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Mirrors > Home > MPE Home > Th. List > lagsubg2 | Structured version Visualization version GIF version |
Description: Lagrange's theorem for finite groups. Call the "order" of a group the cardinal number of the basic set of the group, and "index of a subgroup" the cardinal number of the set of left (or right, this is the same) cosets of this subgroup. Then the order of the group is the (cardinal) product of the order of any of its subgroups by the index of this subgroup. (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
lagsubg.1 | ⊢ 𝑋 = (Base‘𝐺) |
lagsubg.2 | ⊢ ∼ = (𝐺 ~QG 𝑌) |
lagsubg.3 | ⊢ (𝜑 → 𝑌 ∈ (SubGrp‘𝐺)) |
lagsubg.4 | ⊢ (𝜑 → 𝑋 ∈ Fin) |
Ref | Expression |
---|---|
lagsubg2 | ⊢ (𝜑 → (♯‘𝑋) = ((♯‘(𝑋 / ∼ )) · (♯‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lagsubg.3 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (SubGrp‘𝐺)) | |
2 | lagsubg.1 | . . . . 5 ⊢ 𝑋 = (Base‘𝐺) | |
3 | lagsubg.2 | . . . . 5 ⊢ ∼ = (𝐺 ~QG 𝑌) | |
4 | 2, 3 | eqger 18268 | . . . 4 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝜑 → ∼ Er 𝑋) |
6 | lagsubg.4 | . . 3 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
7 | 5, 6 | qshash 15170 | . 2 ⊢ (𝜑 → (♯‘𝑋) = Σ𝑥 ∈ (𝑋 / ∼ )(♯‘𝑥)) |
8 | 2, 3 | eqgen 18271 | . . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑌 ≈ 𝑥) |
9 | 1, 8 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑌 ≈ 𝑥) |
10 | 2 | subgss 18218 | . . . . . . . 8 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋) |
11 | 1, 10 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
12 | 6, 11 | ssfid 8729 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ Fin) |
13 | 12 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑌 ∈ Fin) |
14 | 6 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑋 ∈ Fin) |
15 | 5 | qsss 8347 | . . . . . . . 8 ⊢ (𝜑 → (𝑋 / ∼ ) ⊆ 𝒫 𝑋) |
16 | 15 | sselda 3964 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑥 ∈ 𝒫 𝑋) |
17 | 16 | elpwid 4549 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑥 ⊆ 𝑋) |
18 | 14, 17 | ssfid 8729 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑥 ∈ Fin) |
19 | hashen 13695 | . . . . 5 ⊢ ((𝑌 ∈ Fin ∧ 𝑥 ∈ Fin) → ((♯‘𝑌) = (♯‘𝑥) ↔ 𝑌 ≈ 𝑥)) | |
20 | 13, 18, 19 | syl2anc 584 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → ((♯‘𝑌) = (♯‘𝑥) ↔ 𝑌 ≈ 𝑥)) |
21 | 9, 20 | mpbird 258 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → (♯‘𝑌) = (♯‘𝑥)) |
22 | 21 | sumeq2dv 15048 | . 2 ⊢ (𝜑 → Σ𝑥 ∈ (𝑋 / ∼ )(♯‘𝑌) = Σ𝑥 ∈ (𝑋 / ∼ )(♯‘𝑥)) |
23 | pwfi 8807 | . . . . 5 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin) | |
24 | 6, 23 | sylib 219 | . . . 4 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin) |
25 | 24, 15 | ssfid 8729 | . . 3 ⊢ (𝜑 → (𝑋 / ∼ ) ∈ Fin) |
26 | hashcl 13705 | . . . . 5 ⊢ (𝑌 ∈ Fin → (♯‘𝑌) ∈ ℕ0) | |
27 | 12, 26 | syl 17 | . . . 4 ⊢ (𝜑 → (♯‘𝑌) ∈ ℕ0) |
28 | 27 | nn0cnd 11945 | . . 3 ⊢ (𝜑 → (♯‘𝑌) ∈ ℂ) |
29 | fsumconst 15133 | . . 3 ⊢ (((𝑋 / ∼ ) ∈ Fin ∧ (♯‘𝑌) ∈ ℂ) → Σ𝑥 ∈ (𝑋 / ∼ )(♯‘𝑌) = ((♯‘(𝑋 / ∼ )) · (♯‘𝑌))) | |
30 | 25, 28, 29 | syl2anc 584 | . 2 ⊢ (𝜑 → Σ𝑥 ∈ (𝑋 / ∼ )(♯‘𝑌) = ((♯‘(𝑋 / ∼ )) · (♯‘𝑌))) |
31 | 7, 22, 30 | 3eqtr2d 2859 | 1 ⊢ (𝜑 → (♯‘𝑋) = ((♯‘(𝑋 / ∼ )) · (♯‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 𝒫 cpw 4535 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 Er wer 8275 / cqs 8277 ≈ cen 8494 Fincfn 8497 ℂcc 10523 · cmul 10530 ℕ0cn0 11885 ♯chash 13678 Σcsu 15030 Basecbs 16471 SubGrpcsubg 18211 ~QG cqg 18213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-disj 5023 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-ec 8280 df-qs 8284 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-sum 15031 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-subg 18214 df-eqg 18216 |
This theorem is referenced by: lagsubg 18280 orbsta2 18382 sylow2blem3 18676 sylow3lem3 18683 sylow3lem4 18684 |
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