iscygd - Metamath Proof Explorer

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Theorem iscygd 19931

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 3152	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑋))
5		iscyg.1	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
6		iscyg.2	. . . . . 6 ⊢ · = (._g‘𝐺)
7		eqid 2740	. . . . . 6 ⊢ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}
8	5, 6, 7	iscyggen2 19925	. . . . 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 712	. . 3 ⊢ (𝜑 → 𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵})
11	10	ne0d 4365	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} ≠ ∅)
12	5, 6, 7	iscyg2 19926	. 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 {crab 3443 ∅c0 4352 ↦ cmpt 5249 ran crn 5701 ‘cfv 6575 (class class class)co 7450 ℤcz 12641 Basecbs 17260 Grpcgrp 18975 ._gcmg 19109 CycGrpccyg 19921

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-1st 8032 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-n0 12556 df-z 12642 df-uz 12906 df-fz 13570 df-seq 14055 df-0g 17503 df-mgm 18680 df-sgrp 18759 df-mnd 18775 df-grp 18978 df-minusg 18979 df-mulg 19110 df-cyg 19922

This theorem is referenced by: 0cyg 19937 ghmcyg 19940 cycsubgcyg 19945 ablsimpgcygd 20152 zringcyg 21505 frgpcyg 21617

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