Theorem isnvlem 27465

Description: Lemma for isnv 27467. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
isnvlem.1	⊢ 𝑋 = ran 𝐺
isnvlem.2	⊢ 𝑍 = (GId‘𝐺)

Assertion

Ref	Expression
isnvlem	⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))))

Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑁,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝑍(𝑥,𝑦)

Proof of Theorem isnvlem

Dummy variables 𝑔 𝑛 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nv 27447	. . 3 ⊢ NrmCVec = {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ (⟨𝑔, 𝑠⟩ ∈ CVecOLD ∧ 𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))}
2	1	eleq2i 2693	. 2 ⊢ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ↔ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ (⟨𝑔, 𝑠⟩ ∈ CVecOLD ∧ 𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))})
3		opeq1 4402	. . . . 5 ⊢ (𝑔 = 𝐺 → ⟨𝑔, 𝑠⟩ = ⟨𝐺, 𝑠⟩)
4	3	eleq1d 2686	. . . 4 ⊢ (𝑔 = 𝐺 → (⟨𝑔, 𝑠⟩ ∈ CVecOLD ↔ ⟨𝐺, 𝑠⟩ ∈ CVecOLD))
5		rneq 5351	. . . . . 6 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
6		isnvlem.1	. . . . . 6 ⊢ 𝑋 = ran 𝐺
7	5, 6	syl6eqr 2674	. . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
8	7	feq2d 6031	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑛:ran 𝑔⟶ℝ ↔ 𝑛:𝑋⟶ℝ))
9		fveq2 6191	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (GId‘𝑔) = (GId‘𝐺))
10		isnvlem.2	. . . . . . . . 9 ⊢ 𝑍 = (GId‘𝐺)
11	9, 10	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (GId‘𝑔) = 𝑍)
12	11	eqeq2d 2632	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑥 = (GId‘𝑔) ↔ 𝑥 = 𝑍))
13	12	imbi2d 330	. . . . . 6 ⊢ (𝑔 = 𝐺 → (((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ↔ ((𝑛‘𝑥) = 0 → 𝑥 = 𝑍)))
14		oveq 6656	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
15	14	fveq2d 6195	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑛‘(𝑥𝑔𝑦)) = (𝑛‘(𝑥𝐺𝑦)))
16	15	breq1d 4663	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)) ↔ (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))
17	7, 16	raleqbidv 3152	. . . . . 6 ⊢ (𝑔 = 𝐺 → (∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)) ↔ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))
18	13, 17	3anbi13d 1401	. . . . 5 ⊢ (𝑔 = 𝐺 → ((((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))) ↔ (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)))))
19	7, 18	raleqbidv 3152	. . . 4 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔(((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))) ↔ ∀𝑥 ∈ 𝑋 (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)))))
20	4, 8, 19	3anbi123d 1399	. . 3 ⊢ (𝑔 = 𝐺 → ((⟨𝑔, 𝑠⟩ ∈ CVecOLD ∧ 𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)))) ↔ (⟨𝐺, 𝑠⟩ ∈ CVecOLD ∧ 𝑛:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))))
21		opeq2 4403	. . . . 5 ⊢ (𝑠 = 𝑆 → ⟨𝐺, 𝑠⟩ = ⟨𝐺, 𝑆⟩)
22	21	eleq1d 2686	. . . 4 ⊢ (𝑠 = 𝑆 → (⟨𝐺, 𝑠⟩ ∈ CVecOLD ↔ ⟨𝐺, 𝑆⟩ ∈ CVecOLD))
23		oveq 6656	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝑦𝑠𝑥) = (𝑦𝑆𝑥))
24	23	fveq2d 6195	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑛‘(𝑦𝑠𝑥)) = (𝑛‘(𝑦𝑆𝑥)))
25	24	eqeq1d 2624	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ((𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ↔ (𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥))))
26	25	ralbidv 2986	. . . . . 6 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ↔ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥))))
27	26	3anbi2d 1404	. . . . 5 ⊢ (𝑠 = 𝑆 → ((((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))) ↔ (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)))))
28	27	ralbidv 2986	. . . 4 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑋 (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))) ↔ ∀𝑥 ∈ 𝑋 (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)))))
29	22, 28	3anbi13d 1401	. . 3 ⊢ (𝑠 = 𝑆 → ((⟨𝐺, 𝑠⟩ ∈ CVecOLD ∧ 𝑛:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)))) ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑛:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))))
30		feq1 6026	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑛:𝑋⟶ℝ ↔ 𝑁:𝑋⟶ℝ))
31		fveq1 6190	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛‘𝑥) = (𝑁‘𝑥))
32	31	eqeq1d 2624	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛‘𝑥) = 0 ↔ (𝑁‘𝑥) = 0))
33	32	imbi1d 331	. . . . . 6 ⊢ (𝑛 = 𝑁 → (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ↔ ((𝑁‘𝑥) = 0 → 𝑥 = 𝑍)))
34		fveq1 6190	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛‘(𝑦𝑆𝑥)) = (𝑁‘(𝑦𝑆𝑥)))
35	31	oveq2d 6666	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((abs‘𝑦) · (𝑛‘𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)))
36	34, 35	eqeq12d 2637	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ↔ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥))))
37	36	ralbidv 2986	. . . . . 6 ⊢ (𝑛 = 𝑁 → (∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ↔ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥))))
38		fveq1 6190	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛‘(𝑥𝐺𝑦)) = (𝑁‘(𝑥𝐺𝑦)))
39		fveq1 6190	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑛‘𝑦) = (𝑁‘𝑦))
40	31, 39	oveq12d 6668	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((𝑛‘𝑥) + (𝑛‘𝑦)) = ((𝑁‘𝑥) + (𝑁‘𝑦)))
41	38, 40	breq12d 4666	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)) ↔ (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))
42	41	ralbidv 2986	. . . . . 6 ⊢ (𝑛 = 𝑁 → (∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)) ↔ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))
43	33, 37, 42	3anbi123d 1399	. . . . 5 ⊢ (𝑛 = 𝑁 → ((((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))) ↔ (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))
44	43	ralbidv 2986	. . . 4 ⊢ (𝑛 = 𝑁 → (∀𝑥 ∈ 𝑋 (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))) ↔ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))
45	30, 44	3anbi23d 1402	. . 3 ⊢ (𝑛 = 𝑁 → ((⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑛:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑛‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑛‘(𝑥𝐺𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦)))) ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))))
46	20, 29, 45	eloprabg 6748	. 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ (⟨𝑔, 𝑠⟩ ∈ CVecOLD ∧ 𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))} ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))))
47	2, 46	syl5bb 272	1 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200  ⟨cop 4183   class class class wbr 4653  ran crn 5115  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  {coprab 6651  ℂcc 9934  ℝcr 9935  0cc0 9936   + caddc 9939   · cmul 9941   ≤ cle 10075  abscabs 13974  GIdcgi 27344  CVec_OLDcvc 27413  NrmCVeccnv 27439

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-nv 27447

This theorem is referenced by:  isnv  27467