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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 18322 | . . . 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 15174 | . 2 ⊢ (𝜑 → (♯‘𝑋) = Σ𝑥 ∈ (𝑋 / ∼ )(♯‘𝑥)) |
8 | 2, 3 | eqgen 18325 | . . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑌 ≈ 𝑥) |
9 | 1, 8 | sylan 582 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑌 ≈ 𝑥) |
10 | 2 | subgss 18272 | . . . . . . . 8 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋) |
11 | 1, 10 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
12 | 6, 11 | ssfid 8733 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ Fin) |
13 | 12 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑌 ∈ Fin) |
14 | 6 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑋 ∈ Fin) |
15 | 5 | qsss 8350 | . . . . . . . 8 ⊢ (𝜑 → (𝑋 / ∼ ) ⊆ 𝒫 𝑋) |
16 | 15 | sselda 3965 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑥 ∈ 𝒫 𝑋) |
17 | 16 | elpwid 4551 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑥 ⊆ 𝑋) |
18 | 14, 17 | ssfid 8733 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑥 ∈ Fin) |
19 | hashen 13699 | . . . . 5 ⊢ ((𝑌 ∈ Fin ∧ 𝑥 ∈ Fin) → ((♯‘𝑌) = (♯‘𝑥) ↔ 𝑌 ≈ 𝑥)) | |
20 | 13, 18, 19 | syl2anc 586 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → ((♯‘𝑌) = (♯‘𝑥) ↔ 𝑌 ≈ 𝑥)) |
21 | 9, 20 | mpbird 259 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → (♯‘𝑌) = (♯‘𝑥)) |
22 | 21 | sumeq2dv 15052 | . 2 ⊢ (𝜑 → Σ𝑥 ∈ (𝑋 / ∼ )(♯‘𝑌) = Σ𝑥 ∈ (𝑋 / ∼ )(♯‘𝑥)) |
23 | pwfi 8811 | . . . . 5 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin) | |
24 | 6, 23 | sylib 220 | . . . 4 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin) |
25 | 24, 15 | ssfid 8733 | . . 3 ⊢ (𝜑 → (𝑋 / ∼ ) ∈ Fin) |
26 | hashcl 13709 | . . . . 5 ⊢ (𝑌 ∈ Fin → (♯‘𝑌) ∈ ℕ0) | |
27 | 12, 26 | syl 17 | . . . 4 ⊢ (𝜑 → (♯‘𝑌) ∈ ℕ0) |
28 | 27 | nn0cnd 11949 | . . 3 ⊢ (𝜑 → (♯‘𝑌) ∈ ℂ) |
29 | fsumconst 15137 | . . 3 ⊢ (((𝑋 / ∼ ) ∈ Fin ∧ (♯‘𝑌) ∈ ℂ) → Σ𝑥 ∈ (𝑋 / ∼ )(♯‘𝑌) = ((♯‘(𝑋 / ∼ )) · (♯‘𝑌))) | |
30 | 25, 28, 29 | syl2anc 586 | . 2 ⊢ (𝜑 → Σ𝑥 ∈ (𝑋 / ∼ )(♯‘𝑌) = ((♯‘(𝑋 / ∼ )) · (♯‘𝑌))) |
31 | 7, 22, 30 | 3eqtr2d 2860 | 1 ⊢ (𝜑 → (♯‘𝑋) = ((♯‘(𝑋 / ∼ )) · (♯‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ⊆ wss 3934 𝒫 cpw 4537 class class class wbr 5057 ‘cfv 6348 (class class class)co 7148 Er wer 8278 / cqs 8280 ≈ cen 8498 Fincfn 8501 ℂcc 10527 · cmul 10534 ℕ0cn0 11889 ♯chash 13682 Σcsu 15034 Basecbs 16475 SubGrpcsubg 18265 ~QG cqg 18267 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-inf2 9096 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 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 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-ec 8283 df-qs 8287 df-map 8400 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-sup 8898 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-n0 11890 df-z 11974 df-uz 12236 df-rp 12382 df-fz 12885 df-fzo 13026 df-seq 13362 df-exp 13422 df-hash 13683 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-subg 18268 df-eqg 18270 |
This theorem is referenced by: lagsubg 18334 orbsta2 18436 sylow2blem3 18739 sylow3lem3 18746 sylow3lem4 18747 |
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