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Mirrors > Home > MPE Home > Th. List > 0cyg | Structured version Visualization version GIF version |
Description: The trivial group is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
cygctb.1 | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
0cyg | ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1o) → 𝐺 ∈ CycGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cygctb.1 | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2818 | . 2 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
3 | simpl 483 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1o) → 𝐺 ∈ Grp) | |
4 | eqid 2818 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
5 | 1, 4 | grpidcl 18069 | . . 3 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵) |
6 | 5 | adantr 481 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1o) → (0g‘𝐺) ∈ 𝐵) |
7 | 0z 11980 | . . 3 ⊢ 0 ∈ ℤ | |
8 | en1eqsn 8736 | . . . . . . . 8 ⊢ (((0g‘𝐺) ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {(0g‘𝐺)}) | |
9 | 5, 8 | sylan 580 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1o) → 𝐵 = {(0g‘𝐺)}) |
10 | 9 | eleq2d 2895 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1o) → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ {(0g‘𝐺)})) |
11 | 10 | biimpa 477 | . . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ≈ 1o) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ {(0g‘𝐺)}) |
12 | velsn 4573 | . . . . 5 ⊢ (𝑥 ∈ {(0g‘𝐺)} ↔ 𝑥 = (0g‘𝐺)) | |
13 | 11, 12 | sylib 219 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ≈ 1o) ∧ 𝑥 ∈ 𝐵) → 𝑥 = (0g‘𝐺)) |
14 | 1, 4, 2 | mulg0 18169 | . . . . . 6 ⊢ ((0g‘𝐺) ∈ 𝐵 → (0(.g‘𝐺)(0g‘𝐺)) = (0g‘𝐺)) |
15 | 6, 14 | syl 17 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1o) → (0(.g‘𝐺)(0g‘𝐺)) = (0g‘𝐺)) |
16 | 15 | adantr 481 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ≈ 1o) ∧ 𝑥 ∈ 𝐵) → (0(.g‘𝐺)(0g‘𝐺)) = (0g‘𝐺)) |
17 | 13, 16 | eqtr4d 2856 | . . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ≈ 1o) ∧ 𝑥 ∈ 𝐵) → 𝑥 = (0(.g‘𝐺)(0g‘𝐺))) |
18 | oveq1 7152 | . . . 4 ⊢ (𝑛 = 0 → (𝑛(.g‘𝐺)(0g‘𝐺)) = (0(.g‘𝐺)(0g‘𝐺))) | |
19 | 18 | rspceeqv 3635 | . . 3 ⊢ ((0 ∈ ℤ ∧ 𝑥 = (0(.g‘𝐺)(0g‘𝐺))) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g‘𝐺)(0g‘𝐺))) |
20 | 7, 17, 19 | sylancr 587 | . 2 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ≈ 1o) ∧ 𝑥 ∈ 𝐵) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g‘𝐺)(0g‘𝐺))) |
21 | 1, 2, 3, 6, 20 | iscygd 18935 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1o) → 𝐺 ∈ CycGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 {csn 4557 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 1oc1o 8084 ≈ cen 8494 0cc0 10525 ℤcz 11969 Basecbs 16471 0gc0g 16701 Grpcgrp 18041 .gcmg 18162 CycGrpccyg 18925 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-seq 13358 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-mulg 18163 df-cyg 18926 |
This theorem is referenced by: lt6abl 18944 frgpcyg 20648 |
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