MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psgnghm Structured version   Visualization version   GIF version

Theorem psgnghm 20984
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 2736 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
3 eqid 2736 . . . . . 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 19283 . . . . 5 𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
65fndmi 6606 . . . 4 dom 𝑁 = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
76ssrab3 4040 . . 3 dom 𝑁 ⊆ (Base‘𝑆)
8 psgnghm.f . . . 4 𝐹 = (𝑆s dom 𝑁)
98, 2ressbas2 17120 . . 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 20982 . 2 {1, -1} = (Base‘𝑈)
1310fvexi 6856 . . 3 dom 𝑁 ∈ V
14 eqid 2736 . . . 4 (+g𝑆) = (+g𝑆)
158, 14ressplusg 17171 . . 3 (dom 𝑁 ∈ V → (+g𝑆) = (+g𝐹))
1613, 15ax-mp 5 . 2 (+g𝑆) = (+g𝐹)
17 prex 5389 . . 3 {1, -1} ∈ V
18 eqid 2736 . . . . 5 (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
19 cnfldmul 20802 . . . . 5 · = (.r‘ℂfld)
2018, 19mgpplusg 19900 . . . 4 · = (+g‘(mulGrp‘ℂfld))
2111, 20ressplusg 17171 . . 3 ({1, -1} ∈ V → · = (+g𝑈))
2217, 21ax-mp 5 . 2 · = (+g𝑈)
231, 4psgndmsubg 19284 . . 3 (𝐷𝑉 → dom 𝑁 ∈ (SubGrp‘𝑆))
248subggrp 18931 . . 3 (dom 𝑁 ∈ (SubGrp‘𝑆) → 𝐹 ∈ Grp)
2523, 24syl 17 . 2 (𝐷𝑉𝐹 ∈ Grp)
2611cnmsgngrp 20983 . . 3 𝑈 ∈ Grp
2726a1i 11 . 2 (𝐷𝑉𝑈 ∈ Grp)
28 fnfun 6602 . . . . . 6 (𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} → Fun 𝑁)
295, 28ax-mp 5 . . . . 5 Fun 𝑁
30 funfn 6531 . . . . 5 (Fun 𝑁𝑁 Fn dom 𝑁)
3129, 30mpbi 229 . . . 4 𝑁 Fn dom 𝑁
3231a1i 11 . . 3 (𝐷𝑉𝑁 Fn dom 𝑁)
33 eqid 2736 . . . . . 6 ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
341, 33, 4psgnvali 19290 . . . . 5 (𝑥 ∈ dom 𝑁 → ∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(♯‘𝑧))))
35 lencl 14421 . . . . . . . . . 10 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑧) ∈ ℕ0)
3635nn0zd 12525 . . . . . . . . 9 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑧) ∈ ℤ)
37 m1expcl2 13991 . . . . . . . . . 10 ((♯‘𝑧) ∈ ℤ → (-1↑(♯‘𝑧)) ∈ {-1, 1})
38 prcom 4693 . . . . . . . . . 10 {-1, 1} = {1, -1}
3937, 38eleqtrdi 2848 . . . . . . . . 9 ((♯‘𝑧) ∈ ℤ → (-1↑(♯‘𝑧)) ∈ {1, -1})
40 eleq1a 2833 . . . . . . . . 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 3145 . . . . . 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 3142 . . 3 (𝐷𝑉 → ∀𝑥 ∈ dom 𝑁(𝑁𝑥) ∈ {1, -1})
47 ffnfv 7066 . . 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 14462 . . . . . . 7 ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷))
501, 33, 4psgnvalii 19291 . . . . . . 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 19182 . . . . . . . . . 10 (𝐷𝑉𝑆 ∈ Grp)
5352grpmndd 18760 . . . . . . . . 9 (𝐷𝑉𝑆 ∈ Mnd)
5433, 1, 2symgtrf 19251 . . . . . . . . . . 11 ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆)
55 sswrd 14410 . . . . . . . . . . 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 3940 . . . . . . . . 9 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → 𝑧 ∈ Word (Base‘𝑆))
5856sseli 3940 . . . . . . . . 9 (𝑤 ∈ Word ran (pmTrsp‘𝐷) → 𝑤 ∈ Word (Base‘𝑆))
592, 14gsumccat 18651 . . . . . . . . 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 6846 . . . . . 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 14463 . . . . . . . . 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 7373 . . . . . . 7 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(♯‘(𝑧 ++ 𝑤))) = (-1↑((♯‘𝑧) + (♯‘𝑤))))
66 neg1cn 12267 . . . . . . . . 9 -1 ∈ ℂ
6766a1i 11 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → -1 ∈ ℂ)
68 lencl 14421 . . . . . . . . 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 14053 . . . . . . 7 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑((♯‘𝑧) + (♯‘𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
7265, 71eqtrd 2776 . . . . . 6 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(♯‘(𝑧 ++ 𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
7351, 62, 723eqtr3d 2784 . . . . 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 7366 . . . . . . . 8 ((𝑥 = (𝑆 Σg 𝑧) ∧ 𝑦 = (𝑆 Σg 𝑤)) → (𝑥(+g𝑆)𝑦) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
7574fveq2d 6846 . . . . . . 7 ((𝑥 = (𝑆 Σg 𝑧) ∧ 𝑦 = (𝑆 Σg 𝑤)) → (𝑁‘(𝑥(+g𝑆)𝑦)) = (𝑁‘((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤))))
76 oveq12 7366 . . . . . . 7 (((𝑁𝑥) = (-1↑(♯‘𝑧)) ∧ (𝑁𝑦) = (-1↑(♯‘𝑤))) → ((𝑁𝑥) · (𝑁𝑦)) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
7775, 76eqeqan12d 2750 . . . . . 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 3205 . . 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 19290 . . . . 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 3217 . . . 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 19017 1 (𝐷𝑉𝑁 ∈ (𝐹 GrpHom 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  cdif 3907  wss 3910  {cpr 4588   I cid 5530  dom cdm 5633  ran crn 5634  Fun wfun 6490   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  Fincfn 8883  cc 11049  1c1 11052   + caddc 11054   · cmul 11056  -cneg 11386  0cn0 12413  cz 12499  cexp 13967  chash 14230  Word cword 14402   ++ cconcat 14458  Basecbs 17083  s cress 17112  +gcplusg 17133   Σg cgsu 17322  Mndcmnd 18556  Grpcgrp 18748  SubGrpcsubg 18922   GrpHom cghm 19005  SymGrpcsymg 19148  pmTrspcpmtr 19223  pmSgncpsgn 19271  mulGrpcmgp 19896  fldccnfld 20796
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-addf 11130  ax-mulf 11131
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-ot 4595  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-xnn0 12486  df-z 12500  df-dec 12619  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-word 14403  df-lsw 14451  df-concat 14459  df-s1 14484  df-substr 14529  df-pfx 14559  df-splice 14638  df-reverse 14647  df-s2 14737  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-0g 17323  df-gsum 17324  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-efmnd 18679  df-grp 18751  df-minusg 18752  df-subg 18925  df-ghm 19006  df-gim 19049  df-oppg 19124  df-symg 19149  df-pmtr 19224  df-psgn 19273  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-dvr 20112  df-drng 20187  df-cnfld 20797
This theorem is referenced by:  psgnghm2  20985  evpmss  20990
  Copyright terms: Public domain W3C validator