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Theorem psgnghm 20642
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 2825 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
3 eqid 2825 . . . . . 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 18551 . . . . 5 𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
6 fndm 6451 . . . . 5 (𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} → dom 𝑁 = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin})
75, 6ax-mp 5 . . . 4 dom 𝑁 = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
87ssrab3 4060 . . 3 dom 𝑁 ⊆ (Base‘𝑆)
9 psgnghm.f . . . 4 𝐹 = (𝑆s dom 𝑁)
109, 2ressbas2 16547 . . 3 (dom 𝑁 ⊆ (Base‘𝑆) → dom 𝑁 = (Base‘𝐹))
118, 10ax-mp 5 . 2 dom 𝑁 = (Base‘𝐹)
12 psgnghm.u . . 3 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1})
1312cnmsgnbas 20640 . 2 {1, -1} = (Base‘𝑈)
1411fvexi 6680 . . 3 dom 𝑁 ∈ V
15 eqid 2825 . . . 4 (+g𝑆) = (+g𝑆)
169, 15ressplusg 16604 . . 3 (dom 𝑁 ∈ V → (+g𝑆) = (+g𝐹))
1714, 16ax-mp 5 . 2 (+g𝑆) = (+g𝐹)
18 prex 5328 . . 3 {1, -1} ∈ V
19 eqid 2825 . . . . 5 (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
20 cnfldmul 20469 . . . . 5 · = (.r‘ℂfld)
2119, 20mgpplusg 19165 . . . 4 · = (+g‘(mulGrp‘ℂfld))
2212, 21ressplusg 16604 . . 3 ({1, -1} ∈ V → · = (+g𝑈))
2318, 22ax-mp 5 . 2 · = (+g𝑈)
241, 4psgndmsubg 18552 . . 3 (𝐷𝑉 → dom 𝑁 ∈ (SubGrp‘𝑆))
259subggrp 18214 . . 3 (dom 𝑁 ∈ (SubGrp‘𝑆) → 𝐹 ∈ Grp)
2624, 25syl 17 . 2 (𝐷𝑉𝐹 ∈ Grp)
2712cnmsgngrp 20641 . . 3 𝑈 ∈ Grp
2827a1i 11 . 2 (𝐷𝑉𝑈 ∈ Grp)
29 fnfun 6449 . . . . . 6 (𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} → Fun 𝑁)
305, 29ax-mp 5 . . . . 5 Fun 𝑁
31 funfn 6381 . . . . 5 (Fun 𝑁𝑁 Fn dom 𝑁)
3230, 31mpbi 231 . . . 4 𝑁 Fn dom 𝑁
3332a1i 11 . . 3 (𝐷𝑉𝑁 Fn dom 𝑁)
34 eqid 2825 . . . . . 6 ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
351, 34, 4psgnvali 18558 . . . . 5 (𝑥 ∈ dom 𝑁 → ∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(♯‘𝑧))))
36 lencl 13876 . . . . . . . . . 10 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑧) ∈ ℕ0)
3736nn0zd 12077 . . . . . . . . 9 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑧) ∈ ℤ)
38 m1expcl2 13444 . . . . . . . . . 10 ((♯‘𝑧) ∈ ℤ → (-1↑(♯‘𝑧)) ∈ {-1, 1})
39 prcom 4666 . . . . . . . . . 10 {-1, 1} = {1, -1}
4038, 39syl6eleq 2927 . . . . . . . . 9 ((♯‘𝑧) ∈ ℤ → (-1↑(♯‘𝑧)) ∈ {1, -1})
41 eleq1a 2912 . . . . . . . . 9 ((-1↑(♯‘𝑧)) ∈ {1, -1} → ((𝑁𝑥) = (-1↑(♯‘𝑧)) → (𝑁𝑥) ∈ {1, -1}))
4237, 40, 413syl 18 . . . . . . . 8 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → ((𝑁𝑥) = (-1↑(♯‘𝑧)) → (𝑁𝑥) ∈ {1, -1}))
4342adantld 491 . . . . . . 7 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → ((𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(♯‘𝑧))) → (𝑁𝑥) ∈ {1, -1}))
4443rexlimiv 3284 . . . . . 6 (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(♯‘𝑧))) → (𝑁𝑥) ∈ {1, -1})
4544a1i 11 . . . . 5 (𝐷𝑉 → (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(♯‘𝑧))) → (𝑁𝑥) ∈ {1, -1}))
4635, 45syl5 34 . . . 4 (𝐷𝑉 → (𝑥 ∈ dom 𝑁 → (𝑁𝑥) ∈ {1, -1}))
4746ralrimiv 3185 . . 3 (𝐷𝑉 → ∀𝑥 ∈ dom 𝑁(𝑁𝑥) ∈ {1, -1})
48 ffnfv 6877 . . 3 (𝑁:dom 𝑁⟶{1, -1} ↔ (𝑁 Fn dom 𝑁 ∧ ∀𝑥 ∈ dom 𝑁(𝑁𝑥) ∈ {1, -1}))
4933, 47, 48sylanbrc 583 . 2 (𝐷𝑉𝑁:dom 𝑁⟶{1, -1})
50 ccatcl 13919 . . . . . . 7 ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷))
511, 34, 4psgnvalii 18559 . . . . . . 7 ((𝐷𝑉 ∧ (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷)) → (𝑁‘(𝑆 Σg (𝑧 ++ 𝑤))) = (-1↑(♯‘(𝑧 ++ 𝑤))))
5250, 51sylan2 592 . . . . . 6 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘(𝑆 Σg (𝑧 ++ 𝑤))) = (-1↑(♯‘(𝑧 ++ 𝑤))))
531symggrp 18450 . . . . . . . . . 10 (𝐷𝑉𝑆 ∈ Grp)
54 grpmnd 18042 . . . . . . . . . 10 (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
5553, 54syl 17 . . . . . . . . 9 (𝐷𝑉𝑆 ∈ Mnd)
5634, 1, 2symgtrf 18519 . . . . . . . . . . 11 ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆)
57 sswrd 13862 . . . . . . . . . . 11 (ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆) → Word ran (pmTrsp‘𝐷) ⊆ Word (Base‘𝑆))
5856, 57ax-mp 5 . . . . . . . . . 10 Word ran (pmTrsp‘𝐷) ⊆ Word (Base‘𝑆)
5958sseli 3966 . . . . . . . . 9 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → 𝑧 ∈ Word (Base‘𝑆))
6058sseli 3966 . . . . . . . . 9 (𝑤 ∈ Word ran (pmTrsp‘𝐷) → 𝑤 ∈ Word (Base‘𝑆))
612, 15gsumccat 17991 . . . . . . . . 9 ((𝑆 ∈ Mnd ∧ 𝑧 ∈ Word (Base‘𝑆) ∧ 𝑤 ∈ Word (Base‘𝑆)) → (𝑆 Σg (𝑧 ++ 𝑤)) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
6255, 59, 60, 61syl3an 1154 . . . . . . . 8 ((𝐷𝑉𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (𝑆 Σg (𝑧 ++ 𝑤)) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
63623expb 1114 . . . . . . 7 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑆 Σg (𝑧 ++ 𝑤)) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
6463fveq2d 6670 . . . . . 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))))
65 ccatlen 13920 . . . . . . . . 9 ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (♯‘(𝑧 ++ 𝑤)) = ((♯‘𝑧) + (♯‘𝑤)))
6665adantl 482 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘(𝑧 ++ 𝑤)) = ((♯‘𝑧) + (♯‘𝑤)))
6766oveq2d 7167 . . . . . . 7 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(♯‘(𝑧 ++ 𝑤))) = (-1↑((♯‘𝑧) + (♯‘𝑤))))
68 neg1cn 11743 . . . . . . . . 9 -1 ∈ ℂ
6968a1i 11 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → -1 ∈ ℂ)
70 lencl 13876 . . . . . . . . 9 (𝑤 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑤) ∈ ℕ0)
7170ad2antll 725 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘𝑤) ∈ ℕ0)
7236ad2antrl 724 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘𝑧) ∈ ℕ0)
7369, 71, 72expaddd 13505 . . . . . . 7 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑((♯‘𝑧) + (♯‘𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
7467, 73eqtrd 2860 . . . . . 6 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(♯‘(𝑧 ++ 𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
7552, 64, 743eqtr3d 2868 . . . . 5 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
76 oveq12 7160 . . . . . . . 8 ((𝑥 = (𝑆 Σg 𝑧) ∧ 𝑦 = (𝑆 Σg 𝑤)) → (𝑥(+g𝑆)𝑦) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
7776fveq2d 6670 . . . . . . 7 ((𝑥 = (𝑆 Σg 𝑧) ∧ 𝑦 = (𝑆 Σg 𝑤)) → (𝑁‘(𝑥(+g𝑆)𝑦)) = (𝑁‘((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤))))
78 oveq12 7160 . . . . . . 7 (((𝑁𝑥) = (-1↑(♯‘𝑧)) ∧ (𝑁𝑦) = (-1↑(♯‘𝑤))) → ((𝑁𝑥) · (𝑁𝑦)) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
7977, 78eqeqan12d 2842 . . . . . 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)))))
8079an4s 656 . . . . 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)))))
8175, 80syl5ibrcom 248 . . . 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))))
8281rexlimdvva 3298 . . 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))))
831, 34, 4psgnvali 18558 . . . . 5 (𝑦 ∈ dom 𝑁 → ∃𝑤 ∈ Word ran (pmTrsp‘𝐷)(𝑦 = (𝑆 Σg 𝑤) ∧ (𝑁𝑦) = (-1↑(♯‘𝑤))))
8435, 83anim12i 612 . . . 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)))))
85 reeanv 3372 . . . 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)))))
8684, 85sylibr 235 . . 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)))))
8782, 86impel 506 . 2 ((𝐷𝑉 ∧ (𝑥 ∈ dom 𝑁𝑦 ∈ dom 𝑁)) → (𝑁‘(𝑥(+g𝑆)𝑦)) = ((𝑁𝑥) · (𝑁𝑦)))
8811, 13, 17, 23, 26, 28, 49, 87isghmd 18299 1 (𝐷𝑉𝑁 ∈ (𝐹 GrpHom 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3142  wrex 3143  {crab 3146  Vcvv 3499  cdif 3936  wss 3939  {cpr 4565   I cid 5457  dom cdm 5553  ran crn 5554  Fun wfun 6345   Fn wfn 6346  wf 6347  cfv 6351  (class class class)co 7151  Fincfn 8501  cc 10527  1c1 10530   + caddc 10532   · cmul 10534  -cneg 10863  0cn0 11889  cz 11973  cexp 13422  chash 13683  Word cword 13854   ++ cconcat 13915  Basecbs 16475  s cress 16476  +gcplusg 16557   Σg cgsu 16706  Mndcmnd 17902  Grpcgrp 18035  SubGrpcsubg 18205   GrpHom cghm 18287  SymGrpcsymg 18427  pmTrspcpmtr 18491  pmSgncpsgn 18539  mulGrpcmgp 19161  fldccnfld 20463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-addf 10608  ax-mulf 10609
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-xor 1498  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-ot 4572  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-tpos 7886  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-xnn0 11960  df-z 11974  df-dec 12091  df-uz 12236  df-rp 12383  df-fz 12886  df-fzo 13027  df-seq 13363  df-exp 13423  df-hash 13684  df-word 13855  df-lsw 13908  df-concat 13916  df-s1 13943  df-substr 13996  df-pfx 14026  df-splice 14105  df-reverse 14114  df-s2 14203  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-starv 16572  df-tset 16576  df-ple 16577  df-ds 16579  df-unif 16580  df-0g 16707  df-gsum 16708  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-mhm 17946  df-submnd 17947  df-grp 18038  df-minusg 18039  df-subg 18208  df-ghm 18288  df-gim 18331  df-oppg 18406  df-symg 18428  df-pmtr 18492  df-psgn 18541  df-cmn 18830  df-abl 18831  df-mgp 19162  df-ur 19174  df-ring 19221  df-cring 19222  df-oppr 19295  df-dvdsr 19313  df-unit 19314  df-invr 19344  df-dvr 19355  df-drng 19426  df-cnfld 20464
This theorem is referenced by:  psgnghm2  20643  evpmss  20648
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