ply1fermltlchr - Metamath Proof Explorer

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Theorem ply1fermltlchr 22332

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 2735	. . 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 6910	. . . 4 ⊢ (._g‘𝑁) = (._g‘(mulGrp‘𝑊))
6	3, 5	eqtri 2763	. . 3 ⊢ ↑ = (._g‘(mulGrp‘𝑊))
7		eqid 2735	. . 3 ⊢ (chr‘𝑊) = (chr‘𝑊)
8		ply1fermltlchr.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ CRing)
9		ply1fermltlchr.w	. . . . 5 ⊢ 𝑊 = (Poly₁‘𝐹)
10	9	ply1crng 22216	. . . 4 ⊢ (𝐹 ∈ CRing → 𝑊 ∈ CRing)
11	8, 10	syl 17	. . 3 ⊢ (𝜑 → 𝑊 ∈ CRing)
12	9	ply1chr 22326	. . . . . 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 2793	. . . 4 ⊢ (𝜑 → (chr‘𝑊) = 𝑃)
16		ply1fermltlchr.1	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℙ)
17	15, 16	eqeltrd 2839	. . 3 ⊢ (𝜑 → (chr‘𝑊) ∈ ℙ)
18	8	crngringd 20264	. . . 4 ⊢ (𝜑 → 𝐹 ∈ Ring)
19		ply1fermltlchr.x	. . . . 5 ⊢ 𝑋 = (var₁‘𝐹)
20	19, 9, 1	vr1cl 22235	. . . 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 2735	. . . . . . . 8 ⊢ (ℤRHom‘𝐹) = (ℤRHom‘𝐹)
24	23	zrhrhm 21540	. . . . . . 7 ⊢ (𝐹 ∈ Ring → (ℤRHom‘𝐹) ∈ (ℤ_ring RingHom 𝐹))
25		zringbas 21482	. . . . . . . 8 ⊢ ℤ = (Base‘ℤ_ring)
26		eqid 2735	. . . . . . . 8 ⊢ (Base‘𝐹) = (Base‘𝐹)
27	25, 26	rhmf 20502	. . . . . . 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 22305	. . . . 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 2843	. . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑊))
35	1, 2, 6, 7, 11, 17, 21, 34	freshmansdream 21611	. 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 22217	. . . . . . . . 9 ⊢ (𝐹 ∈ CRing → 𝑊 ∈ AssAlg)
40		eqid 2735	. . . . . . . . . 10 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
41	31, 40	asclrhm 21928	. . . . . . . . 9 ⊢ (𝑊 ∈ AssAlg → 𝐶 ∈ ((Scalar‘𝑊) RingHom 𝑊))
42	8, 39, 41	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ((Scalar‘𝑊) RingHom 𝑊))
43	8	crnggrpd 20265	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ Grp)
44	9	ply1sca 22270	. . . . . . . . . 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 2842	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (𝐹 RingHom 𝑊))
48		eqid 2735	. . . . . . . 8 ⊢ (mulGrp‘𝐹) = (mulGrp‘𝐹)
49	48, 4	rhmmhm 20496	. . . . . . 7 ⊢ (𝐶 ∈ (𝐹 RingHom 𝑊) → 𝐶 ∈ ((mulGrp‘𝐹) MndHom 𝑁))
50	47, 49	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ((mulGrp‘𝐹) MndHom 𝑁))
51		prmnn 16708	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
52		nnnn0 12531	. . . . . . 7 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℕ₀)
53	16, 51, 52	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
54	48, 26	mgpbas 20158	. . . . . . 7 ⊢ (Base‘𝐹) = (Base‘(mulGrp‘𝐹))
55		eqid 2735	. . . . . . 7 ⊢ (._g‘(mulGrp‘𝐹)) = (._g‘(mulGrp‘𝐹))
56	54, 55, 3	mhmmulg 19146	. . . . . 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 1370	. . . . 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 2778	. . . 4 ⊢ (𝜑 → (𝐶‘(𝑃(._g‘(mulGrp‘𝐹))((ℤRHom‘𝐹)‘𝐸))) = (𝑃 ↑ 𝐴))
61		eqid 2735	. . . . . . 7 ⊢ ((ℤRHom‘𝐹)‘𝐸) = ((ℤRHom‘𝐹)‘𝐸)
62	14, 26, 55, 61, 16, 29, 8	fermltlchr 21562	. . . . . 6 ⊢ (𝜑 → (𝑃(._g‘(mulGrp‘𝐹))((ℤRHom‘𝐹)‘𝐸)) = ((ℤRHom‘𝐹)‘𝐸))
63	62	fveq2d 6911	. . . . 5 ⊢ (𝜑 → (𝐶‘(𝑃(._g‘(mulGrp‘𝐹))((ℤRHom‘𝐹)‘𝐸))) = (𝐶‘((ℤRHom‘𝐹)‘𝐸)))
64	63, 22	eqtr4di 2793	. . . 4 ⊢ (𝜑 → (𝐶‘(𝑃(._g‘(mulGrp‘𝐹))((ℤRHom‘𝐹)‘𝐸))) = 𝐴)
65	38, 60, 64	3eqtr2d 2781	. . 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 2783	1 ⊢ (𝜑 → (𝑃 ↑ (𝑋 + 𝐴)) = ((𝑃 ↑ 𝑋) + 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℕcn 12264 ℕ₀cn0 12524 ℤcz 12611 ℙcprime 16705 Basecbs 17245 +_gcplusg 17298 Scalarcsca 17301 MndHom cmhm 18807 Grpcgrp 18964 ._gcmg 19098 mulGrpcmgp 20152 Ringcrg 20251 CRingccrg 20252 RingHom crh 20486 ℤ_ringczring 21475 ℤRHomczrh 21528 chrcchr 21530 AssAlgcasa 21888 algSccascl 21890 var₁cv1 22193 Poly₁cpl1 22194

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-dvds 16288 df-gcd 16529 df-prm 16706 df-phi 16800 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-od 19561 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-srg 20205 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-drng 20748 df-lmod 20877 df-lss 20948 df-cnfld 21383 df-zring 21476 df-zrh 21532 df-chr 21534 df-assa 21891 df-ascl 21893 df-psr 21947 df-mvr 21948 df-mpl 21949 df-opsr 21951 df-psr1 22197 df-vr1 22198 df-ply1 22199 df-coe1 22200

This theorem is referenced by: ply1fermltl 33589 aks6d1c1p2 42091

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