psgneldm2 - Metamath Proof Explorer

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Theorem psgneldm2 18626

Description: The finitary permutations are the span of the transpositions. (Contributed by Stefan O'Rear, 28-Aug-2015.)

**Hypotheses**
Ref	Expression
psgnval.g	⊢ 𝐺 = (SymGrp‘𝐷)
psgnval.t	⊢ 𝑇 = ran (pmTrsp‘𝐷)
psgnval.n	⊢ 𝑁 = (pmSgn‘𝐷)

**Assertion**
Ref	Expression
psgneldm2	⊢ (𝐷 ∈ 𝑉 → (𝑃 ∈ dom 𝑁 ↔ ∃𝑤 ∈ Word 𝑇𝑃 = (𝐺 Σ_g 𝑤)))

Distinct variable groups: 𝑤,𝐺 𝑤,𝑁 𝑤,𝑃 𝑤,𝑇 𝑤,𝐷 Allowed substitution hint: 𝑉(𝑤)

Proof of Theorem psgneldm2 Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		psgnval.g	. . . . . 6 ⊢ 𝐺 = (SymGrp‘𝐷)
2		eqid 2821	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
3		eqid 2821	. . . . . 6 ⊢ {𝑝 ∈ (Base‘𝐺) ∣ dom (𝑝 ∖ I ) ∈ Fin} = {𝑝 ∈ (Base‘𝐺) ∣ dom (𝑝 ∖ I ) ∈ Fin}
4		psgnval.n	. . . . . 6 ⊢ 𝑁 = (pmSgn‘𝐷)
5	1, 2, 3, 4	psgnfn 18623	. . . . 5 ⊢ 𝑁 Fn {𝑝 ∈ (Base‘𝐺) ∣ dom (𝑝 ∖ I ) ∈ Fin}
6		fndm 6449	. . . . 5 ⊢ (𝑁 Fn {𝑝 ∈ (Base‘𝐺) ∣ dom (𝑝 ∖ I ) ∈ Fin} → dom 𝑁 = {𝑝 ∈ (Base‘𝐺) ∣ dom (𝑝 ∖ I ) ∈ Fin})
7	5, 6	ax-mp 5	. . . 4 ⊢ dom 𝑁 = {𝑝 ∈ (Base‘𝐺) ∣ dom (𝑝 ∖ I ) ∈ Fin}
8		psgnval.t	. . . . . 6 ⊢ 𝑇 = ran (pmTrsp‘𝐷)
9		eqid 2821	. . . . . 6 ⊢ (mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺))
10	8, 1, 2, 9	symggen 18592	. . . . 5 ⊢ (𝐷 ∈ 𝑉 → ((mrCls‘(SubMnd‘𝐺))‘𝑇) = {𝑝 ∈ (Base‘𝐺) ∣ dom (𝑝 ∖ I ) ∈ Fin})
11	1	symggrp 18522	. . . . . . 7 ⊢ (𝐷 ∈ 𝑉 → 𝐺 ∈ Grp)
12		grpmnd 18104	. . . . . . 7 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
13	11, 12	syl 17	. . . . . 6 ⊢ (𝐷 ∈ 𝑉 → 𝐺 ∈ Mnd)
14	8, 1, 2	symgtrf 18591	. . . . . 6 ⊢ 𝑇 ⊆ (Base‘𝐺)
15	2, 9	gsumwspan 18005	. . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺)) → ((mrCls‘(SubMnd‘𝐺))‘𝑇) = ran (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σ_g 𝑤)))
16	13, 14, 15	sylancl 588	. . . . 5 ⊢ (𝐷 ∈ 𝑉 → ((mrCls‘(SubMnd‘𝐺))‘𝑇) = ran (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σ_g 𝑤)))
17	10, 16	eqtr3d 2858	. . . 4 ⊢ (𝐷 ∈ 𝑉 → {𝑝 ∈ (Base‘𝐺) ∣ dom (𝑝 ∖ I ) ∈ Fin} = ran (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σ_g 𝑤)))
18	7, 17	syl5eq 2868	. . 3 ⊢ (𝐷 ∈ 𝑉 → dom 𝑁 = ran (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σ_g 𝑤)))
19	18	eleq2d 2898	. 2 ⊢ (𝐷 ∈ 𝑉 → (𝑃 ∈ dom 𝑁 ↔ 𝑃 ∈ ran (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σ_g 𝑤))))
20		eqid 2821	. . 3 ⊢ (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σ_g 𝑤)) = (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σ_g 𝑤))
21		ovex 7183	. . 3 ⊢ (𝐺 Σ_g 𝑤) ∈ V
22	20, 21	elrnmpti 5826	. 2 ⊢ (𝑃 ∈ ran (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σ_g 𝑤)) ↔ ∃𝑤 ∈ Word 𝑇𝑃 = (𝐺 Σ_g 𝑤))
23	19, 22	syl6bb 289	1 ⊢ (𝐷 ∈ 𝑉 → (𝑃 ∈ dom 𝑁 ↔ ∃𝑤 ∈ Word 𝑇𝑃 = (𝐺 Σ_g 𝑤)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 {crab 3142 ∖ cdif 3932 ⊆ wss 3935 ↦ cmpt 5138 I cid 5453 dom cdm 5549 ran crn 5550 Fn wfn 6344 ‘cfv 6349 (class class class)co 7150 Fincfn 8503 Word cword 13855 Basecbs 16477 Σ_g cgsu 16708 mrClscmrc 16848 Mndcmnd 17905 SubMndcsubmnd 17949 Grpcgrp 18097 SymGrpcsymg 18489 pmTrspcpmtr 18563 pmSgncpsgn 18611

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-word 13856 df-concat 13917 df-s1 13944 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-tset 16578 df-0g 16709 df-gsum 16710 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-efmnd 18028 df-grp 18100 df-minusg 18101 df-subg 18270 df-symg 18490 df-pmtr 18564 df-psgn 18613

This theorem is referenced by: psgneldm2i 18627 psgneu 18628

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