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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 2610 | . . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
3 | 1, 2 | iscyg 18104 | . . 3 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) |
4 | 3 | simprbi 479 | . 2 ⊢ (𝐺 ∈ CycGrp → ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵) |
5 | ovex 6577 | . . . . . 6 ⊢ (𝑛(.g‘𝐺)𝑥) ∈ V | |
6 | eqid 2610 | . . . . . 6 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) | |
7 | 5, 6 | fnmpti 5935 | . . . . 5 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) Fn ℤ |
8 | df-fo 5810 | . . . . 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 955 | . . . 4 ⊢ ((𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)):ℤ–onto→𝐵 ↔ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵) |
10 | omelon 8426 | . . . . . . . 8 ⊢ ω ∈ On | |
11 | onenon 8658 | . . . . . . . 8 ⊢ (ω ∈ On → ω ∈ dom card) | |
12 | 10, 11 | ax-mp 5 | . . . . . . 7 ⊢ ω ∈ dom card |
13 | znnen 14780 | . . . . . . . . 9 ⊢ ℤ ≈ ℕ | |
14 | nnenom 12641 | . . . . . . . . 9 ⊢ ℕ ≈ ω | |
15 | 13, 14 | entri 7896 | . . . . . . . 8 ⊢ ℤ ≈ ω |
16 | ennum 8656 | . . . . . . . 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 220 | . . . . . 6 ⊢ ℤ ∈ dom card |
19 | fodomnum 8763 | . . . . . 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 7901 | . . . . . 6 ⊢ ((𝐵 ≼ ℤ ∧ ℤ ≈ ω) → 𝐵 ≼ ω) | |
22 | 15, 21 | mpan2 703 | . . . . 5 ⊢ (𝐵 ≼ ℤ → 𝐵 ≼ ω) |
23 | 20, 22 | syl6 34 | . . . 4 ⊢ ((𝐺 ∈ CycGrp ∧ 𝑥 ∈ 𝐵) → ((𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)):ℤ–onto→𝐵 → 𝐵 ≼ ω)) |
24 | 9, 23 | syl5bir 232 | . . 3 ⊢ ((𝐺 ∈ CycGrp ∧ 𝑥 ∈ 𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵 → 𝐵 ≼ ω)) |
25 | 24 | rexlimdva 3013 | . 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ran crn 5039 Oncon0 5640 Fn wfn 5799 –onto→wfo 5802 ‘cfv 5804 (class class class)co 6549 ωcom 6957 ≈ cen 7838 ≼ cdom 7839 cardccrd 8644 ℕcn 10897 ℤcz 11254 Basecbs 15695 Grpcgrp 17245 .gcmg 17363 CycGrpccyg 18102 |
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 |
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-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 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-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-cyg 18103 |
This theorem is referenced by: (None) |
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