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Theorem psgnghm 19845
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 2621 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
3 eqid 2621 . . . . . 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 17842 . . . . 5 𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
6 fndm 5948 . . . . 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}
8 ssrab2 3666 . . . 4 {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} ⊆ (Base‘𝑆)
97, 8eqsstri 3614 . . 3 dom 𝑁 ⊆ (Base‘𝑆)
10 psgnghm.f . . . 4 𝐹 = (𝑆s dom 𝑁)
1110, 2ressbas2 15852 . . 3 (dom 𝑁 ⊆ (Base‘𝑆) → dom 𝑁 = (Base‘𝐹))
129, 11ax-mp 5 . 2 dom 𝑁 = (Base‘𝐹)
13 psgnghm.u . . 3 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1})
1413cnmsgnbas 19843 . 2 {1, -1} = (Base‘𝑈)
15 fvex 6158 . . . 4 (Base‘𝐹) ∈ V
1612, 15eqeltri 2694 . . 3 dom 𝑁 ∈ V
17 eqid 2621 . . . 4 (+g𝑆) = (+g𝑆)
1810, 17ressplusg 15914 . . 3 (dom 𝑁 ∈ V → (+g𝑆) = (+g𝐹))
1916, 18ax-mp 5 . 2 (+g𝑆) = (+g𝐹)
20 prex 4870 . . 3 {1, -1} ∈ V
21 eqid 2621 . . . . 5 (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
22 cnfldmul 19671 . . . . 5 · = (.r‘ℂfld)
2321, 22mgpplusg 18414 . . . 4 · = (+g‘(mulGrp‘ℂfld))
2413, 23ressplusg 15914 . . 3 ({1, -1} ∈ V → · = (+g𝑈))
2520, 24ax-mp 5 . 2 · = (+g𝑈)
261, 4psgndmsubg 17843 . . 3 (𝐷𝑉 → dom 𝑁 ∈ (SubGrp‘𝑆))
2710subggrp 17518 . . 3 (dom 𝑁 ∈ (SubGrp‘𝑆) → 𝐹 ∈ Grp)
2826, 27syl 17 . 2 (𝐷𝑉𝐹 ∈ Grp)
2913cnmsgngrp 19844 . . 3 𝑈 ∈ Grp
3029a1i 11 . 2 (𝐷𝑉𝑈 ∈ Grp)
31 fnfun 5946 . . . . . 6 (𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} → Fun 𝑁)
325, 31ax-mp 5 . . . . 5 Fun 𝑁
33 funfn 5877 . . . . 5 (Fun 𝑁𝑁 Fn dom 𝑁)
3432, 33mpbi 220 . . . 4 𝑁 Fn dom 𝑁
3534a1i 11 . . 3 (𝐷𝑉𝑁 Fn dom 𝑁)
36 eqid 2621 . . . . . 6 ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
371, 36, 4psgnvali 17849 . . . . 5 (𝑥 ∈ dom 𝑁 → ∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(#‘𝑧))))
38 lencl 13263 . . . . . . . . . . 11 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (#‘𝑧) ∈ ℕ0)
3938nn0zd 11424 . . . . . . . . . 10 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (#‘𝑧) ∈ ℤ)
40 m1expcl2 12822 . . . . . . . . . . 11 ((#‘𝑧) ∈ ℤ → (-1↑(#‘𝑧)) ∈ {-1, 1})
41 prcom 4237 . . . . . . . . . . 11 {-1, 1} = {1, -1}
4240, 41syl6eleq 2708 . . . . . . . . . 10 ((#‘𝑧) ∈ ℤ → (-1↑(#‘𝑧)) ∈ {1, -1})
4339, 42syl 17 . . . . . . . . 9 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (-1↑(#‘𝑧)) ∈ {1, -1})
4443adantl 482 . . . . . . . 8 ((𝐷𝑉𝑧 ∈ Word ran (pmTrsp‘𝐷)) → (-1↑(#‘𝑧)) ∈ {1, -1})
45 eleq1a 2693 . . . . . . . 8 ((-1↑(#‘𝑧)) ∈ {1, -1} → ((𝑁𝑥) = (-1↑(#‘𝑧)) → (𝑁𝑥) ∈ {1, -1}))
4644, 45syl 17 . . . . . . 7 ((𝐷𝑉𝑧 ∈ Word ran (pmTrsp‘𝐷)) → ((𝑁𝑥) = (-1↑(#‘𝑧)) → (𝑁𝑥) ∈ {1, -1}))
4746adantld 483 . . . . . 6 ((𝐷𝑉𝑧 ∈ Word ran (pmTrsp‘𝐷)) → ((𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(#‘𝑧))) → (𝑁𝑥) ∈ {1, -1}))
4847rexlimdva 3024 . . . . 5 (𝐷𝑉 → (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(#‘𝑧))) → (𝑁𝑥) ∈ {1, -1}))
4937, 48syl5 34 . . . 4 (𝐷𝑉 → (𝑥 ∈ dom 𝑁 → (𝑁𝑥) ∈ {1, -1}))
5049ralrimiv 2959 . . 3 (𝐷𝑉 → ∀𝑥 ∈ dom 𝑁(𝑁𝑥) ∈ {1, -1})
51 ffnfv 6343 . . 3 (𝑁:dom 𝑁⟶{1, -1} ↔ (𝑁 Fn dom 𝑁 ∧ ∀𝑥 ∈ dom 𝑁(𝑁𝑥) ∈ {1, -1}))
5235, 50, 51sylanbrc 697 . 2 (𝐷𝑉𝑁:dom 𝑁⟶{1, -1})
531, 36, 4psgnvali 17849 . . . . . 6 (𝑦 ∈ dom 𝑁 → ∃𝑤 ∈ Word ran (pmTrsp‘𝐷)(𝑦 = (𝑆 Σg 𝑤) ∧ (𝑁𝑦) = (-1↑(#‘𝑤))))
5437, 53anim12i 589 . . . . 5 ((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)))))
55 reeanv 3097 . . . . 5 (∃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)))))
5654, 55sylibr 224 . . . 4 ((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)))))
57 ccatcl 13298 . . . . . . . 8 ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷))
581, 36, 4psgnvalii 17850 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷)) → (𝑁‘(𝑆 Σg (𝑧 ++ 𝑤))) = (-1↑(#‘(𝑧 ++ 𝑤))))
5957, 58sylan2 491 . . . . . . 7 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘(𝑆 Σg (𝑧 ++ 𝑤))) = (-1↑(#‘(𝑧 ++ 𝑤))))
601symggrp 17741 . . . . . . . . . . 11 (𝐷𝑉𝑆 ∈ Grp)
61 grpmnd 17350 . . . . . . . . . . 11 (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
6260, 61syl 17 . . . . . . . . . 10 (𝐷𝑉𝑆 ∈ Mnd)
6336, 1, 2symgtrf 17810 . . . . . . . . . . . 12 ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆)
64 sswrd 13252 . . . . . . . . . . . 12 (ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆) → Word ran (pmTrsp‘𝐷) ⊆ Word (Base‘𝑆))
6563, 64ax-mp 5 . . . . . . . . . . 11 Word ran (pmTrsp‘𝐷) ⊆ Word (Base‘𝑆)
6665sseli 3579 . . . . . . . . . 10 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → 𝑧 ∈ Word (Base‘𝑆))
6765sseli 3579 . . . . . . . . . 10 (𝑤 ∈ Word ran (pmTrsp‘𝐷) → 𝑤 ∈ Word (Base‘𝑆))
682, 17gsumccat 17299 . . . . . . . . . 10 ((𝑆 ∈ Mnd ∧ 𝑧 ∈ Word (Base‘𝑆) ∧ 𝑤 ∈ Word (Base‘𝑆)) → (𝑆 Σg (𝑧 ++ 𝑤)) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
6962, 66, 67, 68syl3an 1365 . . . . . . . . 9 ((𝐷𝑉𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (𝑆 Σg (𝑧 ++ 𝑤)) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
70693expb 1263 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑆 Σg (𝑧 ++ 𝑤)) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
7170fveq2d 6152 . . . . . . 7 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘(𝑆 Σg (𝑧 ++ 𝑤))) = (𝑁‘((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤))))
72 ccatlen 13299 . . . . . . . . . 10 ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (#‘(𝑧 ++ 𝑤)) = ((#‘𝑧) + (#‘𝑤)))
7372adantl 482 . . . . . . . . 9 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (#‘(𝑧 ++ 𝑤)) = ((#‘𝑧) + (#‘𝑤)))
7473oveq2d 6620 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(#‘(𝑧 ++ 𝑤))) = (-1↑((#‘𝑧) + (#‘𝑤))))
75 neg1cn 11068 . . . . . . . . . 10 -1 ∈ ℂ
7675a1i 11 . . . . . . . . 9 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → -1 ∈ ℂ)
77 lencl 13263 . . . . . . . . . 10 (𝑤 ∈ Word ran (pmTrsp‘𝐷) → (#‘𝑤) ∈ ℕ0)
7877ad2antll 764 . . . . . . . . 9 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (#‘𝑤) ∈ ℕ0)
7938ad2antrl 763 . . . . . . . . 9 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (#‘𝑧) ∈ ℕ0)
8076, 78, 79expaddd 12950 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑((#‘𝑧) + (#‘𝑤))) = ((-1↑(#‘𝑧)) · (-1↑(#‘𝑤))))
8174, 80eqtrd 2655 . . . . . . 7 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(#‘(𝑧 ++ 𝑤))) = ((-1↑(#‘𝑧)) · (-1↑(#‘𝑤))))
8259, 71, 813eqtr3d 2663 . . . . . 6 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤))) = ((-1↑(#‘𝑧)) · (-1↑(#‘𝑤))))
83 oveq12 6613 . . . . . . . . 9 ((𝑥 = (𝑆 Σg 𝑧) ∧ 𝑦 = (𝑆 Σg 𝑤)) → (𝑥(+g𝑆)𝑦) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
8483fveq2d 6152 . . . . . . . 8 ((𝑥 = (𝑆 Σg 𝑧) ∧ 𝑦 = (𝑆 Σg 𝑤)) → (𝑁‘(𝑥(+g𝑆)𝑦)) = (𝑁‘((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤))))
85 oveq12 6613 . . . . . . . 8 (((𝑁𝑥) = (-1↑(#‘𝑧)) ∧ (𝑁𝑦) = (-1↑(#‘𝑤))) → ((𝑁𝑥) · (𝑁𝑦)) = ((-1↑(#‘𝑧)) · (-1↑(#‘𝑤))))
8684, 85eqeqan12d 2637 . . . . . . 7 (((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)))))
8786an4s 868 . . . . . 6 (((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)))))
8882, 87syl5ibrcom 237 . . . . 5 ((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))))
8988rexlimdvva 3031 . . . 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))))
9056, 89syl5 34 . . 3 (𝐷𝑉 → ((𝑥 ∈ dom 𝑁𝑦 ∈ dom 𝑁) → (𝑁‘(𝑥(+g𝑆)𝑦)) = ((𝑁𝑥) · (𝑁𝑦))))
9190imp 445 . 2 ((𝐷𝑉 ∧ (𝑥 ∈ dom 𝑁𝑦 ∈ dom 𝑁)) → (𝑁‘(𝑥(+g𝑆)𝑦)) = ((𝑁𝑥) · (𝑁𝑦)))
9212, 14, 19, 25, 28, 30, 52, 91isghmd 17590 1 (𝐷𝑉𝑁 ∈ (𝐹 GrpHom 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  cdif 3552  wss 3555  {cpr 4150   I cid 4984  dom cdm 5074  ran crn 5075  Fun wfun 5841   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  Fincfn 7899  cc 9878  1c1 9881   + caddc 9883   · cmul 9885  -cneg 10211  0cn0 11236  cz 11321  cexp 12800  #chash 13057  Word cword 13230   ++ cconcat 13232  Basecbs 15781  s cress 15782  +gcplusg 15862   Σg cgsu 16022  Mndcmnd 17215  Grpcgrp 17343  SubGrpcsubg 17509   GrpHom cghm 17578  SymGrpcsymg 17718  pmTrspcpmtr 17782  pmSgncpsgn 17830  mulGrpcmgp 18410  fldccnfld 19665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-addf 9959  ax-mulf 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-xor 1462  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-ot 4157  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-tpos 7297  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-xnn0 11308  df-z 11322  df-dec 11438  df-uz 11632  df-rp 11777  df-fz 12269  df-fzo 12407  df-seq 12742  df-exp 12801  df-hash 13058  df-word 13238  df-lsw 13239  df-concat 13240  df-s1 13241  df-substr 13242  df-splice 13243  df-reverse 13244  df-s2 13530  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-starv 15877  df-tset 15881  df-ple 15882  df-ds 15885  df-unif 15886  df-0g 16023  df-gsum 16024  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-submnd 17257  df-grp 17346  df-minusg 17347  df-subg 17512  df-ghm 17579  df-gim 17622  df-oppg 17697  df-symg 17719  df-pmtr 17783  df-psgn 17832  df-cmn 18116  df-abl 18117  df-mgp 18411  df-ur 18423  df-ring 18470  df-cring 18471  df-oppr 18544  df-dvdsr 18562  df-unit 18563  df-invr 18593  df-dvr 18604  df-drng 18670  df-cnfld 19666
This theorem is referenced by:  psgnghm2  19846  evpmss  19851
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