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Theorem psgnghm 20201
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 2765 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
3 eqid 2765 . . . . . 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 18188 . . . . 5 𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
6 fndm 6170 . . . . 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 3849 . . . 4 {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} ⊆ (Base‘𝑆)
97, 8eqsstri 3797 . . 3 dom 𝑁 ⊆ (Base‘𝑆)
10 psgnghm.f . . . 4 𝐹 = (𝑆s dom 𝑁)
1110, 2ressbas2 16206 . . 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 20199 . 2 {1, -1} = (Base‘𝑈)
1512fvexi 6391 . . 3 dom 𝑁 ∈ V
16 eqid 2765 . . . 4 (+g𝑆) = (+g𝑆)
1710, 16ressplusg 16268 . . 3 (dom 𝑁 ∈ V → (+g𝑆) = (+g𝐹))
1815, 17ax-mp 5 . 2 (+g𝑆) = (+g𝐹)
19 prex 5067 . . 3 {1, -1} ∈ V
20 eqid 2765 . . . . 5 (mulGrp‘ℂfld) = (mulGrp‘ℂfld)
21 cnfldmul 20028 . . . . 5 · = (.r‘ℂfld)
2220, 21mgpplusg 18763 . . . 4 · = (+g‘(mulGrp‘ℂfld))
2313, 22ressplusg 16268 . . 3 ({1, -1} ∈ V → · = (+g𝑈))
2419, 23ax-mp 5 . 2 · = (+g𝑈)
251, 4psgndmsubg 18189 . . 3 (𝐷𝑉 → dom 𝑁 ∈ (SubGrp‘𝑆))
2610subggrp 17864 . . 3 (dom 𝑁 ∈ (SubGrp‘𝑆) → 𝐹 ∈ Grp)
2725, 26syl 17 . 2 (𝐷𝑉𝐹 ∈ Grp)
2813cnmsgngrp 20200 . . 3 𝑈 ∈ Grp
2928a1i 11 . 2 (𝐷𝑉𝑈 ∈ Grp)
30 fnfun 6168 . . . . . 6 (𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} → Fun 𝑁)
315, 30ax-mp 5 . . . . 5 Fun 𝑁
32 funfn 6100 . . . . 5 (Fun 𝑁𝑁 Fn dom 𝑁)
3331, 32mpbi 221 . . . 4 𝑁 Fn dom 𝑁
3433a1i 11 . . 3 (𝐷𝑉𝑁 Fn dom 𝑁)
35 eqid 2765 . . . . . 6 ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
361, 35, 4psgnvali 18195 . . . . 5 (𝑥 ∈ dom 𝑁 → ∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(♯‘𝑧))))
37 lencl 13508 . . . . . . . . . . 11 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑧) ∈ ℕ0)
3837nn0zd 11730 . . . . . . . . . 10 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑧) ∈ ℤ)
39 m1expcl2 13092 . . . . . . . . . . 11 ((♯‘𝑧) ∈ ℤ → (-1↑(♯‘𝑧)) ∈ {-1, 1})
40 prcom 4424 . . . . . . . . . . 11 {-1, 1} = {1, -1}
4139, 40syl6eleq 2854 . . . . . . . . . 10 ((♯‘𝑧) ∈ ℤ → (-1↑(♯‘𝑧)) ∈ {1, -1})
4238, 41syl 17 . . . . . . . . 9 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (-1↑(♯‘𝑧)) ∈ {1, -1})
4342adantl 473 . . . . . . . 8 ((𝐷𝑉𝑧 ∈ Word ran (pmTrsp‘𝐷)) → (-1↑(♯‘𝑧)) ∈ {1, -1})
44 eleq1a 2839 . . . . . . . 8 ((-1↑(♯‘𝑧)) ∈ {1, -1} → ((𝑁𝑥) = (-1↑(♯‘𝑧)) → (𝑁𝑥) ∈ {1, -1}))
4543, 44syl 17 . . . . . . 7 ((𝐷𝑉𝑧 ∈ Word ran (pmTrsp‘𝐷)) → ((𝑁𝑥) = (-1↑(♯‘𝑧)) → (𝑁𝑥) ∈ {1, -1}))
4645adantld 484 . . . . . 6 ((𝐷𝑉𝑧 ∈ Word ran (pmTrsp‘𝐷)) → ((𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(♯‘𝑧))) → (𝑁𝑥) ∈ {1, -1}))
4746rexlimdva 3178 . . . . 5 (𝐷𝑉 → (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σg 𝑧) ∧ (𝑁𝑥) = (-1↑(♯‘𝑧))) → (𝑁𝑥) ∈ {1, -1}))
4836, 47syl5 34 . . . 4 (𝐷𝑉 → (𝑥 ∈ dom 𝑁 → (𝑁𝑥) ∈ {1, -1}))
4948ralrimiv 3112 . . 3 (𝐷𝑉 → ∀𝑥 ∈ dom 𝑁(𝑁𝑥) ∈ {1, -1})
50 ffnfv 6580 . . 3 (𝑁:dom 𝑁⟶{1, -1} ↔ (𝑁 Fn dom 𝑁 ∧ ∀𝑥 ∈ dom 𝑁(𝑁𝑥) ∈ {1, -1}))
5134, 49, 50sylanbrc 578 . 2 (𝐷𝑉𝑁:dom 𝑁⟶{1, -1})
521, 35, 4psgnvali 18195 . . . . . 6 (𝑦 ∈ dom 𝑁 → ∃𝑤 ∈ Word ran (pmTrsp‘𝐷)(𝑦 = (𝑆 Σg 𝑤) ∧ (𝑁𝑦) = (-1↑(♯‘𝑤))))
5336, 52anim12i 606 . . . . 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)))))
54 reeanv 3254 . . . . 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)))))
5553, 54sylibr 225 . . . 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)))))
56 ccatcl 13548 . . . . . . . 8 ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷))
571, 35, 4psgnvalii 18196 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷)) → (𝑁‘(𝑆 Σg (𝑧 ++ 𝑤))) = (-1↑(♯‘(𝑧 ++ 𝑤))))
5856, 57sylan2 586 . . . . . . 7 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘(𝑆 Σg (𝑧 ++ 𝑤))) = (-1↑(♯‘(𝑧 ++ 𝑤))))
591symggrp 18086 . . . . . . . . . . 11 (𝐷𝑉𝑆 ∈ Grp)
60 grpmnd 17699 . . . . . . . . . . 11 (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
6159, 60syl 17 . . . . . . . . . 10 (𝐷𝑉𝑆 ∈ Mnd)
6235, 1, 2symgtrf 18155 . . . . . . . . . . . 12 ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆)
63 sswrd 13497 . . . . . . . . . . . 12 (ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆) → Word ran (pmTrsp‘𝐷) ⊆ Word (Base‘𝑆))
6462, 63ax-mp 5 . . . . . . . . . . 11 Word ran (pmTrsp‘𝐷) ⊆ Word (Base‘𝑆)
6564sseli 3759 . . . . . . . . . 10 (𝑧 ∈ Word ran (pmTrsp‘𝐷) → 𝑧 ∈ Word (Base‘𝑆))
6664sseli 3759 . . . . . . . . . 10 (𝑤 ∈ Word ran (pmTrsp‘𝐷) → 𝑤 ∈ Word (Base‘𝑆))
672, 16gsumccat 17647 . . . . . . . . . 10 ((𝑆 ∈ Mnd ∧ 𝑧 ∈ Word (Base‘𝑆) ∧ 𝑤 ∈ Word (Base‘𝑆)) → (𝑆 Σg (𝑧 ++ 𝑤)) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
6861, 65, 66, 67syl3an 1199 . . . . . . . . 9 ((𝐷𝑉𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (𝑆 Σg (𝑧 ++ 𝑤)) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
69683expb 1149 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑆 Σg (𝑧 ++ 𝑤)) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
7069fveq2d 6381 . . . . . . 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))))
71 ccatlen 13549 . . . . . . . . . 10 ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (♯‘(𝑧 ++ 𝑤)) = ((♯‘𝑧) + (♯‘𝑤)))
7271adantl 473 . . . . . . . . 9 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘(𝑧 ++ 𝑤)) = ((♯‘𝑧) + (♯‘𝑤)))
7372oveq2d 6860 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(♯‘(𝑧 ++ 𝑤))) = (-1↑((♯‘𝑧) + (♯‘𝑤))))
74 neg1cn 11395 . . . . . . . . . 10 -1 ∈ ℂ
7574a1i 11 . . . . . . . . 9 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → -1 ∈ ℂ)
76 lencl 13508 . . . . . . . . . 10 (𝑤 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑤) ∈ ℕ0)
7776ad2antll 720 . . . . . . . . 9 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘𝑤) ∈ ℕ0)
7837ad2antrl 719 . . . . . . . . 9 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (♯‘𝑧) ∈ ℕ0)
7975, 77, 78expaddd 13220 . . . . . . . 8 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑((♯‘𝑧) + (♯‘𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
8073, 79eqtrd 2799 . . . . . . 7 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(♯‘(𝑧 ++ 𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
8158, 70, 803eqtr3d 2807 . . . . . 6 ((𝐷𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤))) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
82 oveq12 6853 . . . . . . . . 9 ((𝑥 = (𝑆 Σg 𝑧) ∧ 𝑦 = (𝑆 Σg 𝑤)) → (𝑥(+g𝑆)𝑦) = ((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤)))
8382fveq2d 6381 . . . . . . . 8 ((𝑥 = (𝑆 Σg 𝑧) ∧ 𝑦 = (𝑆 Σg 𝑤)) → (𝑁‘(𝑥(+g𝑆)𝑦)) = (𝑁‘((𝑆 Σg 𝑧)(+g𝑆)(𝑆 Σg 𝑤))))
84 oveq12 6853 . . . . . . . 8 (((𝑁𝑥) = (-1↑(♯‘𝑧)) ∧ (𝑁𝑦) = (-1↑(♯‘𝑤))) → ((𝑁𝑥) · (𝑁𝑦)) = ((-1↑(♯‘𝑧)) · (-1↑(♯‘𝑤))))
8583, 84eqeqan12d 2781 . . . . . . 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)))))
8685an4s 650 . . . . . 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)))))
8781, 86syl5ibrcom 238 . . . . 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))))
8887rexlimdvva 3185 . . . 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))))
8955, 88syl5 34 . . 3 (𝐷𝑉 → ((𝑥 ∈ dom 𝑁𝑦 ∈ dom 𝑁) → (𝑁‘(𝑥(+g𝑆)𝑦)) = ((𝑁𝑥) · (𝑁𝑦))))
9089imp 395 . 2 ((𝐷𝑉 ∧ (𝑥 ∈ dom 𝑁𝑦 ∈ dom 𝑁)) → (𝑁‘(𝑥(+g𝑆)𝑦)) = ((𝑁𝑥) · (𝑁𝑦)))
9112, 14, 18, 24, 27, 29, 51, 90isghmd 17936 1 (𝐷𝑉𝑁 ∈ (𝐹 GrpHom 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  cdif 3731  wss 3734  {cpr 4338   I cid 5186  dom cdm 5279  ran crn 5280  Fun wfun 6064   Fn wfn 6065  wf 6066  cfv 6070  (class class class)co 6844  Fincfn 8162  cc 10189  1c1 10192   + caddc 10194   · cmul 10196  -cneg 10523  0cn0 11540  cz 11626  cexp 13070  chash 13324  Word cword 13489   ++ cconcat 13544  Basecbs 16133  s cress 16134  +gcplusg 16217   Σg cgsu 16370  Mndcmnd 17563  Grpcgrp 17692  SubGrpcsubg 17855   GrpHom cghm 17924  SymGrpcsymg 18063  pmTrspcpmtr 18127  pmSgncpsgn 18175  mulGrpcmgp 18759  fldccnfld 20022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268  ax-addf 10270  ax-mulf 10271
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-xor 1634  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-ot 4345  df-uni 4597  df-int 4636  df-iun 4680  df-iin 4681  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-om 7266  df-1st 7368  df-2nd 7369  df-tpos 7557  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-1o 7766  df-2o 7767  df-oadd 7770  df-er 7949  df-map 8064  df-en 8163  df-dom 8164  df-sdom 8165  df-fin 8166  df-card 9018  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-div 10941  df-nn 11277  df-2 11337  df-3 11338  df-4 11339  df-5 11340  df-6 11341  df-7 11342  df-8 11343  df-9 11344  df-n0 11541  df-xnn0 11613  df-z 11627  df-dec 11744  df-uz 11890  df-rp 12032  df-fz 12537  df-fzo 12677  df-seq 13012  df-exp 13071  df-hash 13325  df-word 13490  df-lsw 13537  df-concat 13545  df-s1 13570  df-substr 13620  df-pfx 13665  df-splice 13768  df-reverse 13786  df-s2 13880  df-struct 16135  df-ndx 16136  df-slot 16137  df-base 16139  df-sets 16140  df-ress 16141  df-plusg 16230  df-mulr 16231  df-starv 16232  df-tset 16236  df-ple 16237  df-ds 16239  df-unif 16240  df-0g 16371  df-gsum 16372  df-mre 16515  df-mrc 16516  df-acs 16518  df-mgm 17511  df-sgrp 17553  df-mnd 17564  df-mhm 17604  df-submnd 17605  df-grp 17695  df-minusg 17696  df-subg 17858  df-ghm 17925  df-gim 17968  df-oppg 18042  df-symg 18064  df-pmtr 18128  df-psgn 18177  df-cmn 18464  df-abl 18465  df-mgp 18760  df-ur 18772  df-ring 18819  df-cring 18820  df-oppr 18893  df-dvdsr 18911  df-unit 18912  df-invr 18942  df-dvr 18953  df-drng 19021  df-cnfld 20023
This theorem is referenced by:  psgnghm2  20202  evpmss  20207
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