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 22316

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 6909	. . . 4 ⊢ (._g‘𝑁) = (._g‘(mulGrp‘𝑊))
6	3, 5	eqtri 2765	. . 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 22200	. . . 4 ⊢ (𝐹 ∈ CRing → 𝑊 ∈ CRing)
11	8, 10	syl 17	. . 3 ⊢ (𝜑 → 𝑊 ∈ CRing)
12	9	ply1chr 22310	. . . . . 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 2795	. . . 4 ⊢ (𝜑 → (chr‘𝑊) = 𝑃)
16		ply1fermltlchr.1	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℙ)
17	15, 16	eqeltrd 2841	. . 3 ⊢ (𝜑 → (chr‘𝑊) ∈ ℙ)
18	8	crngringd 20243	. . . 4 ⊢ (𝜑 → 𝐹 ∈ Ring)
19		ply1fermltlchr.x	. . . . 5 ⊢ 𝑋 = (var₁‘𝐹)
20	19, 9, 1	vr1cl 22219	. . . 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 21522	. . . . . . 7 ⊢ (𝐹 ∈ Ring → (ℤRHom‘𝐹) ∈ (ℤ_ring RingHom 𝐹))
25		zringbas 21464	. . . . . . . 8 ⊢ ℤ = (Base‘ℤ_ring)
26		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝐹) = (Base‘𝐹)
27	25, 26	rhmf 20485	. . . . . . 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 7105	. . . . 5 ⊢ (𝜑 → ((ℤRHom‘𝐹)‘𝐸) ∈ (Base‘𝐹))
31		ply1fermltlchr.c	. . . . . 6 ⊢ 𝐶 = (algSc‘𝑊)
32	9, 31, 26, 1	ply1sclcl 22289	. . . . 5 ⊢ ((𝐹 ∈ Ring ∧ ((ℤRHom‘𝐹)‘𝐸) ∈ (Base‘𝐹)) → (𝐶‘((ℤRHom‘𝐹)‘𝐸)) ∈ (Base‘𝑊))
33	18, 30, 32	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐶‘((ℤRHom‘𝐹)‘𝐸)) ∈ (Base‘𝑊))
34	22, 33	eqeltrid 2845	. . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑊))
35	1, 2, 6, 7, 11, 17, 21, 34	freshmansdream 21593	. 2 ⊢ (𝜑 → ((chr‘𝑊) ↑ (𝑋 + 𝐴)) = (((chr‘𝑊) ↑ 𝑋) + ((chr‘𝑊) ↑ 𝐴)))
36	15	oveq1d 7446	. 2 ⊢ (𝜑 → ((chr‘𝑊) ↑ (𝑋 + 𝐴)) = (𝑃 ↑ (𝑋 + 𝐴)))
37	15	oveq1d 7446	. . 3 ⊢ (𝜑 → ((chr‘𝑊) ↑ 𝑋) = (𝑃 ↑ 𝑋))
38	15	oveq1d 7446	. . . 4 ⊢ (𝜑 → ((chr‘𝑊) ↑ 𝐴) = (𝑃 ↑ 𝐴))
39	9	ply1assa 22201	. . . . . . . . 9 ⊢ (𝐹 ∈ CRing → 𝑊 ∈ AssAlg)
40		eqid 2737	. . . . . . . . . 10 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
41	31, 40	asclrhm 21910	. . . . . . . . 9 ⊢ (𝑊 ∈ AssAlg → 𝐶 ∈ ((Scalar‘𝑊) RingHom 𝑊))
42	8, 39, 41	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ((Scalar‘𝑊) RingHom 𝑊))
43	8	crnggrpd 20244	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ Grp)
44	9	ply1sca 22254	. . . . . . . . . 10 ⊢ (𝐹 ∈ Grp → 𝐹 = (Scalar‘𝑊))
45	43, 44	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (Scalar‘𝑊))
46	45	oveq1d 7446	. . . . . . . 8 ⊢ (𝜑 → (𝐹 RingHom 𝑊) = ((Scalar‘𝑊) RingHom 𝑊))
47	42, 46	eleqtrrd 2844	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (𝐹 RingHom 𝑊))
48		eqid 2737	. . . . . . . 8 ⊢ (mulGrp‘𝐹) = (mulGrp‘𝐹)
49	48, 4	rhmmhm 20479	. . . . . . 7 ⊢ (𝐶 ∈ (𝐹 RingHom 𝑊) → 𝐶 ∈ ((mulGrp‘𝐹) MndHom 𝑁))
50	47, 49	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ((mulGrp‘𝐹) MndHom 𝑁))
51		prmnn 16711	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
52		nnnn0 12533	. . . . . . 7 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ₀)
53	16, 51, 52	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
54	48, 26	mgpbas 20142	. . . . . . 7 ⊢ (Base‘𝐹) = (Base‘(mulGrp‘𝐹))
55		eqid 2737	. . . . . . 7 ⊢ (._g‘(mulGrp‘𝐹)) = (._g‘(mulGrp‘𝐹))
56	54, 55, 3	mhmmulg 19133	. . . . . 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 1373	. . . . 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 7447	. . . . 5 ⊢ (𝜑 → (𝑃 ↑ 𝐴) = (𝑃 ↑ (𝐶‘((ℤRHom‘𝐹)‘𝐸))))
60	57, 59	eqtr4d 2780	. . . 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 21544	. . . . . 6 ⊢ (𝜑 → (𝑃(._g‘(mulGrp‘𝐹))((ℤRHom‘𝐹)‘𝐸)) = ((ℤRHom‘𝐹)‘𝐸))
63	62	fveq2d 6910	. . . . 5 ⊢ (𝜑 → (𝐶‘(𝑃(._g‘(mulGrp‘𝐹))((ℤRHom‘𝐹)‘𝐸))) = (𝐶‘((ℤRHom‘𝐹)‘𝐸)))
64	63, 22	eqtr4di 2795	. . . 4 ⊢ (𝜑 → (𝐶‘(𝑃(._g‘(mulGrp‘𝐹))((ℤRHom‘𝐹)‘𝐸))) = 𝐴)
65	38, 60, 64	3eqtr2d 2783	. . 3 ⊢ (𝜑 → ((chr‘𝑊) ↑ 𝐴) = 𝐴)
66	37, 65	oveq12d 7449	. 2 ⊢ (𝜑 → (((chr‘𝑊) ↑ 𝑋) + ((chr‘𝑊) ↑ 𝐴)) = ((𝑃 ↑ 𝑋) + 𝐴))
67	35, 36, 66	3eqtr3d 2785	1 ⊢ (𝜑 → (𝑃 ↑ (𝑋 + 𝐴)) = ((𝑃 ↑ 𝑋) + 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℕcn 12266 ℕ₀cn0 12526 ℤcz 12613 ℙcprime 16708 Basecbs 17247 +_gcplusg 17297 Scalarcsca 17300 MndHom cmhm 18794 Grpcgrp 18951 ._gcmg 19085 mulGrpcmgp 20137 Ringcrg 20230 CRingccrg 20231 RingHom crh 20469 ℤ_ringczring 21457 ℤRHomczrh 21510 chrcchr 21512 AssAlgcasa 21870 algSccascl 21872 var₁cv1 22177 Poly₁cpl1 22178

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-gcd 16532 df-prm 16709 df-phi 16803 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-od 19546 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-srg 20184 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-rhm 20472 df-subrng 20546 df-subrg 20570 df-drng 20731 df-lmod 20860 df-lss 20930 df-cnfld 21365 df-zring 21458 df-zrh 21514 df-chr 21516 df-assa 21873 df-ascl 21875 df-psr 21929 df-mvr 21930 df-mpl 21931 df-opsr 21933 df-psr1 22181 df-vr1 22182 df-ply1 22183 df-coe1 22184

This theorem is referenced by: ply1fermltl 33609 aks6d1c1p2 42110

W3C validator