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Mirrors > Home > MPE Home > Th. List > lagsubg | Structured version Visualization version GIF version |
Description: Lagrange theorem for Groups: the order of any subgroup of a finite group is a divisor of the order of the group. This is Metamath 100 proof #71. (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
lagsubg.1 | ⊢ 𝑋 = (Base‘𝐺) |
Ref | Expression |
---|---|
lagsubg | ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (#‘𝑌) ∥ (#‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 477 | . . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → 𝑋 ∈ Fin) | |
2 | pwfi 8261 | . . . . . . 7 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin) | |
3 | 1, 2 | sylib 208 | . . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → 𝒫 𝑋 ∈ Fin) |
4 | lagsubg.1 | . . . . . . . . 9 ⊢ 𝑋 = (Base‘𝐺) | |
5 | eqid 2622 | . . . . . . . . 9 ⊢ (𝐺 ~QG 𝑌) = (𝐺 ~QG 𝑌) | |
6 | 4, 5 | eqger 17644 | . . . . . . . 8 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝑌) Er 𝑋) |
7 | 6 | adantr 481 | . . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (𝐺 ~QG 𝑌) Er 𝑋) |
8 | 7 | qsss 7808 | . . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (𝑋 / (𝐺 ~QG 𝑌)) ⊆ 𝒫 𝑋) |
9 | ssfi 8180 | . . . . . 6 ⊢ ((𝒫 𝑋 ∈ Fin ∧ (𝑋 / (𝐺 ~QG 𝑌)) ⊆ 𝒫 𝑋) → (𝑋 / (𝐺 ~QG 𝑌)) ∈ Fin) | |
10 | 3, 8, 9 | syl2anc 693 | . . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (𝑋 / (𝐺 ~QG 𝑌)) ∈ Fin) |
11 | hashcl 13147 | . . . . 5 ⊢ ((𝑋 / (𝐺 ~QG 𝑌)) ∈ Fin → (#‘(𝑋 / (𝐺 ~QG 𝑌))) ∈ ℕ0) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (#‘(𝑋 / (𝐺 ~QG 𝑌))) ∈ ℕ0) |
13 | 12 | nn0zd 11480 | . . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (#‘(𝑋 / (𝐺 ~QG 𝑌))) ∈ ℤ) |
14 | id 22 | . . . . . 6 ⊢ (𝑋 ∈ Fin → 𝑋 ∈ Fin) | |
15 | 4 | subgss 17595 | . . . . . 6 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋) |
16 | ssfi 8180 | . . . . . 6 ⊢ ((𝑋 ∈ Fin ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ Fin) | |
17 | 14, 15, 16 | syl2anr 495 | . . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → 𝑌 ∈ Fin) |
18 | hashcl 13147 | . . . . 5 ⊢ (𝑌 ∈ Fin → (#‘𝑌) ∈ ℕ0) | |
19 | 17, 18 | syl 17 | . . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (#‘𝑌) ∈ ℕ0) |
20 | 19 | nn0zd 11480 | . . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (#‘𝑌) ∈ ℤ) |
21 | dvdsmul2 15004 | . . 3 ⊢ (((#‘(𝑋 / (𝐺 ~QG 𝑌))) ∈ ℤ ∧ (#‘𝑌) ∈ ℤ) → (#‘𝑌) ∥ ((#‘(𝑋 / (𝐺 ~QG 𝑌))) · (#‘𝑌))) | |
22 | 13, 20, 21 | syl2anc 693 | . 2 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (#‘𝑌) ∥ ((#‘(𝑋 / (𝐺 ~QG 𝑌))) · (#‘𝑌))) |
23 | simpl 473 | . . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → 𝑌 ∈ (SubGrp‘𝐺)) | |
24 | 4, 5, 23, 1 | lagsubg2 17655 | . 2 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (#‘𝑋) = ((#‘(𝑋 / (𝐺 ~QG 𝑌))) · (#‘𝑌))) |
25 | 22, 24 | breqtrrd 4681 | 1 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (#‘𝑌) ∥ (#‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 𝒫 cpw 4158 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Er wer 7739 / cqs 7741 Fincfn 7955 · cmul 9941 ℕ0cn0 11292 ℤcz 11377 #chash 13117 ∥ cdvds 14983 Basecbs 15857 SubGrpcsubg 17588 ~QG cqg 17590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-dvds 14984 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-subg 17591 df-eqg 17593 |
This theorem is referenced by: oddvds2 17983 fislw 18040 sylow3lem4 18045 ablfacrp2 18466 ablfac1c 18470 ablfac1eu 18472 |
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