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Theorem psgnghm 21066
Description: The sign is a homomorphism from the finitary permutation group to the numeric signs. (Contributed by Stefan O'Rear, 28-Aug-2015.)
Hypotheses
Ref Expression
psgnghm.s 𝑆 = (SymGrp‘𝐷)
psgnghm.n 𝑁 = (pmSgn‘𝐷)
psgnghm.f 𝐹 = (𝑆s dom 𝑁)
psgnghm.u 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1})
Assertion
Ref Expression
psgnghm (𝐷𝑉𝑁 ∈ (𝐹 GrpHom 𝑈))

Proof of Theorem psgnghm
Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psgnghm.s . . . . . 6 𝑆 = (SymGrp‘𝐷)
2 eqid 2731 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
3 eqid 2731 . . . . . 6 {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
4 psgnghm.n . . . . . 6 𝑁 = (pmSgn‘𝐷)
51, 2, 3, 4psgnfn 19333 . . . . 5 𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
65fndmi 6642 . . . 4 dom 𝑁 = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
76ssrab3 4076 . . 3 dom 𝑁 ⊆ (Base‘𝑆)
8 psgnghm.f . . . 4 𝐹 = (𝑆s dom 𝑁)
98, 2ressbas2 17164 . . 3 (dom 𝑁 ⊆ (Base‘𝑆) → dom 𝑁 = (Base‘𝐹))
107, 9ax-mp 5 . 2 dom 𝑁 = (Base‘𝐹)
11 psgnghm.u . . 3 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1})
1211cnmsgnbas 21064 . 2 {1, -1} = (Base‘𝑈)
1310fvexi 6892 . . 3 dom 𝑁 ∈ V
14 eqid 2731 . . . 4 (+g𝑆) = (+g𝑆)
158, 14ressplusg 17217 . . 3 (dom 𝑁 ∈ V → (+g𝑆) = (+g𝐹))
1613, 15ax-mp 5 . 2 (+g𝑆) = (+g𝐹)
17 prex 5425 . . 3 {1, -1} ∈ V
18 eqid 2731 . . . . 5 (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
19 cnfldmul 20884 . . . . 5 · = (.r‘ℂfld)
2018, 19mgpplusg 19950 . . . 4 · = (+g‘(mulGrp‘ℂfld))
2111, 20ressplusg 17217 . . 3 ({1, -1} ∈ V → · = (+g𝑈))
2217, 21ax-mp 5 . 2 · = (+g𝑈)
231, 4psgndmsubg 19334 . . 3 (𝐷𝑉 → dom 𝑁 ∈ (SubGrp‘𝑆))
248subggrp 18981 . . 3 (dom 𝑁 ∈ (SubGrp‘𝑆) → 𝐹 ∈ Grp)
2523, 24syl 17 . 2 (𝐷𝑉𝐹 ∈ Grp)
2611cnmsgngrp 21065 . . 3 𝑈 ∈ Grp
2726a1i 11 . 2 (𝐷𝑉𝑈 ∈ Grp)
28 fnfun 6638 . . . . . 6 (𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} → Fun 𝑁)
295, 28ax-mp 5 . . . . 5 Fun 𝑁
30 funfn 6567 . . . . 5 (Fun 𝑁𝑁 Fn dom 𝑁)
3129, 30mpbi 229 . . . 4 𝑁 Fn dom 𝑁
3231a1i 11 . . 3 (𝐷𝑉𝑁 Fn dom 𝑁)
33 eqid 2731 . . . . . 6 ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
341, 33, 4psgnvali 19340 . . . . 5 (𝑥 ∈ dom 𝑁 → ∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(♯‘𝑧))))
35 lencl 14465 . . . . . . . . . 10 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑧) ∈ ℕ0)
3635nn0zd 12566 . . . . . . . . 9 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑧) ∈ ℤ)
37 m1expcl2 14033 . . . . . . . . . 10 ((♯‘𝑧) ∈ ℤ → (-1↑(♯‘𝑧)) ∈ {-1, 1})
38 prcom 4729 . . . . . . . . . 10 {-1, 1} = {1, -1}
3937, 38eleqtrdi 2842 . . . . . . . . 9 ((♯‘𝑧) ∈ ℤ → (-1↑(♯‘𝑧)) ∈ {1, -1})
40 eleq1a 2827 . . . . . . . . 9 ((-1↑(♯‘𝑧)) ∈ {1, -1} → ((𝑁𝑥) = (-1↑(♯‘𝑧)) → (𝑁𝑥) ∈ {1, -1}))
4136, 39, 403syl 18 . . . . . . . 8 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → ((𝑁𝑥) = (-1↑(♯‘𝑧)) → (𝑁𝑥) ∈ {1, -1}))
4241adantld 491 . . . . . . 7 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → ((𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(♯‘𝑧))) → (𝑁𝑥) ∈ {1, -1}))
4342rexlimiv 3147 . . . . . 6 (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(♯‘𝑧))) → (𝑁𝑥) ∈ {1, -1})
4443a1i 11 . . . . 5 (𝐷𝑉 → (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(♯‘𝑧))) → (𝑁𝑥) ∈ {1, -1}))
4534, 44syl5 34 . . . 4 (𝐷𝑉 → (𝑥 ∈ dom 𝑁 → (𝑁𝑥) ∈ {1, -1}))
4645ralrimiv 3144 . . 3 (𝐷𝑉 → ∀𝑥 ∈ dom 𝑁(𝑁𝑥) ∈ {1, -1})
47 ffnfv 7102 . . 3 (𝑁:dom 𝑁⟶{1, -1} ↔ (𝑁 Fn dom 𝑁 ∧ ∀𝑥 ∈ dom 𝑁(𝑁𝑥) ∈ {1, -1}))
4832, 46, 47sylanbrc 583 . 2 (𝐷𝑉𝑁:dom 𝑁⟶{1, -1})
49 ccatcl 14506 . . . . . . 7 ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷))
501, 33, 4psgnvalii 19341 . . . . . . 7 ((𝐷𝑉 ∧ (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷)) → (𝑁‘(𝑆 Σg (𝑧 ++ 𝑤))) = (-1↑(♯‘(𝑧 ++ 𝑤))))
5149, 50sylan2 593 . . . . . 6 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘(𝑆 Σg (𝑧 ++ 𝑤))) = (-1↑(♯‘(𝑧 ++ 𝑤))))
521symggrp 19232 . . . . . . . . . 10 (𝐷𝑉𝑆 ∈ Grp)
5352grpmndd 18807 . . . . . . . . 9 (𝐷𝑉𝑆 ∈ Mnd)
5433, 1, 2symgtrf 19301 . . . . . . . . . . 11 ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆)
55 sswrd 14454 . . . . . . . . . . 11 (ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆) → Word ran (pmTrsp‘𝐷) ⊆ Word (Base‘𝑆))
5654, 55ax-mp 5 . . . . . . . . . 10 Word ran (pmTrsp‘𝐷) ⊆ Word (Base‘𝑆)
5756sseli 3974 . . . . . . . . 9 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → 𝑧 ∈ Word (Base‘𝑆))
5856sseli 3974 . . . . . . . . 9 (𝑤 ∈ Word ran (pmTrsp‘𝐷) → 𝑤 ∈ Word (Base‘𝑆))
592, 14gsumccat 18697 . . . . . . . . 9 ((𝑆 ∈ Mnd ∧ 𝑧 ∈ Word (Base‘𝑆) ∧ 𝑤 ∈ Word (Base‘𝑆)) → (𝑆 Σg (𝑧 ++ 𝑤)) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
6053, 57, 58, 59syl3an 1160 . . . . . . . 8 ((𝐷𝑉𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (𝑆 Σg (𝑧 ++ 𝑤)) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
61603expb 1120 . . . . . . 7 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑆 Σg (𝑧 ++ 𝑤)) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
6261fveq2d 6882 . . . . . 6 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘(𝑆 Σg (𝑧 ++ 𝑤))) = (𝑁‘((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤))))
63 ccatlen 14507 . . . . . . . . 9 ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (♯‘(𝑧 ++ 𝑤)) = ((♯‘𝑧) + (♯‘𝑤)))
6463adantl 482 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘(𝑧 ++ 𝑤)) = ((♯‘𝑧) + (♯‘𝑤)))
6564oveq2d 7409 . . . . . . 7 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(♯‘(𝑧 ++ 𝑤))) = (-1↑((♯‘𝑧) + (♯‘𝑤))))
66 neg1cn 12308 . . . . . . . . 9 -1 ∈ ℂ
6766a1i 11 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → -1 ∈ ℂ)
68 lencl 14465 . . . . . . . . 9 (𝑤 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑤) ∈ ℕ0)
6968ad2antll 727 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘𝑤) ∈ ℕ0)
7035ad2antrl 726 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘𝑧) ∈ ℕ0)
7167, 69, 70expaddd 14095 . . . . . . 7 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑((♯‘𝑧) + (♯‘𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
7265, 71eqtrd 2771 . . . . . 6 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(♯‘(𝑧 ++ 𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
7351, 62, 723eqtr3d 2779 . . . . 5 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
74 oveq12 7402 . . . . . . . 8 ((𝑥 = (𝑆 Σg 𝑧) ∧ 𝑦 = (𝑆 Σg 𝑤)) → (𝑥(+g𝑆)𝑦) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
7574fveq2d 6882 . . . . . . 7 ((𝑥 = (𝑆 Σg 𝑧) ∧ 𝑦 = (𝑆 Σg 𝑤)) → (𝑁‘(𝑥(+g𝑆)𝑦)) = (𝑁‘((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤))))
76 oveq12 7402 . . . . . . 7 (((𝑁𝑥) = (-1↑(♯‘𝑧)) ∧ (𝑁𝑦) = (-1↑(♯‘𝑤))) → ((𝑁𝑥) · (𝑁𝑦)) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
7775, 76eqeqan12d 2745 . . . . . 6 (((𝑥 = (𝑆 Σg 𝑧) ∧ 𝑦 = (𝑆 Σg 𝑤)) ∧ ((𝑁𝑥) = (-1↑(♯‘𝑧)) ∧ (𝑁𝑦) = (-1↑(♯‘𝑤)))) → ((𝑁‘(𝑥(+g𝑆)𝑦)) = ((𝑁𝑥) · (𝑁𝑦)) ↔ (𝑁‘((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤)))))
7877an4s 658 . . . . 5 (((𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(♯‘𝑧))) ∧ (𝑦 = (𝑆 Σg 𝑤) ∧ (𝑁𝑦) = (-1↑(♯‘𝑤)))) → ((𝑁‘(𝑥(+g𝑆)𝑦)) = ((𝑁𝑥) · (𝑁𝑦)) ↔ (𝑁‘((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤)))))
7973, 78syl5ibrcom 246 . . . 4 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (((𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(♯‘𝑧))) ∧ (𝑦 = (𝑆 Σg 𝑤) ∧ (𝑁𝑦) = (-1↑(♯‘𝑤)))) → (𝑁‘(𝑥(+g𝑆)𝑦)) = ((𝑁𝑥) · (𝑁𝑦))))
8079rexlimdvva 3210 . . 3 (𝐷𝑉 → (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)∃𝑤 ∈ Word ran (pmTrsp‘𝐷)((𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(♯‘𝑧))) ∧ (𝑦 = (𝑆 Σg 𝑤) ∧ (𝑁𝑦) = (-1↑(♯‘𝑤)))) → (𝑁‘(𝑥(+g𝑆)𝑦)) = ((𝑁𝑥) · (𝑁𝑦))))
811, 33, 4psgnvali 19340 . . . . 5 (𝑦 ∈ dom 𝑁 → ∃𝑤 ∈ Word ran (pmTrsp‘𝐷)(𝑦 = (𝑆 Σg 𝑤) ∧ (𝑁𝑦) = (-1↑(♯‘𝑤))))
8234, 81anim12i 613 . . . 4 ((𝑥 ∈ dom 𝑁𝑦 ∈ dom 𝑁) → (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(♯‘𝑧))) ∧ ∃𝑤 ∈ Word ran (pmTrsp‘𝐷)(𝑦 = (𝑆 Σg 𝑤) ∧ (𝑁𝑦) = (-1↑(♯‘𝑤)))))
83 reeanv 3225 . . . 4 (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)∃𝑤 ∈ Word ran (pmTrsp‘𝐷)((𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(♯‘𝑧))) ∧ (𝑦 = (𝑆 Σg 𝑤) ∧ (𝑁𝑦) = (-1↑(♯‘𝑤)))) ↔ (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(♯‘𝑧))) ∧ ∃𝑤 ∈ Word ran (pmTrsp‘𝐷)(𝑦 = (𝑆 Σg 𝑤) ∧ (𝑁𝑦) = (-1↑(♯‘𝑤)))))
8482, 83sylibr 233 . . 3 ((𝑥 ∈ dom 𝑁𝑦 ∈ dom 𝑁) → ∃𝑧 ∈ Word ran (pmTrsp‘𝐷)∃𝑤 ∈ Word ran (pmTrsp‘𝐷)((𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(♯‘𝑧))) ∧ (𝑦 = (𝑆 Σg 𝑤) ∧ (𝑁𝑦) = (-1↑(♯‘𝑤)))))
8580, 84impel 506 . 2 ((𝐷𝑉 ∧ (𝑥 ∈ dom 𝑁𝑦 ∈ dom 𝑁)) → (𝑁‘(𝑥(+g𝑆)𝑦)) = ((𝑁𝑥) · (𝑁𝑦)))
8610, 12, 16, 22, 25, 27, 48, 85isghmd 19067 1 (𝐷𝑉𝑁 ∈ (𝐹 GrpHom 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3060  wrex 3069  {crab 3431  Vcvv 3473  cdif 3941  wss 3944  {cpr 4624   I cid 5566  dom cdm 5669  ran crn 5670  Fun wfun 6526   Fn wfn 6527  wf 6528  cfv 6532  (class class class)co 7393  Fincfn 8922  cc 11090  1c1 11093   + caddc 11095   · cmul 11097  -cneg 11427  0cn0 12454  cz 12540  cexp 14009  chash 14272  Word cword 14446   ++ cconcat 14502  Basecbs 17126  s cress 17155  +gcplusg 17179   Σg cgsu 17368  Mndcmnd 18602  Grpcgrp 18794  SubGrpcsubg 18972   GrpHom cghm 19055  SymGrpcsymg 19198  pmTrspcpmtr 19273  pmSgncpsgn 19321  mulGrpcmgp 19946  fldccnfld 20878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169  ax-addf 11171  ax-mulf 11172
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-xor 1510  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-tp 4627  df-op 4629  df-ot 4631  df-uni 4902  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-isom 6541  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-tpos 8193  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-2o 8449  df-er 8686  df-map 8805  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-card 9916  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-div 11854  df-nn 12195  df-2 12257  df-3 12258  df-4 12259  df-5 12260  df-6 12261  df-7 12262  df-8 12263  df-9 12264  df-n0 12455  df-xnn0 12527  df-z 12541  df-dec 12660  df-uz 12805  df-rp 12957  df-fz 13467  df-fzo 13610  df-seq 13949  df-exp 14010  df-hash 14273  df-word 14447  df-lsw 14495  df-concat 14503  df-s1 14528  df-substr 14573  df-pfx 14603  df-splice 14682  df-reverse 14691  df-s2 14781  df-struct 17062  df-sets 17079  df-slot 17097  df-ndx 17109  df-base 17127  df-ress 17156  df-plusg 17192  df-mulr 17193  df-starv 17194  df-tset 17198  df-ple 17199  df-ds 17201  df-unif 17202  df-0g 17369  df-gsum 17370  df-mre 17512  df-mrc 17513  df-acs 17515  df-mgm 18543  df-sgrp 18592  df-mnd 18603  df-mhm 18647  df-submnd 18648  df-efmnd 18725  df-grp 18797  df-minusg 18798  df-subg 18975  df-ghm 19056  df-gim 19099  df-oppg 19174  df-symg 19199  df-pmtr 19274  df-psgn 19323  df-cmn 19614  df-abl 19615  df-mgp 19947  df-ur 19964  df-ring 20016  df-cring 20017  df-oppr 20102  df-dvdsr 20123  df-unit 20124  df-invr 20154  df-dvr 20165  df-drng 20267  df-cnfld 20879
This theorem is referenced by:  psgnghm2  21067  evpmss  21072
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