ply1fermltlchr - Metamath Proof Explorer

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Theorem ply1fermltlchr 22337

Description: Fermat's little theorem for polynomials in a commutative ring 𝐹 of characteristic 𝑃 prime: we have the polynomial equation (𝑋 + 𝐴)↑𝑃 = ((𝑋↑𝑃) + 𝐴). (Contributed by Thierry Arnoux, 9-Jan-2025.)

**Hypotheses**
Ref	Expression
ply1fermltlchr.w	⊢ 𝑊 = (Poly₁‘𝐹)
ply1fermltlchr.x	⊢ 𝑋 = (var₁‘𝐹)
ply1fermltlchr.l	⊢ + = (+_g‘𝑊)
ply1fermltlchr.n	⊢ 𝑁 = (mulGrp‘𝑊)
ply1fermltlchr.t	⊢ ↑ = (._g‘𝑁)
ply1fermltlchr.c	⊢ 𝐶 = (algSc‘𝑊)
ply1fermltlchr.a	⊢ 𝐴 = (𝐶‘((ℤRHom‘𝐹)‘𝐸))
ply1fermltlchr.p	⊢ 𝑃 = (chr‘𝐹)
ply1fermltlchr.f	⊢ (𝜑 → 𝐹 ∈ CRing)
ply1fermltlchr.1	⊢ (𝜑 → 𝑃 ∈ ℙ)
ply1fermltlchr.2	⊢ (𝜑 → 𝐸 ∈ ℤ)

**Assertion**
Ref	Expression
ply1fermltlchr	⊢ (𝜑 → (𝑃 ↑ (𝑋 + 𝐴)) = ((𝑃 ↑ 𝑋) + 𝐴))

**Proof of Theorem ply1fermltlchr**
Step	Hyp	Ref	Expression
1		eqid 2740	. . 3 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		ply1fermltlchr.l	. . 3 ⊢ + = (+_g‘𝑊)
3		ply1fermltlchr.t	. . . 4 ⊢ ↑ = (._g‘𝑁)
4		ply1fermltlchr.n	. . . . 5 ⊢ 𝑁 = (mulGrp‘𝑊)
5	4	fveq2i 6923	. . . 4 ⊢ (._g‘𝑁) = (._g‘(mulGrp‘𝑊))
6	3, 5	eqtri 2768	. . 3 ⊢ ↑ = (._g‘(mulGrp‘𝑊))
7		eqid 2740	. . 3 ⊢ (chr‘𝑊) = (chr‘𝑊)
8		ply1fermltlchr.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ CRing)
9		ply1fermltlchr.w	. . . . 5 ⊢ 𝑊 = (Poly₁‘𝐹)
10	9	ply1crng 22221	. . . 4 ⊢ (𝐹 ∈ CRing → 𝑊 ∈ CRing)
11	8, 10	syl 17	. . 3 ⊢ (𝜑 → 𝑊 ∈ CRing)
12	9	ply1chr 22331	. . . . . 6 ⊢ (𝐹 ∈ CRing → (chr‘𝑊) = (chr‘𝐹))
13	8, 12	syl 17	. . . . 5 ⊢ (𝜑 → (chr‘𝑊) = (chr‘𝐹))
14		ply1fermltlchr.p	. . . . 5 ⊢ 𝑃 = (chr‘𝐹)
15	13, 14	eqtr4di 2798	. . . 4 ⊢ (𝜑 → (chr‘𝑊) = 𝑃)
16		ply1fermltlchr.1	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℙ)
17	15, 16	eqeltrd 2844	. . 3 ⊢ (𝜑 → (chr‘𝑊) ∈ ℙ)
18	8	crngringd 20273	. . . 4 ⊢ (𝜑 → 𝐹 ∈ Ring)
19		ply1fermltlchr.x	. . . . 5 ⊢ 𝑋 = (var₁‘𝐹)
20	19, 9, 1	vr1cl 22240	. . . 4 ⊢ (𝐹 ∈ Ring → 𝑋 ∈ (Base‘𝑊))
21	18, 20	syl 17	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊))
22		ply1fermltlchr.a	. . . 4 ⊢ 𝐴 = (𝐶‘((ℤRHom‘𝐹)‘𝐸))
23		eqid 2740	. . . . . . . 8 ⊢ (ℤRHom‘𝐹) = (ℤRHom‘𝐹)
24	23	zrhrhm 21545	. . . . . . 7 ⊢ (𝐹 ∈ Ring → (ℤRHom‘𝐹) ∈ (ℤ_ring RingHom 𝐹))
25		zringbas 21487	. . . . . . . 8 ⊢ ℤ = (Base‘ℤ_ring)
26		eqid 2740	. . . . . . . 8 ⊢ (Base‘𝐹) = (Base‘𝐹)
27	25, 26	rhmf 20511	. . . . . . 7 ⊢ ((ℤRHom‘𝐹) ∈ (ℤ_ring RingHom 𝐹) → (ℤRHom‘𝐹):ℤ⟶(Base‘𝐹))
28	18, 24, 27	3syl 18	. . . . . 6 ⊢ (𝜑 → (ℤRHom‘𝐹):ℤ⟶(Base‘𝐹))
29		ply1fermltlchr.2	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℤ)
30	28, 29	ffvelcdmd 7119	. . . . 5 ⊢ (𝜑 → ((ℤRHom‘𝐹)‘𝐸) ∈ (Base‘𝐹))
31		ply1fermltlchr.c	. . . . . 6 ⊢ 𝐶 = (algSc‘𝑊)
32	9, 31, 26, 1	ply1sclcl 22310	. . . . 5 ⊢ ((𝐹 ∈ Ring ∧ ((ℤRHom‘𝐹)‘𝐸) ∈ (Base‘𝐹)) → (𝐶‘((ℤRHom‘𝐹)‘𝐸)) ∈ (Base‘𝑊))
33	18, 30, 32	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝐶‘((ℤRHom‘𝐹)‘𝐸)) ∈ (Base‘𝑊))
34	22, 33	eqeltrid 2848	. . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑊))
35	1, 2, 6, 7, 11, 17, 21, 34	freshmansdream 21616	. 2 ⊢ (𝜑 → ((chr‘𝑊) ↑ (𝑋 + 𝐴)) = (((chr‘𝑊) ↑ 𝑋) + ((chr‘𝑊) ↑ 𝐴)))
36	15	oveq1d 7463	. 2 ⊢ (𝜑 → ((chr‘𝑊) ↑ (𝑋 + 𝐴)) = (𝑃 ↑ (𝑋 + 𝐴)))
37	15	oveq1d 7463	. . 3 ⊢ (𝜑 → ((chr‘𝑊) ↑ 𝑋) = (𝑃 ↑ 𝑋))
38	15	oveq1d 7463	. . . 4 ⊢ (𝜑 → ((chr‘𝑊) ↑ 𝐴) = (𝑃 ↑ 𝐴))
39	9	ply1assa 22222	. . . . . . . . 9 ⊢ (𝐹 ∈ CRing → 𝑊 ∈ AssAlg)
40		eqid 2740	. . . . . . . . . 10 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
41	31, 40	asclrhm 21933	. . . . . . . . 9 ⊢ (𝑊 ∈ AssAlg → 𝐶 ∈ ((Scalar‘𝑊) RingHom 𝑊))
42	8, 39, 41	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ((Scalar‘𝑊) RingHom 𝑊))
43	8	crnggrpd 20274	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ Grp)
44	9	ply1sca 22275	. . . . . . . . . 10 ⊢ (𝐹 ∈ Grp → 𝐹 = (Scalar‘𝑊))
45	43, 44	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (Scalar‘𝑊))
46	45	oveq1d 7463	. . . . . . . 8 ⊢ (𝜑 → (𝐹 RingHom 𝑊) = ((Scalar‘𝑊) RingHom 𝑊))
47	42, 46	eleqtrrd 2847	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (𝐹 RingHom 𝑊))
48		eqid 2740	. . . . . . . 8 ⊢ (mulGrp‘𝐹) = (mulGrp‘𝐹)
49	48, 4	rhmmhm 20505	. . . . . . 7 ⊢ (𝐶 ∈ (𝐹 RingHom 𝑊) → 𝐶 ∈ ((mulGrp‘𝐹) MndHom 𝑁))
50	47, 49	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ((mulGrp‘𝐹) MndHom 𝑁))
51		prmnn 16721	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
52		nnnn0 12560	. . . . . . 7 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ₀)
53	16, 51, 52	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
54	48, 26	mgpbas 20167	. . . . . . 7 ⊢ (Base‘𝐹) = (Base‘(mulGrp‘𝐹))
55		eqid 2740	. . . . . . 7 ⊢ (._g‘(mulGrp‘𝐹)) = (._g‘(mulGrp‘𝐹))
56	54, 55, 3	mhmmulg 19155	. . . . . 6 ⊢ ((𝐶 ∈ ((mulGrp‘𝐹) MndHom 𝑁) ∧ 𝑃 ∈ ℕ₀ ∧ ((ℤRHom‘𝐹)‘𝐸) ∈ (Base‘𝐹)) → (𝐶‘(𝑃(._g‘(mulGrp‘𝐹))((ℤRHom‘𝐹)‘𝐸))) = (𝑃 ↑ (𝐶‘((ℤRHom‘𝐹)‘𝐸))))
57	50, 53, 30, 56	syl3anc 1371	. . . . 5 ⊢ (𝜑 → (𝐶‘(𝑃(._g‘(mulGrp‘𝐹))((ℤRHom‘𝐹)‘𝐸))) = (𝑃 ↑ (𝐶‘((ℤRHom‘𝐹)‘𝐸))))
58	22	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐴 = (𝐶‘((ℤRHom‘𝐹)‘𝐸)))
59	58	oveq2d 7464	. . . . 5 ⊢ (𝜑 → (𝑃 ↑ 𝐴) = (𝑃 ↑ (𝐶‘((ℤRHom‘𝐹)‘𝐸))))
60	57, 59	eqtr4d 2783	. . . 4 ⊢ (𝜑 → (𝐶‘(𝑃(._g‘(mulGrp‘𝐹))((ℤRHom‘𝐹)‘𝐸))) = (𝑃 ↑ 𝐴))
61		eqid 2740	. . . . . . 7 ⊢ ((ℤRHom‘𝐹)‘𝐸) = ((ℤRHom‘𝐹)‘𝐸)
62	14, 26, 55, 61, 16, 29, 8	fermltlchr 21567	. . . . . 6 ⊢ (𝜑 → (𝑃(._g‘(mulGrp‘𝐹))((ℤRHom‘𝐹)‘𝐸)) = ((ℤRHom‘𝐹)‘𝐸))
63	62	fveq2d 6924	. . . . 5 ⊢ (𝜑 → (𝐶‘(𝑃(._g‘(mulGrp‘𝐹))((ℤRHom‘𝐹)‘𝐸))) = (𝐶‘((ℤRHom‘𝐹)‘𝐸)))
64	63, 22	eqtr4di 2798	. . . 4 ⊢ (𝜑 → (𝐶‘(𝑃(._g‘(mulGrp‘𝐹))((ℤRHom‘𝐹)‘𝐸))) = 𝐴)
65	38, 60, 64	3eqtr2d 2786	. . 3 ⊢ (𝜑 → ((chr‘𝑊) ↑ 𝐴) = 𝐴)
66	37, 65	oveq12d 7466	. 2 ⊢ (𝜑 → (((chr‘𝑊) ↑ 𝑋) + ((chr‘𝑊) ↑ 𝐴)) = ((𝑃 ↑ 𝑋) + 𝐴))
67	35, 36, 66	3eqtr3d 2788	1 ⊢ (𝜑 → (𝑃 ↑ (𝑋 + 𝐴)) = ((𝑃 ↑ 𝑋) + 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ℕcn 12293 ℕ₀cn0 12553 ℤcz 12639 ℙcprime 16718 Basecbs 17258 +_gcplusg 17311 Scalarcsca 17314 MndHom cmhm 18816 Grpcgrp 18973 ._gcmg 19107 mulGrpcmgp 20161 Ringcrg 20260 CRingccrg 20261 RingHom crh 20495 ℤ_ringczring 21480 ℤRHomczrh 21533 chrcchr 21535 AssAlgcasa 21893 algSccascl 21895 var₁cv1 22198 Poly₁cpl1 22199

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-ofr 7715 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-xnn0 12626 df-z 12640 df-dec 12759 df-uz 12904 df-rp 13058 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-dvds 16303 df-gcd 16541 df-prm 16719 df-phi 16813 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-0g 17501 df-gsum 17502 df-prds 17507 df-pws 17509 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-ghm 19253 df-cntz 19357 df-od 19570 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-srg 20214 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-rhm 20498 df-subrng 20572 df-subrg 20597 df-drng 20753 df-lmod 20882 df-lss 20953 df-cnfld 21388 df-zring 21481 df-zrh 21537 df-chr 21539 df-assa 21896 df-ascl 21898 df-psr 21952 df-mvr 21953 df-mpl 21954 df-opsr 21956 df-psr1 22202 df-vr1 22203 df-ply1 22204 df-coe1 22205

This theorem is referenced by: ply1fermltl 33574 aks6d1c1p2 42066

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