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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 19209 | . . . 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 15860 | . 2 ⊢ (𝜑 → (♯‘𝑋) = Σ𝑥 ∈ (𝑋 / ∼ )(♯‘𝑥)) |
8 | 2, 3 | eqgen 19212 | . . . . 5 ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑌 ≈ 𝑥) |
9 | 1, 8 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑌 ≈ 𝑥) |
10 | 2 | subgss 19158 | . . . . . . . 8 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝑌 ⊆ 𝑋) |
11 | 1, 10 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
12 | 6, 11 | ssfid 9299 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ Fin) |
13 | 12 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑌 ∈ Fin) |
14 | 6 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑋 ∈ Fin) |
15 | 5 | qsss 8817 | . . . . . . . 8 ⊢ (𝜑 → (𝑋 / ∼ ) ⊆ 𝒫 𝑋) |
16 | 15 | sselda 3995 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑥 ∈ 𝒫 𝑋) |
17 | 16 | elpwid 4614 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑥 ⊆ 𝑋) |
18 | 14, 17 | ssfid 9299 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → 𝑥 ∈ Fin) |
19 | hashen 14383 | . . . . 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 257 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 / ∼ )) → (♯‘𝑌) = (♯‘𝑥)) |
22 | 21 | sumeq2dv 15735 | . 2 ⊢ (𝜑 → Σ𝑥 ∈ (𝑋 / ∼ )(♯‘𝑌) = Σ𝑥 ∈ (𝑋 / ∼ )(♯‘𝑥)) |
23 | pwfi 9355 | . . . . 5 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin) | |
24 | 6, 23 | sylib 218 | . . . 4 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin) |
25 | 24, 15 | ssfid 9299 | . . 3 ⊢ (𝜑 → (𝑋 / ∼ ) ∈ Fin) |
26 | hashcl 14392 | . . . . 5 ⊢ (𝑌 ∈ Fin → (♯‘𝑌) ∈ ℕ0) | |
27 | 12, 26 | syl 17 | . . . 4 ⊢ (𝜑 → (♯‘𝑌) ∈ ℕ0) |
28 | 27 | nn0cnd 12587 | . . 3 ⊢ (𝜑 → (♯‘𝑌) ∈ ℂ) |
29 | fsumconst 15823 | . . 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 2781 | 1 ⊢ (𝜑 → (♯‘𝑋) = ((♯‘(𝑋 / ∼ )) · (♯‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 𝒫 cpw 4605 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Er wer 8741 / cqs 8743 ≈ cen 8981 Fincfn 8984 ℂcc 11151 · cmul 11158 ℕ0cn0 12524 ♯chash 14366 Σcsu 15719 Basecbs 17245 SubGrpcsubg 19151 ~QG cqg 19153 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-ec 8746 df-qs 8750 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-subg 19154 df-eqg 19156 |
This theorem is referenced by: lagsubg 19226 orbsta2 19345 sylow2blem3 19655 sylow3lem3 19662 sylow3lem4 19663 |
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