845 df-mre 16920 df-mrc 16921 df-acs 16923 df-mgm 17923 df-sgrp 17972 df-mnd 17983 df-mhm 18027 df-submnd 18028 df-grp 18177 df-minusg 18178 df-sbg 18179 df-mulg 18297 df-subg 18348 df-nsg 18349 df-eqg 18350 df-ghm 18428 df-cntz 18519 df-od 18728 df-cmn 18980 df-abl 18981 df-mgp 19313 df-ur 19325 df-srg 19329 df-ring 19372 df-cring 19373 df-oppr 19449 df-dvdsr 19467 df-unit 19468 df-invr 19498 df-dvr 19509 df-rnghom 19543 df-drng 19577 df-subrg 19606 df-lmod 19709 df-lss 19777 df-lsp 19817 df-sra 20017 df-rgmod 20018 df-lidl 20019 df-rsp 20020 df-2idl 20078 df-cnfld 20172 df-zring 20244 df-zrh 20278 df-chr 20280 df-zn 20281 df-assa 20623 df-ascl 20625 df-psr 20676 df-mvr 20677 df-mpl 20678 df-opsr 20680 df-psr1 20909 df-vr1 20910 df-ply1 20911 df-coe1 20912

This theorem is referenced by: (None)

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