ply1fermltlchr - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ply1fermltlchr		Structured version Visualization version GIF version

Theorem ply1fermltlchr 22260

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 2737	. . 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 6838	. . . 4 ⊢ (._g‘𝑁) = (._g‘(mulGrp‘𝑊))
6	3, 5	eqtri 2760	. . 3 ⊢ ↑ = (._g‘(mulGrp‘𝑊))
7		eqid 2737	. . 3 ⊢ (chr‘𝑊) = (chr‘𝑊)
8		ply1fermltlchr.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ CRing)
9		ply1fermltlchr.w	. . . . 5 ⊢ 𝑊 = (Poly₁‘𝐹)
10	9	ply1crng 22143	. . . 4 ⊢ (𝐹 ∈ CRing → 𝑊 ∈ CRing)
11	8, 10	syl 17	. . 3 ⊢ (𝜑 → 𝑊 ∈ CRing)
12	9	ply1chr 22254	. . . . . 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 2790	. . . 4 ⊢ (𝜑 → (chr‘𝑊) = 𝑃)
16		ply1fermltlchr.1	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℙ)
17	15, 16	eqeltrd 2837	. . 3 ⊢ (𝜑 → (chr‘𝑊) ∈ ℙ)
18	8	crngringd 20185	. . . 4 ⊢ (𝜑 → 𝐹 ∈ Ring)
19		ply1fermltlchr.x	. . . . 5 ⊢ 𝑋 = (var₁‘𝐹)
20	19, 9, 1	vr1cl 22162	. . . 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 2737	. . . . . . . 8 ⊢ (ℤRHom‘𝐹) = (ℤRHom‘𝐹)
24	23	zrhrhm 21470	. . . . . . 7 ⊢ (𝐹 ∈ Ring → (ℤRHom‘𝐹) ∈ (ℤ_ring RingHom 𝐹))
25		zringbas 21412	. . . . . . . 8 ⊢ ℤ = (Base‘ℤ_ring)
26		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝐹) = (Base‘𝐹)
27	25, 26	rhmf 20424	. . . . . . 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 7032	. . . . 5 ⊢ (𝜑 → ((ℤRHom‘𝐹)‘𝐸) ∈ (Base‘𝐹))
31		ply1fermltlchr.c	. . . . . 6 ⊢ 𝐶 = (algSc‘𝑊)
32	9, 31, 26, 1	ply1sclcl 22232	. . . . 5 ⊢ ((𝐹 ∈ Ring ∧ ((ℤRHom‘𝐹)‘𝐸) ∈ (Base‘𝐹)) → (𝐶‘((ℤRHom‘𝐹)‘𝐸)) ∈ (Base‘𝑊))
33	18, 30, 32	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝐶‘((ℤRHom‘𝐹)‘𝐸)) ∈ (Base‘𝑊))
34	22, 33	eqeltrid 2841	. . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑊))
35	1, 2, 6, 7, 11, 17, 21, 34	freshmansdream 21533	. 2 ⊢ (𝜑 → ((chr‘𝑊) ↑ (𝑋 + 𝐴)) = (((chr‘𝑊) ↑ 𝑋) + ((chr‘𝑊) ↑ 𝐴)))
36	15	oveq1d 7375	. 2 ⊢ (𝜑 → ((chr‘𝑊) ↑ (𝑋 + 𝐴)) = (𝑃 ↑ (𝑋 + 𝐴)))
37	15	oveq1d 7375	. . 3 ⊢ (𝜑 → ((chr‘𝑊) ↑ 𝑋) = (𝑃 ↑ 𝑋))
38	15	oveq1d 7375	. . . 4 ⊢ (𝜑 → ((chr‘𝑊) ↑ 𝐴) = (𝑃 ↑ 𝐴))
39	9	ply1assa 22144	. . . . . . . . 9 ⊢ (𝐹 ∈ CRing → 𝑊 ∈ AssAlg)
40		eqid 2737	. . . . . . . . . 10 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
41	31, 40	asclrhm 21850	. . . . . . . . 9 ⊢ (𝑊 ∈ AssAlg → 𝐶 ∈ ((Scalar‘𝑊) RingHom 𝑊))
42	8, 39, 41	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ((Scalar‘𝑊) RingHom 𝑊))
43	8	crnggrpd 20186	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ Grp)
44	9	ply1sca 22197	. . . . . . . . . 10 ⊢ (𝐹 ∈ Grp → 𝐹 = (Scalar‘𝑊))
45	43, 44	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (Scalar‘𝑊))
46	45	oveq1d 7375	. . . . . . . 8 ⊢ (𝜑 → (𝐹 RingHom 𝑊) = ((Scalar‘𝑊) RingHom 𝑊))
47	42, 46	eleqtrrd 2840	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (𝐹 RingHom 𝑊))
48		eqid 2737	. . . . . . . 8 ⊢ (mulGrp‘𝐹) = (mulGrp‘𝐹)
49	48, 4	rhmmhm 20419	. . . . . . 7 ⊢ (𝐶 ∈ (𝐹 RingHom 𝑊) → 𝐶 ∈ ((mulGrp‘𝐹) MndHom 𝑁))
50	47, 49	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ((mulGrp‘𝐹) MndHom 𝑁))
51		prmnn 16605	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
52		nnnn0 12412	. . . . . . 7 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ₀)
53	16, 51, 52	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
54	48, 26	mgpbas 20084	. . . . . . 7 ⊢ (Base‘𝐹) = (Base‘(mulGrp‘𝐹))
55		eqid 2737	. . . . . . 7 ⊢ (._g‘(mulGrp‘𝐹)) = (._g‘(mulGrp‘𝐹))
56	54, 55, 3	mhmmulg 19049	. . . . . 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 1374	. . . . 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 7376	. . . . 5 ⊢ (𝜑 → (𝑃 ↑ 𝐴) = (𝑃 ↑ (𝐶‘((ℤRHom‘𝐹)‘𝐸))))
60	57, 59	eqtr4d 2775	. . . 4 ⊢ (𝜑 → (𝐶‘(𝑃(._g‘(mulGrp‘𝐹))((ℤRHom‘𝐹)‘𝐸))) = (𝑃 ↑ 𝐴))
61		eqid 2737	. . . . . . 7 ⊢ ((ℤRHom‘𝐹)‘𝐸) = ((ℤRHom‘𝐹)‘𝐸)
62	14, 26, 55, 61, 16, 29, 8	fermltlchr 21488	. . . . . 6 ⊢ (𝜑 → (𝑃(._g‘(mulGrp‘𝐹))((ℤRHom‘𝐹)‘𝐸)) = ((ℤRHom‘𝐹)‘𝐸))
63	62	fveq2d 6839	. . . . 5 ⊢ (𝜑 → (𝐶‘(𝑃(._g‘(mulGrp‘𝐹))((ℤRHom‘𝐹)‘𝐸))) = (𝐶‘((ℤRHom‘𝐹)‘𝐸)))
64	63, 22	eqtr4di 2790	. . . 4 ⊢ (𝜑 → (𝐶‘(𝑃(._g‘(mulGrp‘𝐹))((ℤRHom‘𝐹)‘𝐸))) = 𝐴)
65	38, 60, 64	3eqtr2d 2778	. . 3 ⊢ (𝜑 → ((chr‘𝑊) ↑ 𝐴) = 𝐴)
66	37, 65	oveq12d 7378	. 2 ⊢ (𝜑 → (((chr‘𝑊) ↑ 𝑋) + ((chr‘𝑊) ↑ 𝐴)) = ((𝑃 ↑ 𝑋) + 𝐴))
67	35, 36, 66	3eqtr3d 2780	1 ⊢ (𝜑 → (𝑃 ↑ (𝑋 + 𝐴)) = ((𝑃 ↑ 𝑋) + 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⟶wf 6489 ‘cfv 6493 (class class class)co 7360 ℕcn 12149 ℕ₀cn0 12405 ℤcz 12492 ℙcprime 16602 Basecbs 17140 +_gcplusg 17181 Scalarcsca 17184 MndHom cmhm 18710 Grpcgrp 18867 ._gcmg 19001 mulGrpcmgp 20079 Ringcrg 20172 CRingccrg 20173 RingHom crh 20409 ℤ_ringczring 21405 ℤRHomczrh 21458 chrcchr 21460 AssAlgcasa 21809 algSccascl 21811 var₁cv1 22120 Poly₁cpl1 22121

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109 ax-mulf 11110

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-ofr 7625 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-sup 9349 df-inf 9350 df-oi 9419 df-dju 9817 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-xnn0 12479 df-z 12493 df-dec 12612 df-uz 12756 df-rp 12910 df-fz 13428 df-fzo 13575 df-fl 13716 df-mod 13794 df-seq 13929 df-exp 13989 df-fac 14201 df-bc 14230 df-hash 14258 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-dvds 16184 df-gcd 16426 df-prm 16603 df-phi 16697 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-0g 17365 df-gsum 17366 df-prds 17371 df-pws 17373 df-mre 17509 df-mrc 17510 df-acs 17512 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18712 df-submnd 18713 df-grp 18870 df-minusg 18871 df-sbg 18872 df-mulg 19002 df-subg 19057 df-ghm 19146 df-cntz 19250 df-od 19461 df-cmn 19715 df-abl 19716 df-mgp 20080 df-rng 20092 df-ur 20121 df-srg 20126 df-ring 20174 df-cring 20175 df-oppr 20277 df-dvdsr 20297 df-unit 20298 df-invr 20328 df-dvr 20341 df-rhm 20412 df-subrng 20483 df-subrg 20507 df-drng 20668 df-lmod 20817 df-lss 20887 df-cnfld 21314 df-zring 21406 df-zrh 21462 df-chr 21464 df-assa 21812 df-ascl 21814 df-psr 21869 df-mvr 21870 df-mpl 21871 df-opsr 21873 df-psr1 22124 df-vr1 22125 df-ply1 22126 df-coe1 22127

This theorem is referenced by: ply1fermltl 33669 aks6d1c1p2 42431

W3C validator