psgnfix1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > psgnfix1		Structured version Visualization version GIF version

Theorem psgnfix1 21588

Description: A permutation of a finite set fixing one element is generated by transpositions not involving the fixed element. (Contributed by AV, 13-Jan-2019.)

**Hypotheses**
Ref	Expression
psgnfix.p	⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
psgnfix.t	⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
psgnfix.s	⊢ 𝑆 = (SymGrp‘(𝑁 ∖ {𝐾}))

**Assertion**
Ref	Expression
psgnfix1	⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → ∃𝑤 ∈ Word 𝑇(𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σ_g 𝑤)))

Distinct variable groups: 𝐾,𝑞 𝑤,𝐾 𝑤,𝑁 𝑃,𝑞 𝑤,𝑄 𝑤,𝑆 𝑤,𝑇 Allowed substitution hints: 𝑃(𝑤) 𝑄(𝑞) 𝑆(𝑞) 𝑇(𝑞) 𝑁(𝑞)

**Proof of Theorem psgnfix1**
Step	Hyp	Ref	Expression
1		psgnfix.p	. . . . 5 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
2		eqid 2737	. . . . 5 ⊢ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} = {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}
3		psgnfix.s	. . . . . 6 ⊢ 𝑆 = (SymGrp‘(𝑁 ∖ {𝐾}))
4	3	fveq2i 6837	. . . . 5 ⊢ (Base‘𝑆) = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))
5		eqid 2737	. . . . 5 ⊢ (𝑁 ∖ {𝐾}) = (𝑁 ∖ {𝐾})
6	1, 2, 4, 5	symgfixelsi 19401	. . . 4 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝑄 ↾ (𝑁 ∖ {𝐾})) ∈ (Base‘𝑆))
7	6	adantll 715	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝑄 ↾ (𝑁 ∖ {𝐾})) ∈ (Base‘𝑆))
8		diffi 9102	. . . . 5 ⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝐾}) ∈ Fin)
9	8	ad2antrr 727	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝑁 ∖ {𝐾}) ∈ Fin)
10		eqid 2737	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
11		psgnfix.t	. . . . 5 ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
12	3, 10, 11	psgnfitr 19483	. . . 4 ⊢ ((𝑁 ∖ {𝐾}) ∈ Fin → ((𝑄 ↾ (𝑁 ∖ {𝐾})) ∈ (Base‘𝑆) ↔ ∃𝑤 ∈ Word 𝑇(𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σ_g 𝑤)))
13	9, 12	syl 17	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → ((𝑄 ↾ (𝑁 ∖ {𝐾})) ∈ (Base‘𝑆) ↔ ∃𝑤 ∈ Word 𝑇(𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σ_g 𝑤)))
14	7, 13	mpbid 232	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → ∃𝑤 ∈ Word 𝑇(𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σ_g 𝑤))
15	14	ex 412	1 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → ∃𝑤 ∈ Word 𝑇(𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σ_g 𝑤)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∃wrex 3062 {crab 3390 ∖ cdif 3887 {csn 4568 ran crn 5625 ↾ cres 5626 ‘cfv 6492 (class class class)co 7360 Fincfn 8886 Word cword 14466 Basecbs 17170 Σ_g cgsu 17394 SymGrpcsymg 19335 pmTrspcpmtr 19407

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-map 8768 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-uz 12780 df-fz 13453 df-fzo 13600 df-seq 13955 df-hash 14284 df-word 14467 df-concat 14524 df-s1 14550 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-tset 17230 df-0g 17395 df-gsum 17396 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-efmnd 18828 df-grp 18903 df-minusg 18904 df-subg 19090 df-symg 19336 df-pmtr 19408

This theorem is referenced by: psgndif 21592

W3C validator