psgnfix1 - Metamath Proof Explorer

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Theorem psgnfix1 21913

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 2760	. . . . 5 ⊢ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} = {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}
3		psgnfix.s	. . . . . 6 ⊢ 𝑆 = (SymGrp‘(𝑁 ∖ {𝐾}))
4	3	fveq2i 6893	. . . . 5 ⊢ (Base‘𝑆) = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))
5		eqid 2760	. . . . 5 ⊢ (𝑁 ∖ {𝐾}) = (𝑁 ∖ {𝐾})
6	1, 2, 4, 5	symgfixelsi 19658	. . . 4 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝑄 ↾ (𝑁 ∖ {𝐾})) ∈ (Base‘𝑆))
7	6	adantll 727	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝑄 ↾ (𝑁 ∖ {𝐾})) ∈ (Base‘𝑆))
8		diffi 9197	. . . . 5 ⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝐾}) ∈ Fin)
9	8	ad2antrr 739	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝑁 ∖ {𝐾}) ∈ Fin)
10		eqid 2760	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
11		psgnfix.t	. . . . 5 ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
12	3, 10, 11	psgnfitr 19740	. . . 4 ⊢ ((𝑁 ∖ {𝐾}) ∈ Fin → ((𝑄 ↾ (𝑁 ∖ {𝐾})) ∈ (Base‘𝑆) ↔ ∃𝑤 ∈ Word 𝑇(𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σ_g 𝑤)))
13	9, 12	syl 18	. . 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 235	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → ∃𝑤 ∈ Word 𝑇(𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σ_g 𝑤))
15	14	ex 418	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ∃wrex 3086 {crab 3412 ∖ cdif 4044 {csn 4584 ran crn 5652 ↾ cres 5653 ‘cfv 6543 (class class class)co 7424 Fincfn 8980 Word cword 14665 Basecbs 17396 Σ_g cgsu 17620 SymGrpcsymg 19592 pmTrspcpmtr 19664

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7882 df-1st 8005 df-2nd 8006 df-frecs 8306 df-wrecs 8337 df-recs 8386 df-rdg 8425 df-1o 8483 df-2o 8484 df-er 8724 df-map 8856 df-en 8981 df-dom 8982 df-sdom 8983 df-fin 8984 df-card 10027 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-7 12417 df-8 12418 df-9 12419 df-n0 12614 df-z 12701 df-uz 12973 df-fz 13647 df-fzo 13796 df-seq 14152 df-hash 14482 df-word 14666 df-concat 14723 df-s1 14750 df-struct 17334 df-sets 17351 df-slot 17369 df-ndx 17381 df-base 17397 df-ress 17418 df-plusg 17450 df-tset 17456 df-0g 17621 df-gsum 17622 df-mre 17765 df-mrc 17766 df-acs 17768 df-mgm 18825 df-sgrp 18917 df-mnd 18933 df-submnd 18988 df-efmnd 19074 df-grp 19156 df-minusg 19157 df-subg 19342 df-symg 19593 df-pmtr 19665

This theorem is used by: psgndif 21917

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