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Mirrors > Home > MPE Home > Th. List > gexcl2 | Structured version Visualization version GIF version |
Description: The exponent of a finite group is finite. (Contributed by Mario Carneiro, 24-Apr-2016.) |
Ref | Expression |
---|---|
gexcl2.1 | ⊢ 𝑋 = (Base‘𝐺) |
gexcl2.2 | ⊢ 𝐸 = (gEx‘𝐺) |
Ref | Expression |
---|---|
gexcl2 | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → 𝐸 ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gexcl2.1 | . . . . . 6 ⊢ 𝑋 = (Base‘𝐺) | |
2 | eqid 2724 | . . . . . 6 ⊢ (od‘𝐺) = (od‘𝐺) | |
3 | 1, 2 | odcl2 19481 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑥 ∈ 𝑋) → ((od‘𝐺)‘𝑥) ∈ ℕ) |
4 | 1, 2 | oddvds2 19482 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑥 ∈ 𝑋) → ((od‘𝐺)‘𝑥) ∥ (♯‘𝑋)) |
5 | 3 | nnzd 12584 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑥 ∈ 𝑋) → ((od‘𝐺)‘𝑥) ∈ ℤ) |
6 | 1 | grpbn0 18892 | . . . . . . . . 9 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅) |
7 | 6 | 3ad2ant1 1130 | . . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑥 ∈ 𝑋) → 𝑋 ≠ ∅) |
8 | hashnncl 14327 | . . . . . . . . 9 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅)) | |
9 | 8 | 3ad2ant2 1131 | . . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑥 ∈ 𝑋) → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅)) |
10 | 7, 9 | mpbird 257 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑥 ∈ 𝑋) → (♯‘𝑋) ∈ ℕ) |
11 | dvdsle 16256 | . . . . . . 7 ⊢ ((((od‘𝐺)‘𝑥) ∈ ℤ ∧ (♯‘𝑋) ∈ ℕ) → (((od‘𝐺)‘𝑥) ∥ (♯‘𝑋) → ((od‘𝐺)‘𝑥) ≤ (♯‘𝑋))) | |
12 | 5, 10, 11 | syl2anc 583 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑥 ∈ 𝑋) → (((od‘𝐺)‘𝑥) ∥ (♯‘𝑋) → ((od‘𝐺)‘𝑥) ≤ (♯‘𝑋))) |
13 | 4, 12 | mpd 15 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑥 ∈ 𝑋) → ((od‘𝐺)‘𝑥) ≤ (♯‘𝑋)) |
14 | 10 | nnzd 12584 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑥 ∈ 𝑋) → (♯‘𝑋) ∈ ℤ) |
15 | fznn 13570 | . . . . . 6 ⊢ ((♯‘𝑋) ∈ ℤ → (((od‘𝐺)‘𝑥) ∈ (1...(♯‘𝑋)) ↔ (((od‘𝐺)‘𝑥) ∈ ℕ ∧ ((od‘𝐺)‘𝑥) ≤ (♯‘𝑋)))) | |
16 | 14, 15 | syl 17 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑥 ∈ 𝑋) → (((od‘𝐺)‘𝑥) ∈ (1...(♯‘𝑋)) ↔ (((od‘𝐺)‘𝑥) ∈ ℕ ∧ ((od‘𝐺)‘𝑥) ≤ (♯‘𝑋)))) |
17 | 3, 13, 16 | mpbir2and 710 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑥 ∈ 𝑋) → ((od‘𝐺)‘𝑥) ∈ (1...(♯‘𝑋))) |
18 | 17 | 3expa 1115 | . . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) ∧ 𝑥 ∈ 𝑋) → ((od‘𝐺)‘𝑥) ∈ (1...(♯‘𝑋))) |
19 | 18 | ralrimiva 3138 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → ∀𝑥 ∈ 𝑋 ((od‘𝐺)‘𝑥) ∈ (1...(♯‘𝑋))) |
20 | gexcl2.2 | . . 3 ⊢ 𝐸 = (gEx‘𝐺) | |
21 | 1, 20, 2 | gexcl3 19503 | . 2 ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 ((od‘𝐺)‘𝑥) ∈ (1...(♯‘𝑋))) → 𝐸 ∈ ℕ) |
22 | 19, 21 | syldan 590 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → 𝐸 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 ∅c0 4315 class class class wbr 5139 ‘cfv 6534 (class class class)co 7402 Fincfn 8936 1c1 11108 ≤ cle 11248 ℕcn 12211 ℤcz 12557 ...cfz 13485 ♯chash 14291 ∥ cdvds 16200 Basecbs 17149 Grpcgrp 18859 odcod 19440 gExcgex 19441 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-disj 5105 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-omul 8467 df-er 8700 df-ec 8702 df-qs 8706 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-acn 9934 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12976 df-fz 13486 df-fzo 13629 df-fl 13758 df-mod 13836 df-seq 13968 df-exp 14029 df-fac 14235 df-hash 14292 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-sum 15635 df-dvds 16201 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-0g 17392 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18862 df-minusg 18863 df-sbg 18864 df-mulg 18992 df-subg 19046 df-eqg 19048 df-od 19444 df-gex 19445 |
This theorem is referenced by: cyggexb 19815 pgpfac1lem3a 19994 pgpfaclem3 20001 |
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