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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 17845 | . . . 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 14759 | . 2 ⊢ (𝜑 → (♯‘𝑋) = Σ𝑥 ∈ (𝑋 / ∼ )(♯‘𝑥)) |
8 | 2, 3 | eqgen 17848 | . . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑌 ≈ 𝑥) |
9 | 1, 8 | sylan 569 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑌 ≈ 𝑥) |
10 | 2 | subgss 17796 | . . . . . . . 8 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋) |
11 | 1, 10 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
12 | ssfi 8334 | . . . . . . 7 ⊢ ((𝑋 ∈ Fin ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ Fin) | |
13 | 6, 11, 12 | syl2anc 573 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ Fin) |
14 | 13 | adantr 466 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑌 ∈ Fin) |
15 | 6 | adantr 466 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑋 ∈ Fin) |
16 | 5 | qsss 7958 | . . . . . . . 8 ⊢ (𝜑 → (𝑋 / ∼ ) ⊆ 𝒫 𝑋) |
17 | 16 | sselda 3752 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑥 ∈ 𝒫 𝑋) |
18 | 17 | elpwid 4309 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑥 ⊆ 𝑋) |
19 | ssfi 8334 | . . . . . 6 ⊢ ((𝑋 ∈ Fin ∧ 𝑥 ⊆ 𝑋) → 𝑥 ∈ Fin) | |
20 | 15, 18, 19 | syl2anc 573 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑥 ∈ Fin) |
21 | hashen 13332 | . . . . 5 ⊢ ((𝑌 ∈ Fin ∧ 𝑥 ∈ Fin) → ((♯‘𝑌) = (♯‘𝑥) ↔ 𝑌 ≈ 𝑥)) | |
22 | 14, 20, 21 | syl2anc 573 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → ((♯‘𝑌) = (♯‘𝑥) ↔ 𝑌 ≈ 𝑥)) |
23 | 9, 22 | mpbird 247 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → (♯‘𝑌) = (♯‘𝑥)) |
24 | 23 | sumeq2dv 14634 | . 2 ⊢ (𝜑 → Σ𝑥 ∈ (𝑋 / ∼ )(♯‘𝑌) = Σ𝑥 ∈ (𝑋 / ∼ )(♯‘𝑥)) |
25 | pwfi 8415 | . . . . 5 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin) | |
26 | 6, 25 | sylib 208 | . . . 4 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin) |
27 | ssfi 8334 | . . . 4 ⊢ ((𝒫 𝑋 ∈ Fin ∧ (𝑋 / ∼ ) ⊆ 𝒫 𝑋) → (𝑋 / ∼ ) ∈ Fin) | |
28 | 26, 16, 27 | syl2anc 573 | . . 3 ⊢ (𝜑 → (𝑋 / ∼ ) ∈ Fin) |
29 | hashcl 13342 | . . . . 5 ⊢ (𝑌 ∈ Fin → (♯‘𝑌) ∈ ℕ0) | |
30 | 13, 29 | syl 17 | . . . 4 ⊢ (𝜑 → (♯‘𝑌) ∈ ℕ0) |
31 | 30 | nn0cnd 11553 | . . 3 ⊢ (𝜑 → (♯‘𝑌) ∈ ℂ) |
32 | fsumconst 14722 | . . 3 ⊢ (((𝑋 / ∼ ) ∈ Fin ∧ (♯‘𝑌) ∈ ℂ) → Σ𝑥 ∈ (𝑋 / ∼ )(♯‘𝑌) = ((♯‘(𝑋 / ∼ )) · (♯‘𝑌))) | |
33 | 28, 31, 32 | syl2anc 573 | . 2 ⊢ (𝜑 → Σ𝑥 ∈ (𝑋 / ∼ )(♯‘𝑌) = ((♯‘(𝑋 / ∼ )) · (♯‘𝑌))) |
34 | 7, 24, 33 | 3eqtr2d 2811 | 1 ⊢ (𝜑 → (♯‘𝑋) = ((♯‘(𝑋 / ∼ )) · (♯‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ⊆ wss 3723 𝒫 cpw 4297 class class class wbr 4786 ‘cfv 6029 (class class class)co 6791 Er wer 7891 / cqs 7893 ≈ cen 8104 Fincfn 8107 ℂcc 10134 · cmul 10141 ℕ0cn0 11492 ♯chash 13314 Σcsu 14617 Basecbs 16057 SubGrpcsubg 17789 ~QG cqg 17791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-inf2 8700 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 ax-pre-sup 10214 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-disj 4755 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-isom 6038 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-om 7211 df-1st 7313 df-2nd 7314 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-1o 7711 df-2o 7712 df-oadd 7715 df-er 7894 df-ec 7896 df-qs 7900 df-map 8009 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-sup 8502 df-oi 8569 df-card 8963 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-div 10885 df-nn 11221 df-2 11279 df-3 11280 df-n0 11493 df-z 11578 df-uz 11887 df-rp 12029 df-fz 12527 df-fzo 12667 df-seq 13002 df-exp 13061 df-hash 13315 df-cj 14040 df-re 14041 df-im 14042 df-sqrt 14176 df-abs 14177 df-clim 14420 df-sum 14618 df-ndx 16060 df-slot 16061 df-base 16063 df-sets 16064 df-ress 16065 df-plusg 16155 df-0g 16303 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-grp 17626 df-minusg 17627 df-subg 17792 df-eqg 17794 |
This theorem is referenced by: lagsubg 17857 orbsta2 17947 sylow2blem3 18237 sylow3lem3 18244 sylow3lem4 18245 |
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