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Mirrors > Home > MPE Home > Th. List > cycsubg | Structured version Visualization version GIF version |
Description: The cyclic group generated by 𝐴 is the smallest subgroup containing 𝐴. (Contributed by Mario Carneiro, 13-Jan-2015.) |
Ref | Expression |
---|---|
cycsubg.x | ⊢ 𝑋 = (Base‘𝐺) |
cycsubg.t | ⊢ · = (.g‘𝐺) |
cycsubg.f | ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴)) |
Ref | Expression |
---|---|
cycsubg | ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ran 𝐹 = ∩ {𝑠 ∈ (SubGrp‘𝐺) ∣ 𝐴 ∈ 𝑠}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssintab 4885 | . . . . 5 ⊢ (ran 𝐹 ⊆ ∩ {𝑠 ∣ (𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠)} ↔ ∀𝑠((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) → ran 𝐹 ⊆ 𝑠)) | |
2 | cycsubg.x | . . . . . 6 ⊢ 𝑋 = (Base‘𝐺) | |
3 | cycsubg.t | . . . . . 6 ⊢ · = (.g‘𝐺) | |
4 | cycsubg.f | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴)) | |
5 | 2, 3, 4 | cycsubgss 18344 | . . . . 5 ⊢ ((𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠) → ran 𝐹 ⊆ 𝑠) |
6 | 1, 5 | mpgbir 1796 | . . . 4 ⊢ ran 𝐹 ⊆ ∩ {𝑠 ∣ (𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠)} |
7 | df-rab 3147 | . . . . 5 ⊢ {𝑠 ∈ (SubGrp‘𝐺) ∣ 𝐴 ∈ 𝑠} = {𝑠 ∣ (𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠)} | |
8 | 7 | inteqi 4872 | . . . 4 ⊢ ∩ {𝑠 ∈ (SubGrp‘𝐺) ∣ 𝐴 ∈ 𝑠} = ∩ {𝑠 ∣ (𝑠 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑠)} |
9 | 6, 8 | sseqtrri 4003 | . . 3 ⊢ ran 𝐹 ⊆ ∩ {𝑠 ∈ (SubGrp‘𝐺) ∣ 𝐴 ∈ 𝑠} |
10 | 9 | a1i 11 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ran 𝐹 ⊆ ∩ {𝑠 ∈ (SubGrp‘𝐺) ∣ 𝐴 ∈ 𝑠}) |
11 | 2, 3, 4 | cycsubgcl 18343 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (ran 𝐹 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ran 𝐹)) |
12 | eleq2 2901 | . . . . 5 ⊢ (𝑠 = ran 𝐹 → (𝐴 ∈ 𝑠 ↔ 𝐴 ∈ ran 𝐹)) | |
13 | 12 | elrab 3679 | . . . 4 ⊢ (ran 𝐹 ∈ {𝑠 ∈ (SubGrp‘𝐺) ∣ 𝐴 ∈ 𝑠} ↔ (ran 𝐹 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ran 𝐹)) |
14 | 11, 13 | sylibr 236 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ran 𝐹 ∈ {𝑠 ∈ (SubGrp‘𝐺) ∣ 𝐴 ∈ 𝑠}) |
15 | intss1 4883 | . . 3 ⊢ (ran 𝐹 ∈ {𝑠 ∈ (SubGrp‘𝐺) ∣ 𝐴 ∈ 𝑠} → ∩ {𝑠 ∈ (SubGrp‘𝐺) ∣ 𝐴 ∈ 𝑠} ⊆ ran 𝐹) | |
16 | 14, 15 | syl 17 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ∩ {𝑠 ∈ (SubGrp‘𝐺) ∣ 𝐴 ∈ 𝑠} ⊆ ran 𝐹) |
17 | 10, 16 | eqssd 3983 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ran 𝐹 = ∩ {𝑠 ∈ (SubGrp‘𝐺) ∣ 𝐴 ∈ 𝑠}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {cab 2799 {crab 3142 ⊆ wss 3935 ∩ cint 4868 ↦ cmpt 5138 ran crn 5550 ‘cfv 6349 (class class class)co 7150 ℤcz 11975 Basecbs 16477 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-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-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 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-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-mulg 18219 df-subg 18270 |
This theorem is referenced by: cycsubg2 18347 |
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