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Mirrors > Home > MPE Home > Th. List > gexcl3 | Structured version Visualization version GIF version |
Description: If the order of every group element is bounded by 𝑁, the group has finite exponent. (Contributed by Mario Carneiro, 24-Apr-2016.) |
Ref | Expression |
---|---|
gexod.1 | ⊢ 𝑋 = (Base‘𝐺) |
gexod.2 | ⊢ 𝐸 = (gEx‘𝐺) |
gexod.3 | ⊢ 𝑂 = (od‘𝐺) |
Ref | Expression |
---|---|
gexcl3 | ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) → 𝐸 ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 485 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) → 𝐺 ∈ Grp) | |
2 | gexod.1 | . . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺) | |
3 | 2 | grpbn0 18132 | . . . . . . 7 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅) |
4 | r19.2z 4440 | . . . . . . 7 ⊢ ((𝑋 ≠ ∅ ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) → ∃𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) | |
5 | 3, 4 | sylan 582 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) → ∃𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) |
6 | elfzuz2 12913 | . . . . . . . 8 ⊢ ((𝑂‘𝑥) ∈ (1...𝑁) → 𝑁 ∈ (ℤ≥‘1)) | |
7 | nnuz 12282 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
8 | 6, 7 | eleqtrrdi 2924 | . . . . . . 7 ⊢ ((𝑂‘𝑥) ∈ (1...𝑁) → 𝑁 ∈ ℕ) |
9 | 8 | rexlimivw 3282 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁) → 𝑁 ∈ ℕ) |
10 | 5, 9 | syl 17 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) → 𝑁 ∈ ℕ) |
11 | 10 | nnnn0d 11956 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) → 𝑁 ∈ ℕ0) |
12 | 11 | faccld 13645 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) → (!‘𝑁) ∈ ℕ) |
13 | elfzuzb 12903 | . . . . . . . . 9 ⊢ ((𝑂‘𝑥) ∈ (1...𝑁) ↔ ((𝑂‘𝑥) ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑂‘𝑥)))) | |
14 | elnnuz 12283 | . . . . . . . . . 10 ⊢ ((𝑂‘𝑥) ∈ ℕ ↔ (𝑂‘𝑥) ∈ (ℤ≥‘1)) | |
15 | dvdsfac 15676 | . . . . . . . . . 10 ⊢ (((𝑂‘𝑥) ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘(𝑂‘𝑥))) → (𝑂‘𝑥) ∥ (!‘𝑁)) | |
16 | 14, 15 | sylanbr 584 | . . . . . . . . 9 ⊢ (((𝑂‘𝑥) ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑂‘𝑥))) → (𝑂‘𝑥) ∥ (!‘𝑁)) |
17 | 13, 16 | sylbi 219 | . . . . . . . 8 ⊢ ((𝑂‘𝑥) ∈ (1...𝑁) → (𝑂‘𝑥) ∥ (!‘𝑁)) |
18 | 17 | adantl 484 | . . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) ∧ (𝑂‘𝑥) ∈ (1...𝑁)) → (𝑂‘𝑥) ∥ (!‘𝑁)) |
19 | simpll 765 | . . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) ∧ (𝑂‘𝑥) ∈ (1...𝑁)) → 𝐺 ∈ Grp) | |
20 | simplr 767 | . . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) ∧ (𝑂‘𝑥) ∈ (1...𝑁)) → 𝑥 ∈ 𝑋) | |
21 | 8 | adantl 484 | . . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) ∧ (𝑂‘𝑥) ∈ (1...𝑁)) → 𝑁 ∈ ℕ) |
22 | 21 | nnnn0d 11956 | . . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) ∧ (𝑂‘𝑥) ∈ (1...𝑁)) → 𝑁 ∈ ℕ0) |
23 | 22 | faccld 13645 | . . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) ∧ (𝑂‘𝑥) ∈ (1...𝑁)) → (!‘𝑁) ∈ ℕ) |
24 | 23 | nnzd 12087 | . . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) ∧ (𝑂‘𝑥) ∈ (1...𝑁)) → (!‘𝑁) ∈ ℤ) |
25 | gexod.3 | . . . . . . . . 9 ⊢ 𝑂 = (od‘𝐺) | |
26 | eqid 2821 | . . . . . . . . 9 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
27 | eqid 2821 | . . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
28 | 2, 25, 26, 27 | oddvds 18675 | . . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ (!‘𝑁) ∈ ℤ) → ((𝑂‘𝑥) ∥ (!‘𝑁) ↔ ((!‘𝑁)(.g‘𝐺)𝑥) = (0g‘𝐺))) |
29 | 19, 20, 24, 28 | syl3anc 1367 | . . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) ∧ (𝑂‘𝑥) ∈ (1...𝑁)) → ((𝑂‘𝑥) ∥ (!‘𝑁) ↔ ((!‘𝑁)(.g‘𝐺)𝑥) = (0g‘𝐺))) |
30 | 18, 29 | mpbid 234 | . . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) ∧ (𝑂‘𝑥) ∈ (1...𝑁)) → ((!‘𝑁)(.g‘𝐺)𝑥) = (0g‘𝐺)) |
31 | 30 | ex 415 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((𝑂‘𝑥) ∈ (1...𝑁) → ((!‘𝑁)(.g‘𝐺)𝑥) = (0g‘𝐺))) |
32 | 31 | ralimdva 3177 | . . . 4 ⊢ (𝐺 ∈ Grp → (∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁) → ∀𝑥 ∈ 𝑋 ((!‘𝑁)(.g‘𝐺)𝑥) = (0g‘𝐺))) |
33 | 32 | imp 409 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) → ∀𝑥 ∈ 𝑋 ((!‘𝑁)(.g‘𝐺)𝑥) = (0g‘𝐺)) |
34 | gexod.2 | . . . 4 ⊢ 𝐸 = (gEx‘𝐺) | |
35 | 2, 34, 26, 27 | gexlem2 18707 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (!‘𝑁) ∈ ℕ ∧ ∀𝑥 ∈ 𝑋 ((!‘𝑁)(.g‘𝐺)𝑥) = (0g‘𝐺)) → 𝐸 ∈ (1...(!‘𝑁))) |
36 | 1, 12, 33, 35 | syl3anc 1367 | . 2 ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) → 𝐸 ∈ (1...(!‘𝑁))) |
37 | elfznn 12937 | . 2 ⊢ (𝐸 ∈ (1...(!‘𝑁)) → 𝐸 ∈ ℕ) | |
38 | 36, 37 | syl 17 | 1 ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) → 𝐸 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ∅c0 4291 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 1c1 10538 ℕcn 11638 ℤcz 11982 ℤ≥cuz 12244 ...cfz 12893 !cfa 13634 ∥ cdvds 15607 Basecbs 16483 0gc0g 16713 Grpcgrp 18103 .gcmg 18224 odcod 18652 gExcgex 18653 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-od 18656 df-gex 18657 |
This theorem is referenced by: gexcl2 18714 |
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