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Mirrors > Home > MPE Home > Th. List > cygznlem1 | Structured version Visualization version GIF version |
Description: Lemma for cygzn 20314. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
cygzn.b | ⊢ 𝐵 = (Base‘𝐺) |
cygzn.n | ⊢ 𝑁 = if(𝐵 ∈ Fin, (♯‘𝐵), 0) |
cygzn.y | ⊢ 𝑌 = (ℤ/nℤ‘𝑁) |
cygzn.m | ⊢ · = (.g‘𝐺) |
cygzn.l | ⊢ 𝐿 = (ℤRHom‘𝑌) |
cygzn.e | ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} |
cygzn.g | ⊢ (𝜑 → 𝐺 ∈ CycGrp) |
cygzn.x | ⊢ (𝜑 → 𝑋 ∈ 𝐸) |
Ref | Expression |
---|---|
cygznlem1 | ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝐿‘𝐾) = (𝐿‘𝑀) ↔ (𝐾 · 𝑋) = (𝑀 · 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cygzn.n | . . . . 5 ⊢ 𝑁 = if(𝐵 ∈ Fin, (♯‘𝐵), 0) | |
2 | hashcl 13462 | . . . . . . 7 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0) | |
3 | 2 | adantl 475 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ Fin) → (♯‘𝐵) ∈ ℕ0) |
4 | 0nn0 11659 | . . . . . . 7 ⊢ 0 ∈ ℕ0 | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ Fin) → 0 ∈ ℕ0) |
6 | 3, 5 | ifclda 4340 | . . . . 5 ⊢ (𝜑 → if(𝐵 ∈ Fin, (♯‘𝐵), 0) ∈ ℕ0) |
7 | 1, 6 | syl5eqel 2862 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
8 | 7 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑁 ∈ ℕ0) |
9 | simprl 761 | . . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝐾 ∈ ℤ) | |
10 | simprr 763 | . . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑀 ∈ ℤ) | |
11 | cygzn.y | . . . 4 ⊢ 𝑌 = (ℤ/nℤ‘𝑁) | |
12 | cygzn.l | . . . 4 ⊢ 𝐿 = (ℤRHom‘𝑌) | |
13 | 11, 12 | zndvds 20293 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝐿‘𝐾) = (𝐿‘𝑀) ↔ 𝑁 ∥ (𝐾 − 𝑀))) |
14 | 8, 9, 10, 13 | syl3anc 1439 | . 2 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝐿‘𝐾) = (𝐿‘𝑀) ↔ 𝑁 ∥ (𝐾 − 𝑀))) |
15 | cygzn.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ CycGrp) | |
16 | cyggrp 18677 | . . . . . . 7 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp) | |
17 | 15, 16 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp) |
18 | cygzn.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐸) | |
19 | cygzn.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐺) | |
20 | cygzn.m | . . . . . . 7 ⊢ · = (.g‘𝐺) | |
21 | cygzn.e | . . . . . . 7 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} | |
22 | eqid 2777 | . . . . . . 7 ⊢ (od‘𝐺) = (od‘𝐺) | |
23 | 19, 20, 21, 22 | cyggenod2 18673 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐸) → ((od‘𝐺)‘𝑋) = if(𝐵 ∈ Fin, (♯‘𝐵), 0)) |
24 | 17, 18, 23 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → ((od‘𝐺)‘𝑋) = if(𝐵 ∈ Fin, (♯‘𝐵), 0)) |
25 | 24, 1 | syl6eqr 2831 | . . . 4 ⊢ (𝜑 → ((od‘𝐺)‘𝑋) = 𝑁) |
26 | 25 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((od‘𝐺)‘𝑋) = 𝑁) |
27 | 26 | breq1d 4896 | . 2 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((od‘𝐺)‘𝑋) ∥ (𝐾 − 𝑀) ↔ 𝑁 ∥ (𝐾 − 𝑀))) |
28 | 17 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝐺 ∈ Grp) |
29 | 19, 20, 21 | iscyggen 18668 | . . . . . 6 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵)) |
30 | 29 | simplbi 493 | . . . . 5 ⊢ (𝑋 ∈ 𝐸 → 𝑋 ∈ 𝐵) |
31 | 18, 30 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
32 | 31 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑋 ∈ 𝐵) |
33 | eqid 2777 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
34 | 19, 22, 20, 33 | odcong 18352 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((od‘𝐺)‘𝑋) ∥ (𝐾 − 𝑀) ↔ (𝐾 · 𝑋) = (𝑀 · 𝑋))) |
35 | 28, 32, 9, 10, 34 | syl112anc 1442 | . 2 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((od‘𝐺)‘𝑋) ∥ (𝐾 − 𝑀) ↔ (𝐾 · 𝑋) = (𝑀 · 𝑋))) |
36 | 14, 27, 35 | 3bitr2d 299 | 1 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝐿‘𝐾) = (𝐿‘𝑀) ↔ (𝐾 · 𝑋) = (𝑀 · 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 {crab 3093 ifcif 4306 class class class wbr 4886 ↦ cmpt 4965 ran crn 5356 ‘cfv 6135 (class class class)co 6922 Fincfn 8241 0cc0 10272 − cmin 10606 ℕ0cn0 11642 ℤcz 11728 ♯chash 13435 ∥ cdvds 15387 Basecbs 16255 0gc0g 16486 Grpcgrp 17809 .gcmg 17927 odcod 18328 CycGrpccyg 18665 ℤRHomczrh 20244 ℤ/nℤczn 20247 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-omul 7848 df-er 8026 df-ec 8028 df-qs 8032 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-acn 9101 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-rp 12138 df-fz 12644 df-fl 12912 df-mod 12988 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-dvds 15388 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-0g 16488 df-imas 16554 df-qus 16555 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-grp 17812 df-minusg 17813 df-sbg 17814 df-mulg 17928 df-subg 17975 df-nsg 17976 df-eqg 17977 df-ghm 18042 df-od 18332 df-cmn 18581 df-abl 18582 df-cyg 18666 df-mgp 18877 df-ur 18889 df-ring 18936 df-cring 18937 df-oppr 19010 df-dvdsr 19028 df-rnghom 19104 df-subrg 19170 df-lmod 19257 df-lss 19325 df-lsp 19367 df-sra 19569 df-rgmod 19570 df-lidl 19571 df-rsp 19572 df-2idl 19629 df-cnfld 20143 df-zring 20215 df-zrh 20248 df-zn 20251 |
This theorem is referenced by: cygznlem2a 20311 cygznlem3 20313 |
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