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Mirrors > Home > MPE Home > Th. List > cycsubg2 | Structured version Visualization version GIF version |
Description: The subgroup generated by an element is exhausted by its multiples. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Ref | Expression |
---|---|
cycsubg2.x | ⊢ 𝑋 = (Base‘𝐺) |
cycsubg2.t | ⊢ · = (.g‘𝐺) |
cycsubg2.f | ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴)) |
cycsubg2.k | ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) |
Ref | Expression |
---|---|
cycsubg2 | ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐾‘{𝐴}) = ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssg 4678 | . . . . . 6 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ 𝑦 ↔ {𝐴} ⊆ 𝑦)) | |
2 | 1 | bicomd 226 | . . . . 5 ⊢ (𝐴 ∈ 𝑋 → ({𝐴} ⊆ 𝑦 ↔ 𝐴 ∈ 𝑦)) |
3 | 2 | adantl 485 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ({𝐴} ⊆ 𝑦 ↔ 𝐴 ∈ 𝑦)) |
4 | 3 | rabbidv 3427 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → {𝑦 ∈ (SubGrp‘𝐺) ∣ {𝐴} ⊆ 𝑦} = {𝑦 ∈ (SubGrp‘𝐺) ∣ 𝐴 ∈ 𝑦}) |
5 | 4 | inteqd 4843 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ∩ {𝑦 ∈ (SubGrp‘𝐺) ∣ {𝐴} ⊆ 𝑦} = ∩ {𝑦 ∈ (SubGrp‘𝐺) ∣ 𝐴 ∈ 𝑦}) |
6 | cycsubg2.x | . . . . 5 ⊢ 𝑋 = (Base‘𝐺) | |
7 | 6 | subgacs 18305 | . . . 4 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝑋)) |
8 | 7 | acsmred 16919 | . . 3 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (Moore‘𝑋)) |
9 | snssi 4701 | . . 3 ⊢ (𝐴 ∈ 𝑋 → {𝐴} ⊆ 𝑋) | |
10 | cycsubg2.k | . . . 4 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) | |
11 | 10 | mrcval 16873 | . . 3 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝑋) ∧ {𝐴} ⊆ 𝑋) → (𝐾‘{𝐴}) = ∩ {𝑦 ∈ (SubGrp‘𝐺) ∣ {𝐴} ⊆ 𝑦}) |
12 | 8, 9, 11 | syl2an 598 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐾‘{𝐴}) = ∩ {𝑦 ∈ (SubGrp‘𝐺) ∣ {𝐴} ⊆ 𝑦}) |
13 | cycsubg2.t | . . 3 ⊢ · = (.g‘𝐺) | |
14 | cycsubg2.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴)) | |
15 | 6, 13, 14 | cycsubg 18343 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ran 𝐹 = ∩ {𝑦 ∈ (SubGrp‘𝐺) ∣ 𝐴 ∈ 𝑦}) |
16 | 5, 12, 15 | 3eqtr4d 2843 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐾‘{𝐴}) = ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 ⊆ wss 3881 {csn 4525 ∩ cint 4838 ↦ cmpt 5110 ran crn 5520 ‘cfv 6324 (class class class)co 7135 ℤcz 11969 Basecbs 16475 Moorecmre 16845 mrClscmrc 16846 Grpcgrp 18095 .gcmg 18216 SubGrpcsubg 18265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-seq 13365 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-mulg 18217 df-subg 18268 |
This theorem is referenced by: odf1o1 18689 odf1o2 18690 cycsubgcyg2 19015 pgpfac1lem2 19190 pgpfac1lem3 19192 pgpfac1lem4 19193 |
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