iscygd - Metamath Proof Explorer

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Theorem iscygd 19402

Description: Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)

**Hypotheses**
Ref	Expression
iscyg.1	⊢ 𝐵 = (Base‘𝐺)
iscyg.2	⊢ · = (._g‘𝐺)
iscygd.3	⊢ (𝜑 → 𝐺 ∈ Grp)
iscygd.4	⊢ (𝜑 → 𝑋 ∈ 𝐵)
iscygd.5	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋))

**Assertion**
Ref	Expression
iscygd	⊢ (𝜑 → 𝐺 ∈ CycGrp)

Distinct variable groups: 𝑦,𝑛,𝐵 𝑛,𝑋,𝑦 𝑛,𝐺,𝑦 𝜑,𝑦 · ,𝑛,𝑦 Allowed substitution hint: 𝜑(𝑛)

Proof of Theorem iscygd Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscygd.3	. 2 ⊢ (𝜑 → 𝐺 ∈ Grp)
2		iscygd.4	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3		iscygd.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋))
4	3	ralrimiva 3107	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋))
5		iscyg.1	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
6		iscyg.2	. . . . . 6 ⊢ · = (._g‘𝐺)
7		eqid 2738	. . . . . 6 ⊢ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}
8	5, 6, 7	iscyggen2 19396	. . . . 5 ⊢ (𝐺 ∈ Grp → (𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋))))
9	1, 8	syl 17	. . . 4 ⊢ (𝜑 → (𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋))))
10	2, 4, 9	mpbir2and 709	. . 3 ⊢ (𝜑 → 𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵})
11	10	ne0d 4266	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} ≠ ∅)
12	5, 6, 7	iscyg2 19397	. 2 ⊢ (𝐺 ∈ CycGrp ↔ (𝐺 ∈ Grp ∧ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} ≠ ∅))
13	1, 11, 12	sylanbrc 582	1 ⊢ (𝜑 → 𝐺 ∈ CycGrp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 {crab 3067 ∅c0 4253 ↦ cmpt 5153 ran crn 5581 ‘cfv 6418 (class class class)co 7255 ℤcz 12249 Basecbs 16840 Grpcgrp 18492 ._gcmg 18615 CycGrpccyg 19392

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-seq 13650 df-0g 17069 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-grp 18495 df-minusg 18496 df-mulg 18616 df-cyg 19393

This theorem is referenced by: 0cyg 19409 ghmcyg 19412 cycsubgcyg 19417 ablsimpgcygd 19624 zringcyg 20603 frgpcyg 20693

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