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Theorem lagsubg 19066

Description: Lagrange's 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.)

**Hypothesis**
Ref	Expression
lagsubg.1	⊢ 𝑋 = (Base‘𝐺)

**Assertion**
Ref	Expression
lagsubg	⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (♯‘𝑌) ∥ (♯‘𝑋))

**Proof of Theorem lagsubg**
Step	Hyp	Ref	Expression
1		simpr 485	. . . . . . 7 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → 𝑋 ∈ Fin)
2		pwfi 9174	. . . . . . 7 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
3	1, 2	sylib 217	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → 𝒫 𝑋 ∈ Fin)
4		lagsubg.1	. . . . . . . . 9 ⊢ 𝑋 = (Base‘𝐺)
5		eqid 2732	. . . . . . . . 9 ⊢ (𝐺 ~_QG 𝑌) = (𝐺 ~_QG 𝑌)
6	4, 5	eqger 19052	. . . . . . . 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 8768	. . . . . 6 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (𝑋 / (𝐺 ~_QG 𝑌)) ⊆ 𝒫 𝑋)
9	3, 8	ssfid 9263	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (𝑋 / (𝐺 ~_QG 𝑌)) ∈ Fin)
10		hashcl 14312	. . . . 5 ⊢ ((𝑋 / (𝐺 ~_QG 𝑌)) ∈ Fin → (♯‘(𝑋 / (𝐺 ~_QG 𝑌))) ∈ ℕ₀)
11	9, 10	syl 17	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (♯‘(𝑋 / (𝐺 ~_QG 𝑌))) ∈ ℕ₀)
12	11	nn0zd 12580	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (♯‘(𝑋 / (𝐺 ~_QG 𝑌))) ∈ ℤ)
13		id 22	. . . . . 6 ⊢ (𝑋 ∈ Fin → 𝑋 ∈ Fin)
14	4	subgss 19001	. . . . . 6 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋)
15		ssfi 9169	. . . . . 6 ⊢ ((𝑋 ∈ Fin ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ Fin)
16	13, 14, 15	syl2anr 597	. . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → 𝑌 ∈ Fin)
17		hashcl 14312	. . . . 5 ⊢ (𝑌 ∈ Fin → (♯‘𝑌) ∈ ℕ₀)
18	16, 17	syl 17	. . . 4 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (♯‘𝑌) ∈ ℕ₀)
19	18	nn0zd 12580	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (♯‘𝑌) ∈ ℤ)
20		dvdsmul2 16218	. . 3 ⊢ (((♯‘(𝑋 / (𝐺 ~_QG 𝑌))) ∈ ℤ ∧ (♯‘𝑌) ∈ ℤ) → (♯‘𝑌) ∥ ((♯‘(𝑋 / (𝐺 ~_QG 𝑌))) · (♯‘𝑌)))
21	12, 19, 20	syl2anc 584	. 2 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (♯‘𝑌) ∥ ((♯‘(𝑋 / (𝐺 ~_QG 𝑌))) · (♯‘𝑌)))
22		simpl 483	. . 3 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → 𝑌 ∈ (SubGrp‘𝐺))
23	4, 5, 22, 1	lagsubg2 19065	. 2 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = ((♯‘(𝑋 / (𝐺 ~_QG 𝑌))) · (♯‘𝑌)))
24	21, 23	breqtrrd 5175	1 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin) → (♯‘𝑌) ∥ (♯‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3947 𝒫 cpw 4601 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 Er wer 8696 / cqs 8698 Fincfn 8935 · cmul 11111 ℕ₀cn0 12468 ℤcz 12554 ♯chash 14286 ∥ cdvds 16193 Basecbs 17140 SubGrpcsubg 18994 ~_QG cqg 18996

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-ec 8701 df-qs 8705 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-dvds 16194 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-subg 18997 df-eqg 18999

This theorem is referenced by: oddvds2 19428 fislw 19487 sylow3lem4 19492 ablfacrp2 19931 ablfac1c 19935 ablfac1eu 19937 prmgrpsimpgd 19978

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