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Theorem psgnghm 19926

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 = (𝑆 ↾_s dom 𝑁)
psgnghm.u	⊢ 𝑈 = ((mulGrp‘ℂ_fld) ↾_s {1, -1})

**Assertion**
Ref	Expression
psgnghm	⊢ (𝐷 ∈ 𝑉 → 𝑁 ∈ (𝐹 GrpHom 𝑈))

Proof of Theorem psgnghm Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psgnghm.s	. . . . . 6 ⊢ 𝑆 = (SymGrp‘𝐷)
2		eqid 2622	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2622	. . . . . 6 ⊢ {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
4		psgnghm.n	. . . . . 6 ⊢ 𝑁 = (pmSgn‘𝐷)
5	1, 2, 3, 4	psgnfn 17921	. . . . 5 ⊢ 𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
6		fndm 5990	. . . . 5 ⊢ (𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} → dom 𝑁 = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin})
7	5, 6	ax-mp 5	. . . 4 ⊢ dom 𝑁 = {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin}
8		ssrab2 3687	. . . 4 ⊢ {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} ⊆ (Base‘𝑆)
9	7, 8	eqsstri 3635	. . 3 ⊢ dom 𝑁 ⊆ (Base‘𝑆)
10		psgnghm.f	. . . 4 ⊢ 𝐹 = (𝑆 ↾_s dom 𝑁)
11	10, 2	ressbas2 15931	. . 3 ⊢ (dom 𝑁 ⊆ (Base‘𝑆) → dom 𝑁 = (Base‘𝐹))
12	9, 11	ax-mp 5	. 2 ⊢ dom 𝑁 = (Base‘𝐹)
13		psgnghm.u	. . 3 ⊢ 𝑈 = ((mulGrp‘ℂ_fld) ↾_s {1, -1})
14	13	cnmsgnbas 19924	. 2 ⊢ {1, -1} = (Base‘𝑈)
15		fvex 6201	. . . 4 ⊢ (Base‘𝐹) ∈ V
16	12, 15	eqeltri 2697	. . 3 ⊢ dom 𝑁 ∈ V
17		eqid 2622	. . . 4 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
18	10, 17	ressplusg 15993	. . 3 ⊢ (dom 𝑁 ∈ V → (+_g‘𝑆) = (+_g‘𝐹))
19	16, 18	ax-mp 5	. 2 ⊢ (+_g‘𝑆) = (+_g‘𝐹)
20		prex 4909	. . 3 ⊢ {1, -1} ∈ V
21		eqid 2622	. . . . 5 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
22		cnfldmul 19752	. . . . 5 ⊢ · = (._r‘ℂ_fld)
23	21, 22	mgpplusg 18493	. . . 4 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
24	13, 23	ressplusg 15993	. . 3 ⊢ ({1, -1} ∈ V → · = (+_g‘𝑈))
25	20, 24	ax-mp 5	. 2 ⊢ · = (+_g‘𝑈)
26	1, 4	psgndmsubg 17922	. . 3 ⊢ (𝐷 ∈ 𝑉 → dom 𝑁 ∈ (SubGrp‘𝑆))
27	10	subggrp 17597	. . 3 ⊢ (dom 𝑁 ∈ (SubGrp‘𝑆) → 𝐹 ∈ Grp)
28	26, 27	syl 17	. 2 ⊢ (𝐷 ∈ 𝑉 → 𝐹 ∈ Grp)
29	13	cnmsgngrp 19925	. . 3 ⊢ 𝑈 ∈ Grp
30	29	a1i 11	. 2 ⊢ (𝐷 ∈ 𝑉 → 𝑈 ∈ Grp)
31		fnfun 5988	. . . . . 6 ⊢ (𝑁 Fn {𝑥 ∈ (Base‘𝑆) ∣ dom (𝑥 ∖ I ) ∈ Fin} → Fun 𝑁)
32	5, 31	ax-mp 5	. . . . 5 ⊢ Fun 𝑁
33		funfn 5918	. . . . 5 ⊢ (Fun 𝑁 ↔ 𝑁 Fn dom 𝑁)
34	32, 33	mpbi 220	. . . 4 ⊢ 𝑁 Fn dom 𝑁
35	34	a1i 11	. . 3 ⊢ (𝐷 ∈ 𝑉 → 𝑁 Fn dom 𝑁)
36		eqid 2622	. . . . . 6 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
37	1, 36, 4	psgnvali 17928	. . . . 5 ⊢ (𝑥 ∈ dom 𝑁 → ∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σ_g 𝑧) ∧ (𝑁‘𝑥) = (-1↑(#‘𝑧))))
38		lencl 13324	. . . . . . . . . . 11 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (#‘𝑧) ∈ ℕ₀)
39	38	nn0zd 11480	. . . . . . . . . 10 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (#‘𝑧) ∈ ℤ)
40		m1expcl2 12882	. . . . . . . . . . 11 ⊢ ((#‘𝑧) ∈ ℤ → (-1↑(#‘𝑧)) ∈ {-1, 1})
41		prcom 4267	. . . . . . . . . . 11 ⊢ {-1, 1} = {1, -1}
42	40, 41	syl6eleq 2711	. . . . . . . . . 10 ⊢ ((#‘𝑧) ∈ ℤ → (-1↑(#‘𝑧)) ∈ {1, -1})
43	39, 42	syl 17	. . . . . . . . 9 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → (-1↑(#‘𝑧)) ∈ {1, -1})
44	43	adantl 482	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑧 ∈ Word ran (pmTrsp‘𝐷)) → (-1↑(#‘𝑧)) ∈ {1, -1})
45		eleq1a 2696	. . . . . . . 8 ⊢ ((-1↑(#‘𝑧)) ∈ {1, -1} → ((𝑁‘𝑥) = (-1↑(#‘𝑧)) → (𝑁‘𝑥) ∈ {1, -1}))
46	44, 45	syl 17	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑧 ∈ Word ran (pmTrsp‘𝐷)) → ((𝑁‘𝑥) = (-1↑(#‘𝑧)) → (𝑁‘𝑥) ∈ {1, -1}))
47	46	adantld 483	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑧 ∈ Word ran (pmTrsp‘𝐷)) → ((𝑥 = (𝑆 Σ_g 𝑧) ∧ (𝑁‘𝑥) = (-1↑(#‘𝑧))) → (𝑁‘𝑥) ∈ {1, -1}))
48	47	rexlimdva 3031	. . . . 5 ⊢ (𝐷 ∈ 𝑉 → (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σ_g 𝑧) ∧ (𝑁‘𝑥) = (-1↑(#‘𝑧))) → (𝑁‘𝑥) ∈ {1, -1}))
49	37, 48	syl5 34	. . . 4 ⊢ (𝐷 ∈ 𝑉 → (𝑥 ∈ dom 𝑁 → (𝑁‘𝑥) ∈ {1, -1}))
50	49	ralrimiv 2965	. . 3 ⊢ (𝐷 ∈ 𝑉 → ∀𝑥 ∈ dom 𝑁(𝑁‘𝑥) ∈ {1, -1})
51		ffnfv 6388	. . 3 ⊢ (𝑁:dom 𝑁⟶{1, -1} ↔ (𝑁 Fn dom 𝑁 ∧ ∀𝑥 ∈ dom 𝑁(𝑁‘𝑥) ∈ {1, -1}))
52	35, 50, 51	sylanbrc 698	. 2 ⊢ (𝐷 ∈ 𝑉 → 𝑁:dom 𝑁⟶{1, -1})
53	1, 36, 4	psgnvali 17928	. . . . . 6 ⊢ (𝑦 ∈ dom 𝑁 → ∃𝑤 ∈ Word ran (pmTrsp‘𝐷)(𝑦 = (𝑆 Σ_g 𝑤) ∧ (𝑁‘𝑦) = (-1↑(#‘𝑤))))
54	37, 53	anim12i 590	. . . . 5 ⊢ ((𝑥 ∈ dom 𝑁 ∧ 𝑦 ∈ dom 𝑁) → (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σ_g 𝑧) ∧ (𝑁‘𝑥) = (-1↑(#‘𝑧))) ∧ ∃𝑤 ∈ Word ran (pmTrsp‘𝐷)(𝑦 = (𝑆 Σ_g 𝑤) ∧ (𝑁‘𝑦) = (-1↑(#‘𝑤)))))
55		reeanv 3107	. . . . 5 ⊢ (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)∃𝑤 ∈ Word ran (pmTrsp‘𝐷)((𝑥 = (𝑆 Σ_g 𝑧) ∧ (𝑁‘𝑥) = (-1↑(#‘𝑧))) ∧ (𝑦 = (𝑆 Σ_g 𝑤) ∧ (𝑁‘𝑦) = (-1↑(#‘𝑤)))) ↔ (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)(𝑥 = (𝑆 Σ_g 𝑧) ∧ (𝑁‘𝑥) = (-1↑(#‘𝑧))) ∧ ∃𝑤 ∈ Word ran (pmTrsp‘𝐷)(𝑦 = (𝑆 Σ_g 𝑤) ∧ (𝑁‘𝑦) = (-1↑(#‘𝑤)))))
56	54, 55	sylibr 224	. . . 4 ⊢ ((𝑥 ∈ dom 𝑁 ∧ 𝑦 ∈ dom 𝑁) → ∃𝑧 ∈ Word ran (pmTrsp‘𝐷)∃𝑤 ∈ Word ran (pmTrsp‘𝐷)((𝑥 = (𝑆 Σ_g 𝑧) ∧ (𝑁‘𝑥) = (-1↑(#‘𝑧))) ∧ (𝑦 = (𝑆 Σ_g 𝑤) ∧ (𝑁‘𝑦) = (-1↑(#‘𝑤)))))
57		ccatcl 13359	. . . . . . . 8 ⊢ ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷))
58	1, 36, 4	psgnvalii 17929	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ++ 𝑤) ∈ Word ran (pmTrsp‘𝐷)) → (𝑁‘(𝑆 Σ_g (𝑧 ++ 𝑤))) = (-1↑(#‘(𝑧 ++ 𝑤))))
59	57, 58	sylan2 491	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘(𝑆 Σ_g (𝑧 ++ 𝑤))) = (-1↑(#‘(𝑧 ++ 𝑤))))
60	1	symggrp 17820	. . . . . . . . . . 11 ⊢ (𝐷 ∈ 𝑉 → 𝑆 ∈ Grp)
61		grpmnd 17429	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
62	60, 61	syl 17	. . . . . . . . . 10 ⊢ (𝐷 ∈ 𝑉 → 𝑆 ∈ Mnd)
63	36, 1, 2	symgtrf 17889	. . . . . . . . . . . 12 ⊢ ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆)
64		sswrd 13313	. . . . . . . . . . . 12 ⊢ (ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆) → Word ran (pmTrsp‘𝐷) ⊆ Word (Base‘𝑆))
65	63, 64	ax-mp 5	. . . . . . . . . . 11 ⊢ Word ran (pmTrsp‘𝐷) ⊆ Word (Base‘𝑆)
66	65	sseli 3599	. . . . . . . . . 10 ⊢ (𝑧 ∈ Word ran (pmTrsp‘𝐷) → 𝑧 ∈ Word (Base‘𝑆))
67	65	sseli 3599	. . . . . . . . . 10 ⊢ (𝑤 ∈ Word ran (pmTrsp‘𝐷) → 𝑤 ∈ Word (Base‘𝑆))
68	2, 17	gsumccat 17378	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Mnd ∧ 𝑧 ∈ Word (Base‘𝑆) ∧ 𝑤 ∈ Word (Base‘𝑆)) → (𝑆 Σ_g (𝑧 ++ 𝑤)) = ((𝑆 Σ_g 𝑧)(+_g‘𝑆)(𝑆 Σ_g 𝑤)))
69	62, 66, 67, 68	syl3an 1368	. . . . . . . . 9 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (𝑆 Σ_g (𝑧 ++ 𝑤)) = ((𝑆 Σ_g 𝑧)(+_g‘𝑆)(𝑆 Σ_g 𝑤)))
70	69	3expb 1266	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑆 Σ_g (𝑧 ++ 𝑤)) = ((𝑆 Σ_g 𝑧)(+_g‘𝑆)(𝑆 Σ_g 𝑤)))
71	70	fveq2d 6195	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘(𝑆 Σ_g (𝑧 ++ 𝑤))) = (𝑁‘((𝑆 Σ_g 𝑧)(+_g‘𝑆)(𝑆 Σ_g 𝑤))))
72		ccatlen 13360	. . . . . . . . . 10 ⊢ ((𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷)) → (#‘(𝑧 ++ 𝑤)) = ((#‘𝑧) + (#‘𝑤)))
73	72	adantl 482	. . . . . . . . 9 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (#‘(𝑧 ++ 𝑤)) = ((#‘𝑧) + (#‘𝑤)))
74	73	oveq2d 6666	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(#‘(𝑧 ++ 𝑤))) = (-1↑((#‘𝑧) + (#‘𝑤))))
75		neg1cn 11124	. . . . . . . . . 10 ⊢ -1 ∈ ℂ
76	75	a1i 11	. . . . . . . . 9 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → -1 ∈ ℂ)
77		lencl 13324	. . . . . . . . . 10 ⊢ (𝑤 ∈ Word ran (pmTrsp‘𝐷) → (#‘𝑤) ∈ ℕ₀)
78	77	ad2antll 765	. . . . . . . . 9 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (#‘𝑤) ∈ ℕ₀)
79	38	ad2antrl 764	. . . . . . . . 9 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (#‘𝑧) ∈ ℕ₀)
80	76, 78, 79	expaddd 13010	. . . . . . . 8 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑((#‘𝑧) + (#‘𝑤))) = ((-1↑(#‘𝑧)) · (-1↑(#‘𝑤))))
81	74, 80	eqtrd 2656	. . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (-1↑(#‘(𝑧 ++ 𝑤))) = ((-1↑(#‘𝑧)) · (-1↑(#‘𝑤))))
82	59, 71, 81	3eqtr3d 2664	. . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (𝑁‘((𝑆 Σ_g 𝑧)(+_g‘𝑆)(𝑆 Σ_g 𝑤))) = ((-1↑(#‘𝑧)) · (-1↑(#‘𝑤))))
83		oveq12 6659	. . . . . . . . 9 ⊢ ((𝑥 = (𝑆 Σ_g 𝑧) ∧ 𝑦 = (𝑆 Σ_g 𝑤)) → (𝑥(+_g‘𝑆)𝑦) = ((𝑆 Σ_g 𝑧)(+_g‘𝑆)(𝑆 Σ_g 𝑤)))
84	83	fveq2d 6195	. . . . . . . 8 ⊢ ((𝑥 = (𝑆 Σ_g 𝑧) ∧ 𝑦 = (𝑆 Σ_g 𝑤)) → (𝑁‘(𝑥(+_g‘𝑆)𝑦)) = (𝑁‘((𝑆 Σ_g 𝑧)(+_g‘𝑆)(𝑆 Σ_g 𝑤))))
85		oveq12 6659	. . . . . . . 8 ⊢ (((𝑁‘𝑥) = (-1↑(#‘𝑧)) ∧ (𝑁‘𝑦) = (-1↑(#‘𝑤))) → ((𝑁‘𝑥) · (𝑁‘𝑦)) = ((-1↑(#‘𝑧)) · (-1↑(#‘𝑤))))
86	84, 85	eqeqan12d 2638	. . . . . . 7 ⊢ (((𝑥 = (𝑆 Σ_g 𝑧) ∧ 𝑦 = (𝑆 Σ_g 𝑤)) ∧ ((𝑁‘𝑥) = (-1↑(#‘𝑧)) ∧ (𝑁‘𝑦) = (-1↑(#‘𝑤)))) → ((𝑁‘(𝑥(+_g‘𝑆)𝑦)) = ((𝑁‘𝑥) · (𝑁‘𝑦)) ↔ (𝑁‘((𝑆 Σ_g 𝑧)(+_g‘𝑆)(𝑆 Σ_g 𝑤))) = ((-1↑(#‘𝑧)) · (-1↑(#‘𝑤)))))
87	86	an4s 869	. . . . . 6 ⊢ (((𝑥 = (𝑆 Σ_g 𝑧) ∧ (𝑁‘𝑥) = (-1↑(#‘𝑧))) ∧ (𝑦 = (𝑆 Σ_g 𝑤) ∧ (𝑁‘𝑦) = (-1↑(#‘𝑤)))) → ((𝑁‘(𝑥(+_g‘𝑆)𝑦)) = ((𝑁‘𝑥) · (𝑁‘𝑦)) ↔ (𝑁‘((𝑆 Σ_g 𝑧)(+_g‘𝑆)(𝑆 Σ_g 𝑤))) = ((-1↑(#‘𝑧)) · (-1↑(#‘𝑤)))))
88	82, 87	syl5ibrcom 237	. . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑧 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑤 ∈ Word ran (pmTrsp‘𝐷))) → (((𝑥 = (𝑆 Σ_g 𝑧) ∧ (𝑁‘𝑥) = (-1↑(#‘𝑧))) ∧ (𝑦 = (𝑆 Σ_g 𝑤) ∧ (𝑁‘𝑦) = (-1↑(#‘𝑤)))) → (𝑁‘(𝑥(+_g‘𝑆)𝑦)) = ((𝑁‘𝑥) · (𝑁‘𝑦))))
89	88	rexlimdvva 3038	. . . 4 ⊢ (𝐷 ∈ 𝑉 → (∃𝑧 ∈ Word ran (pmTrsp‘𝐷)∃𝑤 ∈ Word ran (pmTrsp‘𝐷)((𝑥 = (𝑆 Σ_g 𝑧) ∧ (𝑁‘𝑥) = (-1↑(#‘𝑧))) ∧ (𝑦 = (𝑆 Σ_g 𝑤) ∧ (𝑁‘𝑦) = (-1↑(#‘𝑤)))) → (𝑁‘(𝑥(+_g‘𝑆)𝑦)) = ((𝑁‘𝑥) · (𝑁‘𝑦))))
90	56, 89	syl5 34	. . 3 ⊢ (𝐷 ∈ 𝑉 → ((𝑥 ∈ dom 𝑁 ∧ 𝑦 ∈ dom 𝑁) → (𝑁‘(𝑥(+_g‘𝑆)𝑦)) = ((𝑁‘𝑥) · (𝑁‘𝑦))))
91	90	imp 445	. 2 ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑥 ∈ dom 𝑁 ∧ 𝑦 ∈ dom 𝑁)) → (𝑁‘(𝑥(+_g‘𝑆)𝑦)) = ((𝑁‘𝑥) · (𝑁‘𝑦)))
92	12, 14, 19, 25, 28, 30, 52, 91	isghmd 17669	1 ⊢ (𝐷 ∈ 𝑉 → 𝑁 ∈ (𝐹 GrpHom 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 {crab 2916 Vcvv 3200 ∖ cdif 3571 ⊆ wss 3574 {cpr 4179 I cid 5023 dom cdm 5114 ran crn 5115 Fun wfun 5882 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℂcc 9934 1c1 9937 + caddc 9939 · cmul 9941 -cneg 10267 ℕ₀cn0 11292 ℤcz 11377 ↑cexp 12860 #chash 13117 Word cword 13291 ++ cconcat 13293 Basecbs 15857 ↾_s cress 15858 +_gcplusg 15941 Σ_g cgsu 16101 Mndcmnd 17294 Grpcgrp 17422 SubGrpcsubg 17588 GrpHom cghm 17657 SymGrpcsymg 17797 pmTrspcpmtr 17861 pmSgncpsgn 17909 mulGrpcmgp 18489 ℂ_fldccnfld 19746

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-xor 1465 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-substr 13303 df-splice 13304 df-reverse 13305 df-s2 13593 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-gsum 16103 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-subg 17591 df-ghm 17658 df-gim 17701 df-oppg 17776 df-symg 17798 df-pmtr 17862 df-psgn 17911 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-cnfld 19747

This theorem is referenced by: psgnghm2 19927 evpmss 19932

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