psgnfix2 - Metamath Proof Explorer

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Theorem psgnfix2 21481

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.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
psgnfix2	⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → ∃𝑤 ∈ Word 𝑅𝑄 = (𝑍 Σ_g 𝑤)))

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

**Proof of Theorem psgnfix2**
Step	Hyp	Ref	Expression
1		elrabi 3670	. . . 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 6885	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘(SymGrp‘𝑁))
6	4, 5	eqtr4i 2755	. . . . . 6 ⊢ 𝑃 = (Base‘𝑍)
7		psgnfix.r	. . . . . 6 ⊢ 𝑅 = ran (pmTrsp‘𝑁)
8	3, 6, 7	psgnfitr 19433	. . . . 5 ⊢ (𝑁 ∈ Fin → (𝑄 ∈ 𝑃 ↔ ∃𝑤 ∈ Word 𝑅𝑄 = (𝑍 Σ_g 𝑤)))
9	8	bicomd 222	. . . 4 ⊢ (𝑁 ∈ Fin → (∃𝑤 ∈ Word 𝑅𝑄 = (𝑍 Σ_g 𝑤) ↔ 𝑄 ∈ 𝑃))
10	9	ad2antrr 723	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (∃𝑤 ∈ Word 𝑅𝑄 = (𝑍 Σ_g 𝑤) ↔ 𝑄 ∈ 𝑃))
11	2, 10	mpbird 257	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → ∃𝑤 ∈ Word 𝑅𝑄 = (𝑍 Σ_g 𝑤))
12	11	ex 412	1 ⊢ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) → (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → ∃𝑤 ∈ Word 𝑅𝑄 = (𝑍 Σ_g 𝑤)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3062 {crab 3424 ∖ cdif 3938 {csn 4621 ran crn 5668 ‘cfv 6534 (class class class)co 7402 Fincfn 8936 Word cword 14466 Basecbs 17149 Σ_g cgsu 17391 SymGrpcsymg 19282 pmTrspcpmtr 19357

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13486 df-fzo 13629 df-seq 13968 df-hash 14292 df-word 14467 df-concat 14523 df-s1 14548 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-tset 17221 df-0g 17392 df-gsum 17393 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18710 df-efmnd 18790 df-grp 18862 df-minusg 18863 df-subg 19046 df-symg 19283 df-pmtr 19358

This theorem is referenced by: psgndif 21484

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