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Theorem psgnghm 20128
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 2760 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
3 eqid 2760 . . . . . 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 18121 . . . . 5 𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
6 fndm 6151 . . . . 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 3828 . . . 4 {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} ⊆ (Base‘𝑆)
97, 8eqsstri 3776 . . 3 dom 𝑁 ⊆ (Base‘𝑆)
10 psgnghm.f . . . 4 𝐹 = (𝑆s dom 𝑁)
1110, 2ressbas2 16133 . . 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 20126 . 2 {1, -1} = (Base‘𝑈)
15 fvex 6362 . . . 4 (Base‘𝐹) ∈ V
1612, 15eqeltri 2835 . . 3 dom 𝑁 ∈ V
17 eqid 2760 . . . 4 (+g𝑆) = (+g𝑆)
1810, 17ressplusg 16195 . . 3 (dom 𝑁 ∈ V → (+g𝑆) = (+g𝐹))
1916, 18ax-mp 5 . 2 (+g𝑆) = (+g𝐹)
20 prex 5058 . . 3 {1, -1} ∈ V
21 eqid 2760 . . . . 5 (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
22 cnfldmul 19954 . . . . 5 · = (.r‘ℂfld)
2321, 22mgpplusg 18693 . . . 4 · = (+g‘(mulGrp‘ℂfld))
2413, 23ressplusg 16195 . . 3 ({1, -1} ∈ V → · = (+g𝑈))
2520, 24ax-mp 5 . 2 · = (+g𝑈)
261, 4psgndmsubg 18122 . . 3 (𝐷𝑉 → dom 𝑁 ∈ (SubGrp‘𝑆))
2710subggrp 17798 . . 3 (dom 𝑁 ∈ (SubGrp‘𝑆) → 𝐹 ∈ Grp)
2826, 27syl 17 . 2 (𝐷𝑉𝐹 ∈ Grp)
2913cnmsgngrp 20127 . . 3 𝑈 ∈ Grp
3029a1i 11 . 2 (𝐷𝑉𝑈 ∈ Grp)
31 fnfun 6149 . . . . . 6 (𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} → Fun 𝑁)
325, 31ax-mp 5 . . . . 5 Fun 𝑁
33 funfn 6079 . . . . 5 (Fun 𝑁𝑁 Fn dom 𝑁)
3432, 33mpbi 220 . . . 4 𝑁 Fn dom 𝑁
3534a1i 11 . . 3 (𝐷𝑉𝑁 Fn dom 𝑁)
36 eqid 2760 . . . . . 6 ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
371, 36, 4psgnvali 18128 . . . . 5 (𝑥 ∈ dom 𝑁 → ∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(♯‘𝑧))))
38 lencl 13510 . . . . . . . . . . 11 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑧) ∈ ℕ0)
3938nn0zd 11672 . . . . . . . . . 10 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑧) ∈ ℤ)
40 m1expcl2 13076 . . . . . . . . . . 11 ((♯‘𝑧) ∈ ℤ → (-1↑(♯‘𝑧)) ∈ {-1, 1})
41 prcom 4411 . . . . . . . . . . 11 {-1, 1} = {1, -1}
4240, 41syl6eleq 2849 . . . . . . . . . 10 ((♯‘𝑧) ∈ ℤ → (-1↑(♯‘𝑧)) ∈ {1, -1})
4339, 42syl 17 . . . . . . . . 9 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (-1↑(♯‘𝑧)) ∈ {1, -1})
4443adantl 473 . . . . . . . 8 ((𝐷𝑉𝑧 ∈ Word ran (pmTrsp‘𝐷)) → (-1↑(♯‘𝑧)) ∈ {1, -1})
45 eleq1a 2834 . . . . . . . 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 484 . . . . . 6 ((𝐷𝑉𝑧 ∈ Word ran (pmTrsp‘𝐷)) → ((𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(♯‘𝑧))) → (𝑁𝑥) ∈ {1, -1}))
4847rexlimdva 3169 . . . . 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 3103 . . 3 (𝐷𝑉 → ∀𝑥 ∈ dom 𝑁(𝑁𝑥) ∈ {1, -1})
51 ffnfv 6551 . . 3 (𝑁:dom 𝑁⟶{1, -1} ↔ (𝑁 Fn dom 𝑁 ∧ ∀𝑥 ∈ dom 𝑁(𝑁𝑥) ∈ {1, -1}))
5235, 50, 51sylanbrc 701 . 2 (𝐷𝑉𝑁:dom 𝑁⟶{1, -1})
531, 36, 4psgnvali 18128 . . . . . 6 (𝑦 ∈ dom 𝑁 → ∃𝑤 ∈ Word ran (pmTrsp‘𝐷)(𝑦 = (𝑆 Σg 𝑤) ∧ (𝑁𝑦) = (-1↑(♯‘𝑤))))
5437, 53anim12i 591 . . . . 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 3245 . . . . 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 13546 . . . . . . . 8 ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷))
581, 36, 4psgnvalii 18129 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷)) → (𝑁‘(𝑆 Σg (𝑧 ++ 𝑤))) = (-1↑(♯‘(𝑧 ++ 𝑤))))
5957, 58sylan2 492 . . . . . . 7 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘(𝑆 Σg (𝑧 ++ 𝑤))) = (-1↑(♯‘(𝑧 ++ 𝑤))))
601symggrp 18020 . . . . . . . . . . 11 (𝐷𝑉𝑆 ∈ Grp)
61 grpmnd 17630 . . . . . . . . . . 11 (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
6260, 61syl 17 . . . . . . . . . 10 (𝐷𝑉𝑆 ∈ Mnd)
6336, 1, 2symgtrf 18089 . . . . . . . . . . . 12 ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆)
64 sswrd 13499 . . . . . . . . . . . 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 3740 . . . . . . . . . 10 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → 𝑧 ∈ Word (Base‘𝑆))
6765sseli 3740 . . . . . . . . . 10 (𝑤 ∈ Word ran (pmTrsp‘𝐷) → 𝑤 ∈ Word (Base‘𝑆))
682, 17gsumccat 17579 . . . . . . . . . 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 1164 . . . . . . . . 9 ((𝐷𝑉𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (𝑆 Σg (𝑧 ++ 𝑤)) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
70693expb 1114 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑆 Σg (𝑧 ++ 𝑤)) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
7170fveq2d 6356 . . . . . . 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 13547 . . . . . . . . . 10 ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (♯‘(𝑧 ++ 𝑤)) = ((♯‘𝑧) + (♯‘𝑤)))
7372adantl 473 . . . . . . . . 9 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘(𝑧 ++ 𝑤)) = ((♯‘𝑧) + (♯‘𝑤)))
7473oveq2d 6829 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(♯‘(𝑧 ++ 𝑤))) = (-1↑((♯‘𝑧) + (♯‘𝑤))))
75 neg1cn 11316 . . . . . . . . . 10 -1 ∈ ℂ
7675a1i 11 . . . . . . . . 9 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → -1 ∈ ℂ)
77 lencl 13510 . . . . . . . . . 10 (𝑤 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑤) ∈ ℕ0)
7877ad2antll 767 . . . . . . . . 9 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘𝑤) ∈ ℕ0)
7938ad2antrl 766 . . . . . . . . 9 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘𝑧) ∈ ℕ0)
8076, 78, 79expaddd 13204 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑((♯‘𝑧) + (♯‘𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
8174, 80eqtrd 2794 . . . . . . 7 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(♯‘(𝑧 ++ 𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
8259, 71, 813eqtr3d 2802 . . . . . 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 6822 . . . . . . . . 9 ((𝑥 = (𝑆 Σg 𝑧) ∧ 𝑦 = (𝑆 Σg 𝑤)) → (𝑥(+g𝑆)𝑦) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
8483fveq2d 6356 . . . . . . . 8 ((𝑥 = (𝑆 Σg 𝑧) ∧ 𝑦 = (𝑆 Σg 𝑤)) → (𝑁‘(𝑥(+g𝑆)𝑦)) = (𝑁‘((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤))))
85 oveq12 6822 . . . . . . . 8 (((𝑁𝑥) = (-1↑(♯‘𝑧)) ∧ (𝑁𝑦) = (-1↑(♯‘𝑤))) → ((𝑁𝑥) · (𝑁𝑦)) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
8684, 85eqeqan12d 2776 . . . . . . 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 904 . . . . . 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 3176 . . . 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 444 . 2 ((𝐷𝑉 ∧ (𝑥 ∈ dom 𝑁𝑦 ∈ dom 𝑁)) → (𝑁‘(𝑥(+g𝑆)𝑦)) = ((𝑁𝑥) · (𝑁𝑦)))
9212, 14, 19, 25, 28, 30, 52, 91isghmd 17870 1 (𝐷𝑉𝑁 ∈ (𝐹 GrpHom 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  {crab 3054  Vcvv 3340  cdif 3712  wss 3715  {cpr 4323   I cid 5173  dom cdm 5266  ran crn 5267  Fun wfun 6043   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6813  Fincfn 8121  cc 10126  1c1 10129   + caddc 10131   · cmul 10133  -cneg 10459  0cn0 11484  cz 11569  cexp 13054  chash 13311  Word cword 13477   ++ cconcat 13479  Basecbs 16059  s cress 16060  +gcplusg 16143   Σg cgsu 16303  Mndcmnd 17495  Grpcgrp 17623  SubGrpcsubg 17789   GrpHom cghm 17858  SymGrpcsymg 17997  pmTrspcpmtr 18061  pmSgncpsgn 18109  mulGrpcmgp 18689  fldccnfld 19948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-addf 10207  ax-mulf 10208
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-xor 1614  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-ot 4330  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-tpos 7521  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-xnn0 11556  df-z 11570  df-dec 11686  df-uz 11880  df-rp 12026  df-fz 12520  df-fzo 12660  df-seq 12996  df-exp 13055  df-hash 13312  df-word 13485  df-lsw 13486  df-concat 13487  df-s1 13488  df-substr 13489  df-splice 13490  df-reverse 13491  df-s2 13793  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-mulr 16157  df-starv 16158  df-tset 16162  df-ple 16163  df-ds 16166  df-unif 16167  df-0g 16304  df-gsum 16305  df-mre 16448  df-mrc 16449  df-acs 16451  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-mhm 17536  df-submnd 17537  df-grp 17626  df-minusg 17627  df-subg 17792  df-ghm 17859  df-gim 17902  df-oppg 17976  df-symg 17998  df-pmtr 18062  df-psgn 18111  df-cmn 18395  df-abl 18396  df-mgp 18690  df-ur 18702  df-ring 18749  df-cring 18750  df-oppr 18823  df-dvdsr 18841  df-unit 18842  df-invr 18872  df-dvr 18883  df-drng 18951  df-cnfld 19949
This theorem is referenced by:  psgnghm2  20129  evpmss  20134
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