![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2731 | . . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
3 | 1, 2 | iscyg 19795 | . . 3 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵)) |
4 | 3 | simprbi 496 | . 2 ⊢ (𝐺 ∈ CycGrp → ∃𝑥 ∈ 𝐵 ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵) |
5 | ovex 7445 | . . . . . 6 ⊢ (𝑛(.g‘𝐺)𝑥) ∈ V | |
6 | eqid 2731 | . . . . . 6 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) | |
7 | 5, 6 | fnmpti 6693 | . . . . 5 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) Fn ℤ |
8 | df-fo 6549 | . . . . 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 706 | . . . 4 ⊢ ((𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)):ℤ–onto→𝐵 ↔ ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵) |
10 | omelon 9647 | . . . . . . . 8 ⊢ ω ∈ On | |
11 | onenon 9950 | . . . . . . . 8 ⊢ (ω ∈ On → ω ∈ dom card) | |
12 | 10, 11 | ax-mp 5 | . . . . . . 7 ⊢ ω ∈ dom card |
13 | znnen 16162 | . . . . . . . . 9 ⊢ ℤ ≈ ℕ | |
14 | nnenom 13952 | . . . . . . . . 9 ⊢ ℕ ≈ ω | |
15 | 13, 14 | entri 9010 | . . . . . . . 8 ⊢ ℤ ≈ ω |
16 | ennum 9948 | . . . . . . . 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 230 | . . . . . 6 ⊢ ℤ ∈ dom card |
19 | fodomnum 10058 | . . . . . 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 9015 | . . . . . 6 ⊢ ((𝐵 ≼ ℤ ∧ ℤ ≈ ω) → 𝐵 ≼ ω) | |
22 | 15, 21 | mpan2 688 | . . . . 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 | biimtrrid 242 | . . 3 ⊢ ((𝐺 ∈ CycGrp ∧ 𝑥 ∈ 𝐵) → (ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = 𝐵 → 𝐵 ≼ ω)) |
25 | 24 | rexlimdva 3154 | . 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 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∃wrex 3069 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5676 ran crn 5677 Oncon0 6364 Fn wfn 6538 –onto→wfo 6541 ‘cfv 6543 (class class class)co 7412 ωcom 7859 ≈ cen 8942 ≼ cdom 8943 cardccrd 9936 ℕcn 12219 ℤcz 12565 Basecbs 17151 Grpcgrp 18861 .gcmg 18993 CycGrpccyg 19793 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-oadd 8476 df-omul 8477 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-oi 9511 df-card 9940 df-acn 9943 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-n0 12480 df-z 12566 df-uz 12830 df-cyg 19794 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |