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Theorem psgnghm 19974
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 2651 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
3 eqid 2651 . . . . . 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 17967 . . . . 5 𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
6 fndm 6028 . . . . 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 3720 . . . 4 {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} ⊆ (Base‘𝑆)
97, 8eqsstri 3668 . . 3 dom 𝑁 ⊆ (Base‘𝑆)
10 psgnghm.f . . . 4 𝐹 = (𝑆s dom 𝑁)
1110, 2ressbas2 15978 . . 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 19972 . 2 {1, -1} = (Base‘𝑈)
15 fvex 6239 . . . 4 (Base‘𝐹) ∈ V
1612, 15eqeltri 2726 . . 3 dom 𝑁 ∈ V
17 eqid 2651 . . . 4 (+g𝑆) = (+g𝑆)
1810, 17ressplusg 16040 . . 3 (dom 𝑁 ∈ V → (+g𝑆) = (+g𝐹))
1916, 18ax-mp 5 . 2 (+g𝑆) = (+g𝐹)
20 prex 4939 . . 3 {1, -1} ∈ V
21 eqid 2651 . . . . 5 (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
22 cnfldmul 19800 . . . . 5 · = (.r‘ℂfld)
2321, 22mgpplusg 18539 . . . 4 · = (+g‘(mulGrp‘ℂfld))
2413, 23ressplusg 16040 . . 3 ({1, -1} ∈ V → · = (+g𝑈))
2520, 24ax-mp 5 . 2 · = (+g𝑈)
261, 4psgndmsubg 17968 . . 3 (𝐷𝑉 → dom 𝑁 ∈ (SubGrp‘𝑆))
2710subggrp 17644 . . 3 (dom 𝑁 ∈ (SubGrp‘𝑆) → 𝐹 ∈ Grp)
2826, 27syl 17 . 2 (𝐷𝑉𝐹 ∈ Grp)
2913cnmsgngrp 19973 . . 3 𝑈 ∈ Grp
3029a1i 11 . 2 (𝐷𝑉𝑈 ∈ Grp)
31 fnfun 6026 . . . . . 6 (𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} → Fun 𝑁)
325, 31ax-mp 5 . . . . 5 Fun 𝑁
33 funfn 5956 . . . . 5 (Fun 𝑁𝑁 Fn dom 𝑁)
3432, 33mpbi 220 . . . 4 𝑁 Fn dom 𝑁
3534a1i 11 . . 3 (𝐷𝑉𝑁 Fn dom 𝑁)
36 eqid 2651 . . . . . 6 ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
371, 36, 4psgnvali 17974 . . . . 5 (𝑥 ∈ dom 𝑁 → ∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(#‘𝑧))))
38 lencl 13356 . . . . . . . . . . 11 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (#‘𝑧) ∈ ℕ0)
3938nn0zd 11518 . . . . . . . . . 10 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (#‘𝑧) ∈ ℤ)
40 m1expcl2 12922 . . . . . . . . . . 11 ((#‘𝑧) ∈ ℤ → (-1↑(#‘𝑧)) ∈ {-1, 1})
41 prcom 4299 . . . . . . . . . . 11 {-1, 1} = {1, -1}
4240, 41syl6eleq 2740 . . . . . . . . . 10 ((#‘𝑧) ∈ ℤ → (-1↑(#‘𝑧)) ∈ {1, -1})
4339, 42syl 17 . . . . . . . . 9 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (-1↑(#‘𝑧)) ∈ {1, -1})
4443adantl 481 . . . . . . . 8 ((𝐷𝑉𝑧 ∈ Word ran (pmTrsp‘𝐷)) → (-1↑(#‘𝑧)) ∈ {1, -1})
45 eleq1a 2725 . . . . . . . 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 482 . . . . . 6 ((𝐷𝑉𝑧 ∈ Word ran (pmTrsp‘𝐷)) → ((𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(#‘𝑧))) → (𝑁𝑥) ∈ {1, -1}))
4847rexlimdva 3060 . . . . 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 2994 . . 3 (𝐷𝑉 → ∀𝑥 ∈ dom 𝑁(𝑁𝑥) ∈ {1, -1})
51 ffnfv 6428 . . 3 (𝑁:dom 𝑁⟶{1, -1} ↔ (𝑁 Fn dom 𝑁 ∧ ∀𝑥 ∈ dom 𝑁(𝑁𝑥) ∈ {1, -1}))
5235, 50, 51sylanbrc 699 . 2 (𝐷𝑉𝑁:dom 𝑁⟶{1, -1})
531, 36, 4psgnvali 17974 . . . . . 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 3136 . . . . 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 13392 . . . . . . . 8 ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷))
581, 36, 4psgnvalii 17975 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷)) → (𝑁‘(𝑆 Σg (𝑧 ++ 𝑤))) = (-1↑(#‘(𝑧 ++ 𝑤))))
5957, 58sylan2 490 . . . . . . 7 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘(𝑆 Σg (𝑧 ++ 𝑤))) = (-1↑(#‘(𝑧 ++ 𝑤))))
601symggrp 17866 . . . . . . . . . . 11 (𝐷𝑉𝑆 ∈ Grp)
61 grpmnd 17476 . . . . . . . . . . 11 (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
6260, 61syl 17 . . . . . . . . . 10 (𝐷𝑉𝑆 ∈ Mnd)
6336, 1, 2symgtrf 17935 . . . . . . . . . . . 12 ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆)
64 sswrd 13345 . . . . . . . . . . . 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 3632 . . . . . . . . . 10 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → 𝑧 ∈ Word (Base‘𝑆))
6765sseli 3632 . . . . . . . . . 10 (𝑤 ∈ Word ran (pmTrsp‘𝐷) → 𝑤 ∈ Word (Base‘𝑆))
682, 17gsumccat 17425 . . . . . . . . . 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 1408 . . . . . . . . 9 ((𝐷𝑉𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (𝑆 Σg (𝑧 ++ 𝑤)) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
70693expb 1285 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑆 Σg (𝑧 ++ 𝑤)) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
7170fveq2d 6233 . . . . . . 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 13393 . . . . . . . . . 10 ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (#‘(𝑧 ++ 𝑤)) = ((#‘𝑧) + (#‘𝑤)))
7372adantl 481 . . . . . . . . 9 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (#‘(𝑧 ++ 𝑤)) = ((#‘𝑧) + (#‘𝑤)))
7473oveq2d 6706 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(#‘(𝑧 ++ 𝑤))) = (-1↑((#‘𝑧) + (#‘𝑤))))
75 neg1cn 11162 . . . . . . . . . 10 -1 ∈ ℂ
7675a1i 11 . . . . . . . . 9 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → -1 ∈ ℂ)
77 lencl 13356 . . . . . . . . . 10 (𝑤 ∈ Word ran (pmTrsp‘𝐷) → (#‘𝑤) ∈ ℕ0)
7877ad2antll 765 . . . . . . . . 9 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (#‘𝑤) ∈ ℕ0)
7938ad2antrl 764 . . . . . . . . 9 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (#‘𝑧) ∈ ℕ0)
8076, 78, 79expaddd 13050 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑((#‘𝑧) + (#‘𝑤))) = ((-1↑(#‘𝑧)) · (-1↑(#‘𝑤))))
8174, 80eqtrd 2685 . . . . . . 7 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(#‘(𝑧 ++ 𝑤))) = ((-1↑(#‘𝑧)) · (-1↑(#‘𝑤))))
8259, 71, 813eqtr3d 2693 . . . . . 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 6699 . . . . . . . . 9 ((𝑥 = (𝑆 Σg 𝑧) ∧ 𝑦 = (𝑆 Σg 𝑤)) → (𝑥(+g𝑆)𝑦) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
8483fveq2d 6233 . . . . . . . 8 ((𝑥 = (𝑆 Σg 𝑧) ∧ 𝑦 = (𝑆 Σg 𝑤)) → (𝑁‘(𝑥(+g𝑆)𝑦)) = (𝑁‘((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤))))
85 oveq12 6699 . . . . . . . 8 (((𝑁𝑥) = (-1↑(#‘𝑧)) ∧ (𝑁𝑦) = (-1↑(#‘𝑤))) → ((𝑁𝑥) · (𝑁𝑦)) = ((-1↑(#‘𝑧)) · (-1↑(#‘𝑤))))
8684, 85eqeqan12d 2667 . . . . . . 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 886 . . . . . 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 3067 . . . 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 17716 1 (𝐷𝑉𝑁 ∈ (𝐹 GrpHom 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cdif 3604  wss 3607  {cpr 4212   I cid 5052  dom cdm 5143  ran crn 5144  Fun wfun 5920   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  Fincfn 7997  cc 9972  1c1 9975   + caddc 9977   · cmul 9979  -cneg 10305  0cn0 11330  cz 11415  cexp 12900  #chash 13157  Word cword 13323   ++ cconcat 13325  Basecbs 15904  s cress 15905  +gcplusg 15988   Σg cgsu 16148  Mndcmnd 17341  Grpcgrp 17469  SubGrpcsubg 17635   GrpHom cghm 17704  SymGrpcsymg 17843  pmTrspcpmtr 17907  pmSgncpsgn 17955  mulGrpcmgp 18535  fldccnfld 19794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-xor 1505  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-ot 4219  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-tpos 7397  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-xnn0 11402  df-z 11416  df-dec 11532  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-word 13331  df-lsw 13332  df-concat 13333  df-s1 13334  df-substr 13335  df-splice 13336  df-reverse 13337  df-s2 13639  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-0g 16149  df-gsum 16150  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-subg 17638  df-ghm 17705  df-gim 17748  df-oppg 17822  df-symg 17844  df-pmtr 17908  df-psgn 17957  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-cring 18596  df-oppr 18669  df-dvdsr 18687  df-unit 18688  df-invr 18718  df-dvr 18729  df-drng 18797  df-cnfld 19795
This theorem is referenced by:  psgnghm2  19975  evpmss  19980
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