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Mirrors > Home > MPE Home > Th. List > cygznlem1 | Structured version Visualization version GIF version |
Description: Lemma for cygzn 21607. (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 14392 | . . . . . . 7 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0) | |
3 | 2 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ Fin) → (♯‘𝐵) ∈ ℕ0) |
4 | 0nn0 12539 | . . . . . . 7 ⊢ 0 ∈ ℕ0 | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ Fin) → 0 ∈ ℕ0) |
6 | 3, 5 | ifclda 4566 | . . . . 5 ⊢ (𝜑 → if(𝐵 ∈ Fin, (♯‘𝐵), 0) ∈ ℕ0) |
7 | 1, 6 | eqeltrid 2843 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
8 | 7 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑁 ∈ ℕ0) |
9 | simprl 771 | . . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝐾 ∈ ℤ) | |
10 | simprr 773 | . . 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 21586 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝐿‘𝐾) = (𝐿‘𝑀) ↔ 𝑁 ∥ (𝐾 − 𝑀))) |
14 | 8, 9, 10, 13 | syl3anc 1370 | . 2 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝐿‘𝐾) = (𝐿‘𝑀) ↔ 𝑁 ∥ (𝐾 − 𝑀))) |
15 | cygzn.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ CycGrp) | |
16 | cyggrp 19923 | . . . . . . 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 2735 | . . . . . . 7 ⊢ (od‘𝐺) = (od‘𝐺) | |
23 | 19, 20, 21, 22 | cyggenod2 19918 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐸) → ((od‘𝐺)‘𝑋) = if(𝐵 ∈ Fin, (♯‘𝐵), 0)) |
24 | 17, 18, 23 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → ((od‘𝐺)‘𝑋) = if(𝐵 ∈ Fin, (♯‘𝐵), 0)) |
25 | 24, 1 | eqtr4di 2793 | . . . 4 ⊢ (𝜑 → ((od‘𝐺)‘𝑋) = 𝑁) |
26 | 25 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((od‘𝐺)‘𝑋) = 𝑁) |
27 | 26 | breq1d 5158 | . 2 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((od‘𝐺)‘𝑋) ∥ (𝐾 − 𝑀) ↔ 𝑁 ∥ (𝐾 − 𝑀))) |
28 | 17 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝐺 ∈ Grp) |
29 | 19, 20, 21 | iscyggen 19913 | . . . . . 6 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵)) |
30 | 29 | simplbi 497 | . . . . 5 ⊢ (𝑋 ∈ 𝐸 → 𝑋 ∈ 𝐵) |
31 | 18, 30 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
32 | 31 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑋 ∈ 𝐵) |
33 | eqid 2735 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
34 | 19, 22, 20, 33 | odcong 19582 | . . 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 1373 | . 2 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((od‘𝐺)‘𝑋) ∥ (𝐾 − 𝑀) ↔ (𝐾 · 𝑋) = (𝑀 · 𝑋))) |
36 | 14, 27, 35 | 3bitr2d 307 | 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 ifcif 4531 class class class wbr 5148 ↦ cmpt 5231 ran crn 5690 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 0cc0 11153 − cmin 11490 ℕ0cn0 12524 ℤcz 12611 ♯chash 14366 ∥ cdvds 16287 Basecbs 17245 0gc0g 17486 Grpcgrp 18964 .gcmg 19098 odcod 19557 CycGrpccyg 19910 ℤRHomczrh 21528 ℤ/nℤczn 21531 |
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-inf2 9679 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-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-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-omul 8510 df-er 8744 df-ec 8746 df-qs 8750 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-acn 9980 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-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fz 13545 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-dvds 16288 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-0g 17488 df-imas 17555 df-qus 17556 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-nsg 19155 df-eqg 19156 df-ghm 19244 df-od 19561 df-cmn 19815 df-abl 19816 df-cyg 19911 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-lmod 20877 df-lss 20948 df-lsp 20988 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 df-2idl 21278 df-cnfld 21383 df-zring 21476 df-zrh 21532 df-zn 21535 |
This theorem is referenced by: cygznlem2a 21604 cygznlem3 21606 |
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