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Mirrors > Home > MPE Home > Th. List > cycsubg2cl | Structured version Visualization version GIF version |
Description: Any multiple of an element is contained in the generated cyclic subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
Ref | Expression |
---|---|
cycsubg2cl.x | ⊢ 𝑋 = (Base‘𝐺) |
cycsubg2cl.t | ⊢ · = (.g‘𝐺) |
cycsubg2cl.k | ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) |
Ref | Expression |
---|---|
cycsubg2cl | ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → (𝑁 · 𝐴) ∈ (𝐾‘{𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cycsubg2cl.x | . . . . . 6 ⊢ 𝑋 = (Base‘𝐺) | |
2 | 1 | subgacs 18312 | . . . . 5 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝑋)) |
3 | 2 | acsmred 16926 | . . . 4 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (Moore‘𝑋)) |
4 | 3 | 3ad2ant1 1129 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → (SubGrp‘𝐺) ∈ (Moore‘𝑋)) |
5 | simp2 1133 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → 𝐴 ∈ 𝑋) | |
6 | 5 | snssd 4741 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → {𝐴} ⊆ 𝑋) |
7 | cycsubg2cl.k | . . . 4 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) | |
8 | 7 | mrccl 16881 | . . 3 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝑋) ∧ {𝐴} ⊆ 𝑋) → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺)) |
9 | 4, 6, 8 | syl2anc 586 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺)) |
10 | simp3 1134 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ) | |
11 | 4, 7, 6 | mrcssidd 16895 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → {𝐴} ⊆ (𝐾‘{𝐴})) |
12 | snssg 4716 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ (𝐾‘{𝐴}) ↔ {𝐴} ⊆ (𝐾‘{𝐴}))) | |
13 | 12 | 3ad2ant2 1130 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → (𝐴 ∈ (𝐾‘{𝐴}) ↔ {𝐴} ⊆ (𝐾‘{𝐴}))) |
14 | 11, 13 | mpbird 259 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → 𝐴 ∈ (𝐾‘{𝐴})) |
15 | cycsubg2cl.t | . . 3 ⊢ · = (.g‘𝐺) | |
16 | 15 | subgmulgcl 18291 | . 2 ⊢ (((𝐾‘{𝐴}) ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ (𝐾‘{𝐴})) → (𝑁 · 𝐴) ∈ (𝐾‘{𝐴})) |
17 | 9, 10, 14, 16 | syl3anc 1367 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → (𝑁 · 𝐴) ∈ (𝐾‘{𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 {csn 4566 ‘cfv 6354 (class class class)co 7155 ℤcz 11980 Basecbs 16482 Moorecmre 16852 mrClscmrc 16853 Grpcgrp 18102 .gcmg 18223 SubGrpcsubg 18272 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-seq 13369 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-0g 16714 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-grp 18105 df-minusg 18106 df-mulg 18224 df-subg 18275 |
This theorem is referenced by: odngen 18701 |
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