psgnfix2 - Metamath Proof Explorer

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Theorem psgnfix2 21651

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 3646	. . . 4 ⊢ (𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} → 𝑄 ∈ 𝑃)
2	1	adantl 485	. . 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 6870	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘(SymGrp‘𝑁))
6	4, 5	eqtr4i 2788	. . . . . 6 ⊢ 𝑃 = (Base‘𝑍)
7		psgnfix.r	. . . . . 6 ⊢ 𝑅 = ran (pmTrsp‘𝑁)
8	3, 6, 7	psgnfitr 19557	. . . . 5 ⊢ (𝑁 ∈ Fin → (𝑄 ∈ 𝑃 ↔ ∃𝑤 ∈ Word 𝑅𝑄 = (𝑍 Σ_g 𝑤)))
9	8	bicomd 225	. . . 4 ⊢ (𝑁 ∈ Fin → (∃𝑤 ∈ Word 𝑅𝑄 = (𝑍 Σ_g 𝑤) ↔ 𝑄 ∈ 𝑃))
10	9	ad2antrr 736	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (∃𝑤 ∈ Word 𝑅𝑄 = (𝑍 Σ_g 𝑤) ↔ 𝑄 ∈ 𝑃))
11	2, 10	mpbird 259	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → ∃𝑤 ∈ Word 𝑅𝑄 = (𝑍 Σ_g 𝑤))
12	11	ex 416	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 208 ∧ wa 399 = wceq 1560 ∈ wcel 2142 ∃wrex 3086 {crab 3414 ∖ cdif 3901 {csn 4582 ran crn 5648 ‘cfv 6521 (class class class)co 7396 Fincfn 8927 Word cword 14526 Basecbs 17245 Σ_g cgsu 17469 SymGrpcsymg 19409 pmTrspcpmtr 19481

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-iin 4952 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8678 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-uz 12840 df-fz 13513 df-fzo 13660 df-seq 14015 df-hash 14344 df-word 14527 df-concat 14584 df-s1 14610 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-tset 17305 df-0g 17470 df-gsum 17471 df-mre 17614 df-mrc 17615 df-acs 17617 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-submnd 18818 df-efmnd 18903 df-grp 18978 df-minusg 18979 df-subg 19165 df-symg 19410 df-pmtr 19482

This theorem is referenced by: psgndif 21654

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