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Mirrors > Home > MPE Home > Th. List > psgnfix2 | Structured version Visualization version GIF version |
Description: A permutation of a finite set fixing one element is generated by transpositions not involving the fixed element. (Contributed by AV, 17-Jan-2019.) |
Ref | Expression |
---|---|
psgnfix.p | ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) |
psgnfix.t | ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾})) |
psgnfix.s | ⊢ 𝑆 = (SymGrp‘(𝑁 ∖ {𝐾})) |
psgnfix.z | ⊢ 𝑍 = (SymGrp‘𝑁) |
psgnfix.r | ⊢ 𝑅 = ran (pmTrsp‘𝑁) |
Ref | Expression |
---|---|
psgnfix2 | ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → ∃𝑤 ∈ Word 𝑅𝑄 = (𝑍 Σg 𝑤))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrabi 3676 | . . . 4 ⊢ (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → 𝑄 ∈ 𝑃) | |
2 | 1 | adantl 481 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → 𝑄 ∈ 𝑃) |
3 | psgnfix.z | . . . . . 6 ⊢ 𝑍 = (SymGrp‘𝑁) | |
4 | psgnfix.p | . . . . . . 7 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) | |
5 | 3 | fveq2i 6900 | . . . . . . 7 ⊢ (Base‘𝑍) = (Base‘(SymGrp‘𝑁)) |
6 | 4, 5 | eqtr4i 2759 | . . . . . 6 ⊢ 𝑃 = (Base‘𝑍) |
7 | psgnfix.r | . . . . . 6 ⊢ 𝑅 = ran (pmTrsp‘𝑁) | |
8 | 3, 6, 7 | psgnfitr 19471 | . . . . 5 ⊢ (𝑁 ∈ Fin → (𝑄 ∈ 𝑃 ↔ ∃𝑤 ∈ Word 𝑅𝑄 = (𝑍 Σg 𝑤))) |
9 | 8 | bicomd 222 | . . . 4 ⊢ (𝑁 ∈ Fin → (∃𝑤 ∈ Word 𝑅𝑄 = (𝑍 Σg 𝑤) ↔ 𝑄 ∈ 𝑃)) |
10 | 9 | ad2antrr 725 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (∃𝑤 ∈ Word 𝑅𝑄 = (𝑍 Σg 𝑤) ↔ 𝑄 ∈ 𝑃)) |
11 | 2, 10 | mpbird 257 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → ∃𝑤 ∈ Word 𝑅𝑄 = (𝑍 Σg 𝑤)) |
12 | 11 | ex 412 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → ∃𝑤 ∈ Word 𝑅𝑄 = (𝑍 Σg 𝑤))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∃wrex 3067 {crab 3429 ∖ cdif 3944 {csn 4629 ran crn 5679 ‘cfv 6548 (class class class)co 7420 Fincfn 8963 Word cword 14496 Basecbs 17179 Σg cgsu 17421 SymGrpcsymg 19320 pmTrspcpmtr 19395 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-seq 13999 df-hash 14322 df-word 14497 df-concat 14553 df-s1 14578 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-tset 17251 df-0g 17422 df-gsum 17423 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-efmnd 18820 df-grp 18892 df-minusg 18893 df-subg 19077 df-symg 19321 df-pmtr 19396 |
This theorem is referenced by: psgndif 21533 |
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