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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 2821 | . . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
3 | 1, 2 | iscyg 18998 | . . 3 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) |
4 | 3 | simprbi 499 | . 2 ⊢ (𝐺 ∈ CycGrp → ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵) |
5 | ovex 7189 | . . . . . 6 ⊢ (𝑛(.g‘𝐺)𝑥) ∈ V | |
6 | eqid 2821 | . . . . . 6 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) | |
7 | 5, 6 | fnmpti 6491 | . . . . 5 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) Fn ℤ |
8 | df-fo 6361 | . . . . 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 707 | . . . 4 ⊢ ((𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)):ℤ–onto→𝐵 ↔ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵) |
10 | omelon 9109 | . . . . . . . 8 ⊢ ω ∈ On | |
11 | onenon 9378 | . . . . . . . 8 ⊢ (ω ∈ On → ω ∈ dom card) | |
12 | 10, 11 | ax-mp 5 | . . . . . . 7 ⊢ ω ∈ dom card |
13 | znnen 15565 | . . . . . . . . 9 ⊢ ℤ ≈ ℕ | |
14 | nnenom 13349 | . . . . . . . . 9 ⊢ ℕ ≈ ω | |
15 | 13, 14 | entri 8563 | . . . . . . . 8 ⊢ ℤ ≈ ω |
16 | ennum 9376 | . . . . . . . 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 233 | . . . . . 6 ⊢ ℤ ∈ dom card |
19 | fodomnum 9483 | . . . . . 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 8568 | . . . . . 6 ⊢ ((𝐵 ≼ ℤ ∧ ℤ ≈ ω) → 𝐵 ≼ ω) | |
22 | 15, 21 | mpan2 689 | . . . . 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 245 | . . 3 ⊢ ((𝐺 ∈ CycGrp ∧ 𝑥 ∈ 𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵 → 𝐵 ≼ ω)) |
25 | 24 | rexlimdva 3284 | . 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 class class class wbr 5066 ↦ cmpt 5146 dom cdm 5555 ran crn 5556 Oncon0 6191 Fn wfn 6350 –onto→wfo 6353 ‘cfv 6355 (class class class)co 7156 ωcom 7580 ≈ cen 8506 ≼ cdom 8507 cardccrd 9364 ℕcn 11638 ℤcz 11982 Basecbs 16483 Grpcgrp 18103 .gcmg 18224 CycGrpccyg 18996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-omul 8107 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-oi 8974 df-card 9368 df-acn 9371 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-cyg 18997 |
This theorem is referenced by: (None) |
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