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| Mirrors > Home > MPE Home > Th. List > ply1fermltlchr | Structured version Visualization version GIF version | ||
| 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.) |
| 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) |
| Ref | Expression |
|---|---|
| ply1fermltlchr | ⊢ (𝜑 → (𝑃 ↑ (𝑋 + 𝐴)) = ((𝑃 ↑ 𝑋) + 𝐴)) |
| 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) ∈ (ℤring RingHom 𝐹)) |
| 25 | zringbas 21464 | . . . . . . . 8 ⊢ ℤ = (Base‘ℤring) | |
| 26 | eqid 2737 | . . . . . . . 8 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
| 27 | 25, 26 | rhmf 20485 | . . . . . . 7 ⊢ ((ℤRHom‘𝐹) ∈ (ℤring 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 ℤringczring 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 |
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