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Mirrors > Home > MPE Home > Th. List > cygznlem1 | Structured version Visualization version GIF version |
Description: Lemma for cygzn 19738. (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 13009 | . . . . . . 7 ⊢ (𝐵 ∈ Fin → (#‘𝐵) ∈ ℕ0) | |
3 | 2 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ Fin) → (#‘𝐵) ∈ ℕ0) |
4 | 0nn0 11184 | . . . . . . 7 ⊢ 0 ∈ ℕ0 | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ Fin) → 0 ∈ ℕ0) |
6 | 3, 5 | ifclda 4070 | . . . . 5 ⊢ (𝜑 → if(𝐵 ∈ Fin, (#‘𝐵), 0) ∈ ℕ0) |
7 | 1, 6 | syl5eqel 2692 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
8 | 7 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑁 ∈ ℕ0) |
9 | simprl 790 | . . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝐾 ∈ ℤ) | |
10 | simprr 792 | . . 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 19717 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝐿‘𝐾) = (𝐿‘𝑀) ↔ 𝑁 ∥ (𝐾 − 𝑀))) |
14 | 8, 9, 10, 13 | syl3anc 1318 | . 2 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝐿‘𝐾) = (𝐿‘𝑀) ↔ 𝑁 ∥ (𝐾 − 𝑀))) |
15 | cygzn.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ CycGrp) | |
16 | cyggrp 18114 | . . . . . . 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 2610 | . . . . . . 7 ⊢ (od‘𝐺) = (od‘𝐺) | |
23 | 19, 20, 21, 22 | cyggenod2 18110 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐸) → ((od‘𝐺)‘𝑋) = if(𝐵 ∈ Fin, (#‘𝐵), 0)) |
24 | 17, 18, 23 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → ((od‘𝐺)‘𝑋) = if(𝐵 ∈ Fin, (#‘𝐵), 0)) |
25 | 24, 1 | syl6eqr 2662 | . . . 4 ⊢ (𝜑 → ((od‘𝐺)‘𝑋) = 𝑁) |
26 | 25 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((od‘𝐺)‘𝑋) = 𝑁) |
27 | 26 | breq1d 4593 | . 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 18105 | . . . . . 6 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵)) |
30 | 29 | simplbi 475 | . . . . 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 2610 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
34 | 19, 22, 20, 33 | odcong 17791 | . . 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 1322 | . 2 ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((od‘𝐺)‘𝑋) ∥ (𝐾 − 𝑀) ↔ (𝐾 · 𝑋) = (𝑀 · 𝑋))) |
36 | 14, 27, 35 | 3bitr2d 295 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 ifcif 4036 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 0cc0 9815 − cmin 10145 ℕ0cn0 11169 ℤcz 11254 #chash 12979 ∥ cdvds 14821 Basecbs 15695 0gc0g 15923 Grpcgrp 17245 .gcmg 17363 odcod 17767 CycGrpccyg 18102 ℤRHomczrh 19667 ℤ/nℤczn 19670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-fz 12198 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-imas 15991 df-qus 15992 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-nsg 17415 df-eqg 17416 df-ghm 17481 df-od 17771 df-cmn 18018 df-abl 18019 df-cyg 18103 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-rnghom 18538 df-subrg 18601 df-lmod 18688 df-lss 18754 df-lsp 18793 df-sra 18993 df-rgmod 18994 df-lidl 18995 df-rsp 18996 df-2idl 19053 df-cnfld 19568 df-zring 19638 df-zrh 19671 df-zn 19674 |
This theorem is referenced by: cygznlem2a 19735 cygznlem3 19737 |
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