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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 4710 | . . . . . 6 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ 𝑦 ↔ {𝐴} ⊆ 𝑦)) | |
2 | 1 | bicomd 225 | . . . . 5 ⊢ (𝐴 ∈ 𝑋 → ({𝐴} ⊆ 𝑦 ↔ 𝐴 ∈ 𝑦)) |
3 | 2 | adantl 484 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ({𝐴} ⊆ 𝑦 ↔ 𝐴 ∈ 𝑦)) |
4 | 3 | rabbidv 3480 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → {𝑦 ∈ (SubGrp‘𝐺) ∣ {𝐴} ⊆ 𝑦} = {𝑦 ∈ (SubGrp‘𝐺) ∣ 𝐴 ∈ 𝑦}) |
5 | 4 | inteqd 4873 | . 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 18307 | . . . 4 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝑋)) |
8 | 7 | acsmred 16921 | . . 3 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (Moore‘𝑋)) |
9 | snssi 4734 | . . 3 ⊢ (𝐴 ∈ 𝑋 → {𝐴} ⊆ 𝑋) | |
10 | cycsubg2.k | . . . 4 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) | |
11 | 10 | mrcval 16875 | . . 3 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝑋) ∧ {𝐴} ⊆ 𝑋) → (𝐾‘{𝐴}) = ∩ {𝑦 ∈ (SubGrp‘𝐺) ∣ {𝐴} ⊆ 𝑦}) |
12 | 8, 9, 11 | syl2an 597 | . 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 18345 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ran 𝐹 = ∩ {𝑦 ∈ (SubGrp‘𝐺) ∣ 𝐴 ∈ 𝑦}) |
16 | 5, 12, 15 | 3eqtr4d 2866 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐾‘{𝐴}) = ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 ⊆ wss 3935 {csn 4560 ∩ cint 4868 ↦ cmpt 5138 ran crn 5550 ‘cfv 6349 (class class class)co 7150 ℤcz 11975 Basecbs 16477 Moorecmre 16847 mrClscmrc 16848 Grpcgrp 18097 .gcmg 18218 SubGrpcsubg 18267 |
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 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
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 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-seq 13364 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-0g 16709 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-mulg 18219 df-subg 18270 |
This theorem is referenced by: odf1o1 18691 odf1o2 18692 cycsubgcyg2 19016 pgpfac1lem2 19191 pgpfac1lem3 19193 pgpfac1lem4 19194 |
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