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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 2823 | . . . 4 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
3 | 1, 2 | iscyg 19000 | . . 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 7191 | . . . . . 6 ⊢ (𝑛(.g‘𝐺)𝑥) ∈ V | |
6 | eqid 2823 | . . . . . 6 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) | |
7 | 5, 6 | fnmpti 6493 | . . . . 5 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝐺)𝑥)) Fn ℤ |
8 | df-fo 6363 | . . . . 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 9111 | . . . . . . . 8 ⊢ ω ∈ On | |
11 | onenon 9380 | . . . . . . . 8 ⊢ (ω ∈ On → ω ∈ dom card) | |
12 | 10, 11 | ax-mp 5 | . . . . . . 7 ⊢ ω ∈ dom card |
13 | znnen 15567 | . . . . . . . . 9 ⊢ ℤ ≈ ℕ | |
14 | nnenom 13351 | . . . . . . . . 9 ⊢ ℕ ≈ ω | |
15 | 13, 14 | entri 8565 | . . . . . . . 8 ⊢ ℤ ≈ ω |
16 | ennum 9378 | . . . . . . . 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 9485 | . . . . . 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 8570 | . . . . . 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 3286 | . 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 3141 class class class wbr 5068 ↦ cmpt 5148 dom cdm 5557 ran crn 5558 Oncon0 6193 Fn wfn 6352 –onto→wfo 6355 ‘cfv 6357 (class class class)co 7158 ωcom 7582 ≈ cen 8508 ≼ cdom 8509 cardccrd 9366 ℕcn 11640 ℤcz 11984 Basecbs 16485 Grpcgrp 18105 .gcmg 18226 CycGrpccyg 18998 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-omul 8109 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-oi 8976 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-cyg 18999 |
This theorem is referenced by: (None) |
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