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Mirrors > Home > MPE Home > Th. List > cygctb | Structured version Visualization version GIF version |
Description: A cyclic group is countable. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
cygctb.1 | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
cygctb | ⊢ (𝐺 ∈ CycGrp → 𝐵 ≼ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cygctb.1 | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2622 | . . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
3 | 1, 2 | iscyg 18281 | . . 3 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) |
4 | 3 | simprbi 480 | . 2 ⊢ (𝐺 ∈ CycGrp → ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵) |
5 | ovex 6678 | . . . . . 6 ⊢ (𝑛(.g‘𝐺)𝑥) ∈ V | |
6 | eqid 2622 | . . . . . 6 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) | |
7 | 5, 6 | fnmpti 6022 | . . . . 5 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) Fn ℤ |
8 | df-fo 5894 | . . . . 5 ⊢ ((𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)):ℤ–onto→𝐵 ↔ ((𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) Fn ℤ ∧ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) | |
9 | 7, 8 | mpbiran 953 | . . . 4 ⊢ ((𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)):ℤ–onto→𝐵 ↔ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵) |
10 | omelon 8543 | . . . . . . . 8 ⊢ ω ∈ On | |
11 | onenon 8775 | . . . . . . . 8 ⊢ (ω ∈ On → ω ∈ dom card) | |
12 | 10, 11 | ax-mp 5 | . . . . . . 7 ⊢ ω ∈ dom card |
13 | znnen 14941 | . . . . . . . . 9 ⊢ ℤ ≈ ℕ | |
14 | nnenom 12779 | . . . . . . . . 9 ⊢ ℕ ≈ ω | |
15 | 13, 14 | entri 8010 | . . . . . . . 8 ⊢ ℤ ≈ ω |
16 | ennum 8773 | . . . . . . . 8 ⊢ (ℤ ≈ ω → (ℤ ∈ dom card ↔ ω ∈ dom card)) | |
17 | 15, 16 | ax-mp 5 | . . . . . . 7 ⊢ (ℤ ∈ dom card ↔ ω ∈ dom card) |
18 | 12, 17 | mpbir 221 | . . . . . 6 ⊢ ℤ ∈ dom card |
19 | fodomnum 8880 | . . . . . 6 ⊢ (ℤ ∈ dom card → ((𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)):ℤ–onto→𝐵 → 𝐵 ≼ ℤ)) | |
20 | 18, 19 | mp1i 13 | . . . . 5 ⊢ ((𝐺 ∈ CycGrp ∧ 𝑥 ∈ 𝐵) → ((𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)):ℤ–onto→𝐵 → 𝐵 ≼ ℤ)) |
21 | domentr 8015 | . . . . . 6 ⊢ ((𝐵 ≼ ℤ ∧ ℤ ≈ ω) → 𝐵 ≼ ω) | |
22 | 15, 21 | mpan2 707 | . . . . 5 ⊢ (𝐵 ≼ ℤ → 𝐵 ≼ ω) |
23 | 20, 22 | syl6 35 | . . . 4 ⊢ ((𝐺 ∈ CycGrp ∧ 𝑥 ∈ 𝐵) → ((𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)):ℤ–onto→𝐵 → 𝐵 ≼ ω)) |
24 | 9, 23 | syl5bir 233 | . . 3 ⊢ ((𝐺 ∈ CycGrp ∧ 𝑥 ∈ 𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵 → 𝐵 ≼ ω)) |
25 | 24 | rexlimdva 3031 | . 2 ⊢ (𝐺 ∈ CycGrp → (∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵 → 𝐵 ≼ ω)) |
26 | 4, 25 | mpd 15 | 1 ⊢ (𝐺 ∈ CycGrp → 𝐵 ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 ran crn 5115 Oncon0 5723 Fn wfn 5883 –onto→wfo 5886 ‘cfv 5888 (class class class)co 6650 ωcom 7065 ≈ cen 7952 ≼ cdom 7953 cardccrd 8761 ℕcn 11020 ℤcz 11377 Basecbs 15857 Grpcgrp 17422 .gcmg 17540 CycGrpccyg 18279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-oi 8415 df-card 8765 df-acn 8768 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-cyg 18280 |
This theorem is referenced by: (None) |
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