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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 2610 | . 2 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
3 | simpl 472 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1𝑜) → 𝐺 ∈ Grp) | |
4 | eqid 2610 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
5 | 1, 4 | grpidcl 17273 | . . 3 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵) |
6 | 5 | adantr 480 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1𝑜) → (0g‘𝐺) ∈ 𝐵) |
7 | 0z 11265 | . . 3 ⊢ 0 ∈ ℤ | |
8 | en1eqsn 8075 | . . . . . . . 8 ⊢ (((0g‘𝐺) ∈ 𝐵 ∧ 𝐵 ≈ 1𝑜) → 𝐵 = {(0g‘𝐺)}) | |
9 | 5, 8 | sylan 487 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1𝑜) → 𝐵 = {(0g‘𝐺)}) |
10 | 9 | eleq2d 2673 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1𝑜) → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ {(0g‘𝐺)})) |
11 | 10 | biimpa 500 | . . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ≈ 1𝑜) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ {(0g‘𝐺)}) |
12 | velsn 4141 | . . . . 5 ⊢ (𝑥 ∈ {(0g‘𝐺)} ↔ 𝑥 = (0g‘𝐺)) | |
13 | 11, 12 | sylib 207 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ≈ 1𝑜) ∧ 𝑥 ∈ 𝐵) → 𝑥 = (0g‘𝐺)) |
14 | 1, 4, 2 | mulg0 17369 | . . . . . 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 480 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ≈ 1𝑜) ∧ 𝑥 ∈ 𝐵) → (0(.g‘𝐺)(0g‘𝐺)) = (0g‘𝐺)) |
17 | 13, 16 | eqtr4d 2647 | . . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ≈ 1𝑜) ∧ 𝑥 ∈ 𝐵) → 𝑥 = (0(.g‘𝐺)(0g‘𝐺))) |
18 | oveq1 6556 | . . . . 5 ⊢ (𝑛 = 0 → (𝑛(.g‘𝐺)(0g‘𝐺)) = (0(.g‘𝐺)(0g‘𝐺))) | |
19 | 18 | eqeq2d 2620 | . . . 4 ⊢ (𝑛 = 0 → (𝑥 = (𝑛(.g‘𝐺)(0g‘𝐺)) ↔ 𝑥 = (0(.g‘𝐺)(0g‘𝐺)))) |
20 | 19 | rspcev 3282 | . . 3 ⊢ ((0 ∈ ℤ ∧ 𝑥 = (0(.g‘𝐺)(0g‘𝐺))) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g‘𝐺)(0g‘𝐺))) |
21 | 7, 17, 20 | sylancr 694 | . 2 ⊢ (((𝐺 ∈ Grp ∧ 𝐵 ≈ 1𝑜) ∧ 𝑥 ∈ 𝐵) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g‘𝐺)(0g‘𝐺))) |
22 | 1, 2, 3, 6, 21 | iscygd 18112 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1𝑜) → 𝐺 ∈ CycGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 {csn 4125 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 1𝑜c1o 7440 ≈ cen 7838 0cc0 9815 ℤcz 11254 Basecbs 15695 0gc0g 15923 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-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-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-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-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 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-fz 12198 df-seq 12664 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-mulg 17364 df-cyg 18103 |
This theorem is referenced by: lt6abl 18119 frgpcyg 19741 |
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