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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 19118 | . . . . 5 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝑋)) |
3 | 2 | acsmred 17633 | . . . 4 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (Moore‘𝑋)) |
4 | 3 | 3ad2ant1 1130 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → (SubGrp‘𝐺) ∈ (Moore‘𝑋)) |
5 | simp2 1134 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → 𝐴 ∈ 𝑋) | |
6 | 5 | snssd 4808 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → {𝐴} ⊆ 𝑋) |
7 | cycsubg2cl.k | . . . 4 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) | |
8 | 7 | mrccl 17588 | . . 3 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝑋) ∧ {𝐴} ⊆ 𝑋) → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺)) |
9 | 4, 6, 8 | syl2anc 582 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺)) |
10 | simp3 1135 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ) | |
11 | 4, 7, 6 | mrcssidd 17602 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → {𝐴} ⊆ (𝐾‘{𝐴})) |
12 | snssg 4783 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ (𝐾‘{𝐴}) ↔ {𝐴} ⊆ (𝐾‘{𝐴}))) | |
13 | 12 | 3ad2ant2 1131 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → (𝐴 ∈ (𝐾‘{𝐴}) ↔ {𝐴} ⊆ (𝐾‘{𝐴}))) |
14 | 11, 13 | mpbird 256 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → 𝐴 ∈ (𝐾‘{𝐴})) |
15 | cycsubg2cl.t | . . 3 ⊢ · = (.g‘𝐺) | |
16 | 15 | subgmulgcl 19096 | . 2 ⊢ (((𝐾‘{𝐴}) ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ (𝐾‘{𝐴})) → (𝑁 · 𝐴) ∈ (𝐾‘{𝐴})) |
17 | 9, 10, 14, 16 | syl3anc 1368 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → (𝑁 · 𝐴) ∈ (𝐾‘{𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⊆ wss 3940 {csn 4624 ‘cfv 6542 (class class class)co 7415 ℤcz 12586 Basecbs 17177 Moorecmre 17559 mrClscmrc 17560 Grpcgrp 18892 .gcmg 19025 SubGrpcsubg 19077 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-seq 13997 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-0g 17420 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18738 df-grp 18895 df-minusg 18896 df-mulg 19026 df-subg 19080 |
This theorem is referenced by: odngen 19534 |
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