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Mirrors > Home > MPE Home > Th. List > cygznlem1 | Structured version Visualization version GIF version |
Description: Lemma for cygzn 20711. (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 13711 | . . . . . . 7 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0) | |
3 | 2 | adantl 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ Fin) → (♯‘𝐵) ∈ ℕ0) |
4 | 0nn0 11906 | . . . . . . 7 ⊢ 0 ∈ ℕ0 | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ Fin) → 0 ∈ ℕ0) |
6 | 3, 5 | ifclda 4500 | . . . . 5 ⊢ (𝜑 → if(𝐵 ∈ Fin, (♯‘𝐵), 0) ∈ ℕ0) |
7 | 1, 6 | eqeltrid 2917 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
8 | 7 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑁 ∈ ℕ0) |
9 | simprl 769 | . . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝐾 ∈ ℤ) | |
10 | simprr 771 | . . 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 20690 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝐿‘𝐾) = (𝐿‘𝑀) ↔ 𝑁 ∥ (𝐾 − 𝑀))) |
14 | 8, 9, 10, 13 | syl3anc 1367 | . 2 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝐿‘𝐾) = (𝐿‘𝑀) ↔ 𝑁 ∥ (𝐾 − 𝑀))) |
15 | cygzn.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ CycGrp) | |
16 | cyggrp 19003 | . . . . . . 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 2821 | . . . . . . 7 ⊢ (od‘𝐺) = (od‘𝐺) | |
23 | 19, 20, 21, 22 | cyggenod2 18998 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐸) → ((od‘𝐺)‘𝑋) = if(𝐵 ∈ Fin, (♯‘𝐵), 0)) |
24 | 17, 18, 23 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → ((od‘𝐺)‘𝑋) = if(𝐵 ∈ Fin, (♯‘𝐵), 0)) |
25 | 24, 1 | syl6eqr 2874 | . . . 4 ⊢ (𝜑 → ((od‘𝐺)‘𝑋) = 𝑁) |
26 | 25 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((od‘𝐺)‘𝑋) = 𝑁) |
27 | 26 | breq1d 5068 | . 2 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((od‘𝐺)‘𝑋) ∥ (𝐾 − 𝑀) ↔ 𝑁 ∥ (𝐾 − 𝑀))) |
28 | 17 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝐺 ∈ Grp) |
29 | 19, 20, 21 | iscyggen 18993 | . . . . . 6 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵)) |
30 | 29 | simplbi 500 | . . . . 5 ⊢ (𝑋 ∈ 𝐸 → 𝑋 ∈ 𝐵) |
31 | 18, 30 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
32 | 31 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑋 ∈ 𝐵) |
33 | eqid 2821 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
34 | 19, 22, 20, 33 | odcong 18671 | . . 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 1370 | . 2 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((od‘𝐺)‘𝑋) ∥ (𝐾 − 𝑀) ↔ (𝐾 · 𝑋) = (𝑀 · 𝑋))) |
36 | 14, 27, 35 | 3bitr2d 309 | 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 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 ifcif 4466 class class class wbr 5058 ↦ cmpt 5138 ran crn 5550 ‘cfv 6349 (class class class)co 7150 Fincfn 8503 0cc0 10531 − cmin 10864 ℕ0cn0 11891 ℤcz 11975 ♯chash 13684 ∥ cdvds 15601 Basecbs 16477 0gc0g 16707 Grpcgrp 18097 .gcmg 18218 odcod 18646 CycGrpccyg 18990 ℤRHomczrh 20641 ℤ/nℤczn 20644 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-omul 8101 df-er 8283 df-ec 8285 df-qs 8289 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-acn 9365 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-fz 12887 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-dvds 15602 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-0g 16709 df-imas 16775 df-qus 16776 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-nsg 18271 df-eqg 18272 df-ghm 18350 df-od 18650 df-cmn 18902 df-abl 18903 df-cyg 18991 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-oppr 19367 df-dvdsr 19385 df-rnghom 19461 df-subrg 19527 df-lmod 19630 df-lss 19698 df-lsp 19738 df-sra 19938 df-rgmod 19939 df-lidl 19940 df-rsp 19941 df-2idl 19999 df-cnfld 20540 df-zring 20612 df-zrh 20645 df-zn 20648 |
This theorem is referenced by: cygznlem2a 20708 cygznlem3 20710 |
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