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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 482 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) → 𝐺 ∈ Grp) | |
2 | gexod.1 | . . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺) | |
3 | 2 | grpbn0 18894 | . . . . . . 7 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅) |
4 | r19.2z 4489 | . . . . . . 7 ⊢ ((𝑋 ≠ ∅ ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) → ∃𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) | |
5 | 3, 4 | sylan 579 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) → ∃𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) |
6 | elfzuz2 13509 | . . . . . . . 8 ⊢ ((𝑂‘𝑥) ∈ (1...𝑁) → 𝑁 ∈ (ℤ≥‘1)) | |
7 | nnuz 12866 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
8 | 6, 7 | eleqtrrdi 2838 | . . . . . . 7 ⊢ ((𝑂‘𝑥) ∈ (1...𝑁) → 𝑁 ∈ ℕ) |
9 | 8 | rexlimivw 3145 | . . . . . 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 12533 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) → 𝑁 ∈ ℕ0) |
12 | 11 | faccld 14247 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) → (!‘𝑁) ∈ ℕ) |
13 | elfzuzb 13498 | . . . . . . . . 9 ⊢ ((𝑂‘𝑥) ∈ (1...𝑁) ↔ ((𝑂‘𝑥) ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑂‘𝑥)))) | |
14 | elnnuz 12867 | . . . . . . . . . 10 ⊢ ((𝑂‘𝑥) ∈ ℕ ↔ (𝑂‘𝑥) ∈ (ℤ≥‘1)) | |
15 | dvdsfac 16274 | . . . . . . . . . 10 ⊢ (((𝑂‘𝑥) ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘(𝑂‘𝑥))) → (𝑂‘𝑥) ∥ (!‘𝑁)) | |
16 | 14, 15 | sylanbr 581 | . . . . . . . . 9 ⊢ (((𝑂‘𝑥) ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑂‘𝑥))) → (𝑂‘𝑥) ∥ (!‘𝑁)) |
17 | 13, 16 | sylbi 216 | . . . . . . . 8 ⊢ ((𝑂‘𝑥) ∈ (1...𝑁) → (𝑂‘𝑥) ∥ (!‘𝑁)) |
18 | 17 | adantl 481 | . . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) ∧ (𝑂‘𝑥) ∈ (1...𝑁)) → (𝑂‘𝑥) ∥ (!‘𝑁)) |
19 | simpll 764 | . . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) ∧ (𝑂‘𝑥) ∈ (1...𝑁)) → 𝐺 ∈ Grp) | |
20 | simplr 766 | . . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) ∧ (𝑂‘𝑥) ∈ (1...𝑁)) → 𝑥 ∈ 𝑋) | |
21 | 8 | adantl 481 | . . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) ∧ (𝑂‘𝑥) ∈ (1...𝑁)) → 𝑁 ∈ ℕ) |
22 | 21 | nnnn0d 12533 | . . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) ∧ (𝑂‘𝑥) ∈ (1...𝑁)) → 𝑁 ∈ ℕ0) |
23 | 22 | faccld 14247 | . . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) ∧ (𝑂‘𝑥) ∈ (1...𝑁)) → (!‘𝑁) ∈ ℕ) |
24 | 23 | nnzd 12586 | . . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) ∧ (𝑂‘𝑥) ∈ (1...𝑁)) → (!‘𝑁) ∈ ℤ) |
25 | gexod.3 | . . . . . . . . 9 ⊢ 𝑂 = (od‘𝐺) | |
26 | eqid 2726 | . . . . . . . . 9 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
27 | eqid 2726 | . . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
28 | 2, 25, 26, 27 | oddvds 19465 | . . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ (!‘𝑁) ∈ ℤ) → ((𝑂‘𝑥) ∥ (!‘𝑁) ↔ ((!‘𝑁)(.g‘𝐺)𝑥) = (0g‘𝐺))) |
29 | 19, 20, 24, 28 | syl3anc 1368 | . . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) ∧ (𝑂‘𝑥) ∈ (1...𝑁)) → ((𝑂‘𝑥) ∥ (!‘𝑁) ↔ ((!‘𝑁)(.g‘𝐺)𝑥) = (0g‘𝐺))) |
30 | 18, 29 | mpbid 231 | . . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) ∧ (𝑂‘𝑥) ∈ (1...𝑁)) → ((!‘𝑁)(.g‘𝐺)𝑥) = (0g‘𝐺)) |
31 | 30 | ex 412 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → ((𝑂‘𝑥) ∈ (1...𝑁) → ((!‘𝑁)(.g‘𝐺)𝑥) = (0g‘𝐺))) |
32 | 31 | ralimdva 3161 | . . . 4 ⊢ (𝐺 ∈ Grp → (∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁) → ∀𝑥 ∈ 𝑋 ((!‘𝑁)(.g‘𝐺)𝑥) = (0g‘𝐺))) |
33 | 32 | imp 406 | . . 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 19500 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (!‘𝑁) ∈ ℕ ∧ ∀𝑥 ∈ 𝑋 ((!‘𝑁)(.g‘𝐺)𝑥) = (0g‘𝐺)) → 𝐸 ∈ (1...(!‘𝑁))) |
36 | 1, 12, 33, 35 | syl3anc 1368 | . 2 ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) → 𝐸 ∈ (1...(!‘𝑁))) |
37 | elfznn 13533 | . 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 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∀wral 3055 ∃wrex 3064 ∅c0 4317 class class class wbr 5141 ‘cfv 6536 (class class class)co 7404 1c1 11110 ℕcn 12213 ℤcz 12559 ℤ≥cuz 12823 ...cfz 13487 !cfa 14236 ∥ cdvds 16202 Basecbs 17151 0gc0g 17392 Grpcgrp 18861 .gcmg 18993 odcod 19442 gExcgex 19443 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-rp 12978 df-fz 13488 df-fl 13760 df-mod 13838 df-seq 13970 df-exp 14031 df-fac 14237 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-dvds 16203 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18994 df-od 19446 df-gex 19447 |
This theorem is referenced by: gexcl2 19507 |
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