psgneldm2 - Metamath Proof Explorer

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Theorem psgneldm2 18368

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 2795	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
3		eqid 2795	. . . . . 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 18365	. . . . 5 ⊢ 𝑁 Fn {𝑝 ∈ (Base‘𝐺) ∣ dom (𝑝 ∖ I ) ∈ Fin}
6		fndm 6330	. . . . 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 2795	. . . . . 6 ⊢ (mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺))
10	8, 1, 2, 9	symggen 18334	. . . . 5 ⊢ (𝐷 ∈ 𝑉 → ((mrCls‘(SubMnd‘𝐺))‘𝑇) = {𝑝 ∈ (Base‘𝐺) ∣ dom (𝑝 ∖ I ) ∈ Fin})
11	1	symggrp 18264	. . . . . . 7 ⊢ (𝐷 ∈ 𝑉 → 𝐺 ∈ Grp)
12		grpmnd 17873	. . . . . . 7 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
13	11, 12	syl 17	. . . . . 6 ⊢ (𝐷 ∈ 𝑉 → 𝐺 ∈ Mnd)
14	8, 1, 2	symgtrf 18333	. . . . . 6 ⊢ 𝑇 ⊆ (Base‘𝐺)
15	2, 9	gsumwspan 17827	. . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺)) → ((mrCls‘(SubMnd‘𝐺))‘𝑇) = ran (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σ_g 𝑤)))
16	13, 14, 15	sylancl 586	. . . . 5 ⊢ (𝐷 ∈ 𝑉 → ((mrCls‘(SubMnd‘𝐺))‘𝑇) = ran (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σ_g 𝑤)))
17	10, 16	eqtr3d 2833	. . . 4 ⊢ (𝐷 ∈ 𝑉 → {𝑝 ∈ (Base‘𝐺) ∣ dom (𝑝 ∖ I ) ∈ Fin} = ran (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σ_g 𝑤)))
18	7, 17	syl5eq 2843	. . 3 ⊢ (𝐷 ∈ 𝑉 → dom 𝑁 = ran (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σ_g 𝑤)))
19	18	eleq2d 2868	. 2 ⊢ (𝐷 ∈ 𝑉 → (𝑃 ∈ dom 𝑁 ↔ 𝑃 ∈ ran (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σ_g 𝑤))))
20		eqid 2795	. . 3 ⊢ (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σ_g 𝑤)) = (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σ_g 𝑤))
21		ovex 7053	. . 3 ⊢ (𝐺 Σ_g 𝑤) ∈ V
22	20, 21	elrnmpti 5719	. 2 ⊢ (𝑃 ∈ ran (𝑤 ∈ Word 𝑇 ↦ (𝐺 Σ_g 𝑤)) ↔ ∃𝑤 ∈ Word 𝑇𝑃 = (𝐺 Σ_g 𝑤))
23	19, 22	syl6bb 288	1 ⊢ (𝐷 ∈ 𝑉 → (𝑃 ∈ dom 𝑁 ↔ ∃𝑤 ∈ Word 𝑇𝑃 = (𝐺 Σ_g 𝑤)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 = wceq 1522 ∈ wcel 2081 ∃wrex 3106 {crab 3109 ∖ cdif 3860 ⊆ wss 3863 ↦ cmpt 5045 I cid 5352 dom cdm 5448 ran crn 5449 Fn wfn 6225 ‘cfv 6230 (class class class)co 7021 Fincfn 8362 Word cword 13712 Basecbs 16317 Σ_g cgsu 16548 mrClscmrc 16688 Mndcmnd 17738 SubMndcsubmnd 17778 Grpcgrp 17866 SymGrpcsymg 18241 pmTrspcpmtr 18305 pmSgncpsgn 18353

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-iin 4832 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-se 5408 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-isom 6239 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-1st 7550 df-2nd 7551 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-2o 7959 df-oadd 7962 df-er 8144 df-map 8263 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-card 9219 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-nn 11492 df-2 11553 df-3 11554 df-4 11555 df-5 11556 df-6 11557 df-7 11558 df-8 11559 df-9 11560 df-n0 11751 df-z 11835 df-uz 12099 df-fz 12748 df-fzo 12889 df-seq 13225 df-hash 13546 df-word 13713 df-concat 13774 df-s1 13799 df-struct 16319 df-ndx 16320 df-slot 16321 df-base 16323 df-sets 16324 df-ress 16325 df-plusg 16412 df-tset 16418 df-0g 16549 df-gsum 16550 df-mre 16691 df-mrc 16692 df-acs 16694 df-mgm 17686 df-sgrp 17728 df-mnd 17739 df-submnd 17780 df-grp 17869 df-minusg 17870 df-subg 18035 df-symg 18242 df-pmtr 18306 df-psgn 18355

This theorem is referenced by: psgneldm2i 18369 psgneu 18370

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