cxpcn3 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cxpcn3		Structured version Visualization version GIF version

Theorem cxpcn3 26492

Description: Extend continuity of the complex power function to a base of zero, as long as the exponent has strictly positive real part. (Contributed by Mario Carneiro, 2-May-2016.)

**Hypotheses**
Ref	Expression
cxpcn3.d	⊢ 𝐷 = (^◡ℜ “ ℝ⁺)
cxpcn3.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)
cxpcn3.k	⊢ 𝐾 = (𝐽 ↾_t (0[,)+∞))
cxpcn3.l	⊢ 𝐿 = (𝐽 ↾_t 𝐷)

**Assertion**
Ref	Expression
cxpcn3	⊢ (𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ ((𝐾 ×_t 𝐿) Cn 𝐽)

Distinct variable groups: 𝑥,𝑦,𝐽 𝑥,𝐷,𝑦 Allowed substitution hints: 𝐾(𝑥,𝑦) 𝐿(𝑥,𝑦)

Proof of Theorem cxpcn3 Dummy variables 𝑎 𝑏 𝑑 𝑒 𝑢 𝑣 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rge0ssre 13437	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ
2		ax-resscn 11169	. . . . . . 7 ⊢ ℝ ⊆ ℂ
3	1, 2	sstri 3990	. . . . . 6 ⊢ (0[,)+∞) ⊆ ℂ
4	3	sseli 3977	. . . . 5 ⊢ (𝑥 ∈ (0[,)+∞) → 𝑥 ∈ ℂ)
5		cxpcn3.d	. . . . . . 7 ⊢ 𝐷 = (^◡ℜ “ ℝ⁺)
6		cnvimass 6079	. . . . . . . 8 ⊢ (^◡ℜ “ ℝ⁺) ⊆ dom ℜ
7		ref 15063	. . . . . . . . 9 ⊢ ℜ:ℂ⟶ℝ
8	7	fdmi 6728	. . . . . . . 8 ⊢ dom ℜ = ℂ
9	6, 8	sseqtri 4017	. . . . . . 7 ⊢ (^◡ℜ “ ℝ⁺) ⊆ ℂ
10	5, 9	eqsstri 4015	. . . . . 6 ⊢ 𝐷 ⊆ ℂ
11	10	sseli 3977	. . . . 5 ⊢ (𝑦 ∈ 𝐷 → 𝑦 ∈ ℂ)
12		cxpcl 26418	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥↑_𝑐𝑦) ∈ ℂ)
13	4, 11, 12	syl2an 594	. . . 4 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ 𝐷) → (𝑥↑_𝑐𝑦) ∈ ℂ)
14	13	rgen2 3195	. . 3 ⊢ ∀𝑥 ∈ (0[,)+∞)∀𝑦 ∈ 𝐷 (𝑥↑_𝑐𝑦) ∈ ℂ
15		eqid 2730	. . . 4 ⊢ (𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) = (𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))
16	15	fmpo 8056	. . 3 ⊢ (∀𝑥 ∈ (0[,)+∞)∀𝑦 ∈ 𝐷 (𝑥↑_𝑐𝑦) ∈ ℂ ↔ (𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)):((0[,)+∞) × 𝐷)⟶ℂ)
17	14, 16	mpbi 229	. 2 ⊢ (𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)):((0[,)+∞) × 𝐷)⟶ℂ
18		cxpcn3.j	. . . . . . . . . . . 12 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
19	18	cnfldtopon 24519	. . . . . . . . . . 11 ⊢ 𝐽 ∈ (TopOn‘ℂ)
20		rpre 12986	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
21		rpge0 12991	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ 𝑥)
22		elrege0 13435	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
23	20, 21, 22	sylanbrc 581	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ (0[,)+∞))
24	23	ssriv 3985	. . . . . . . . . . . 12 ⊢ ℝ⁺ ⊆ (0[,)+∞)
25	24, 3	sstri 3990	. . . . . . . . . . 11 ⊢ ℝ⁺ ⊆ ℂ
26		resttopon 22885	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ ℝ⁺ ⊆ ℂ) → (𝐽 ↾_t ℝ⁺) ∈ (TopOn‘ℝ⁺))
27	19, 25, 26	mp2an 688	. . . . . . . . . 10 ⊢ (𝐽 ↾_t ℝ⁺) ∈ (TopOn‘ℝ⁺)
28	27	toponrestid 22643	. . . . . . . . 9 ⊢ (𝐽 ↾_t ℝ⁺) = ((𝐽 ↾_t ℝ⁺) ↾_t ℝ⁺)
29	27	a1i 11	. . . . . . . . 9 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 < 𝑢) → (𝐽 ↾_t ℝ⁺) ∈ (TopOn‘ℝ⁺))
30		ssid 4003	. . . . . . . . . 10 ⊢ ℝ⁺ ⊆ ℝ⁺
31	30	a1i 11	. . . . . . . . 9 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 < 𝑢) → ℝ⁺ ⊆ ℝ⁺)
32		cxpcn3.l	. . . . . . . . 9 ⊢ 𝐿 = (𝐽 ↾_t 𝐷)
33	19	a1i 11	. . . . . . . . 9 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 < 𝑢) → 𝐽 ∈ (TopOn‘ℂ))
34	10	a1i 11	. . . . . . . . 9 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 < 𝑢) → 𝐷 ⊆ ℂ)
35		eqid 2730	. . . . . . . . . . 11 ⊢ (𝐽 ↾_t ℝ⁺) = (𝐽 ↾_t ℝ⁺)
36	18, 35	cxpcn2 26490	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺, 𝑦 ∈ ℂ ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐽 ↾_t ℝ⁺) ×_t 𝐽) Cn 𝐽)
37	36	a1i 11	. . . . . . . . 9 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 < 𝑢) → (𝑥 ∈ ℝ⁺, 𝑦 ∈ ℂ ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐽 ↾_t ℝ⁺) ×_t 𝐽) Cn 𝐽))
38	28, 29, 31, 32, 33, 34, 37	cnmpt2res 23401	. . . . . . . 8 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 < 𝑢) → (𝑥 ∈ ℝ⁺, 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐽 ↾_t ℝ⁺) ×_t 𝐿) Cn 𝐽))
39		elrege0 13435	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ (0[,)+∞) ↔ (𝑢 ∈ ℝ ∧ 0 ≤ 𝑢))
40	39	simplbi 496	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ (0[,)+∞) → 𝑢 ∈ ℝ)
41	40	adantr 479	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) → 𝑢 ∈ ℝ)
42	41	adantr 479	. . . . . . . . . 10 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 < 𝑢) → 𝑢 ∈ ℝ)
43		simpr 483	. . . . . . . . . 10 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 < 𝑢) → 0 < 𝑢)
44	42, 43	elrpd 13017	. . . . . . . . 9 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 < 𝑢) → 𝑢 ∈ ℝ⁺)
45		simplr 765	. . . . . . . . 9 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 < 𝑢) → 𝑣 ∈ 𝐷)
46	44, 45	opelxpd 5714	. . . . . . . 8 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 < 𝑢) → ⟨𝑢, 𝑣⟩ ∈ (ℝ⁺ × 𝐷))
47		resttopon 22885	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝐷 ⊆ ℂ) → (𝐽 ↾_t 𝐷) ∈ (TopOn‘𝐷))
48	19, 10, 47	mp2an 688	. . . . . . . . . . . 12 ⊢ (𝐽 ↾_t 𝐷) ∈ (TopOn‘𝐷)
49	32, 48	eqeltri 2827	. . . . . . . . . . 11 ⊢ 𝐿 ∈ (TopOn‘𝐷)
50		txtopon 23315	. . . . . . . . . . 11 ⊢ (((𝐽 ↾_t ℝ⁺) ∈ (TopOn‘ℝ⁺) ∧ 𝐿 ∈ (TopOn‘𝐷)) → ((𝐽 ↾_t ℝ⁺) ×_t 𝐿) ∈ (TopOn‘(ℝ⁺ × 𝐷)))
51	27, 49, 50	mp2an 688	. . . . . . . . . 10 ⊢ ((𝐽 ↾_t ℝ⁺) ×_t 𝐿) ∈ (TopOn‘(ℝ⁺ × 𝐷))
52	51	toponunii 22638	. . . . . . . . 9 ⊢ (ℝ⁺ × 𝐷) = ∪ ((𝐽 ↾_t ℝ⁺) ×_t 𝐿)
53	52	cncnpi 23002	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ⁺, 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐽 ↾_t ℝ⁺) ×_t 𝐿) Cn 𝐽) ∧ ⟨𝑢, 𝑣⟩ ∈ (ℝ⁺ × 𝐷)) → (𝑥 ∈ ℝ⁺, 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ ((((𝐽 ↾_t ℝ⁺) ×_t 𝐿) CnP 𝐽)‘⟨𝑢, 𝑣⟩))
54	38, 46, 53	syl2anc 582	. . . . . . 7 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 < 𝑢) → (𝑥 ∈ ℝ⁺, 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ ((((𝐽 ↾_t ℝ⁺) ×_t 𝐿) CnP 𝐽)‘⟨𝑢, 𝑣⟩))
55		ssid 4003	. . . . . . . 8 ⊢ 𝐷 ⊆ 𝐷
56		resmpo 7530	. . . . . . . 8 ⊢ ((ℝ⁺ ⊆ (0[,)+∞) ∧ 𝐷 ⊆ 𝐷) → ((𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ↾ (ℝ⁺ × 𝐷)) = (𝑥 ∈ ℝ⁺, 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)))
57	24, 55, 56	mp2an 688	. . . . . . 7 ⊢ ((𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ↾ (ℝ⁺ × 𝐷)) = (𝑥 ∈ ℝ⁺, 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))
58		cxpcn3.k	. . . . . . . . . . . 12 ⊢ 𝐾 = (𝐽 ↾_t (0[,)+∞))
59		resttopon 22885	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ (0[,)+∞) ⊆ ℂ) → (𝐽 ↾_t (0[,)+∞)) ∈ (TopOn‘(0[,)+∞)))
60	19, 3, 59	mp2an 688	. . . . . . . . . . . 12 ⊢ (𝐽 ↾_t (0[,)+∞)) ∈ (TopOn‘(0[,)+∞))
61	58, 60	eqeltri 2827	. . . . . . . . . . 11 ⊢ 𝐾 ∈ (TopOn‘(0[,)+∞))
62		ioorp 13406	. . . . . . . . . . . . . 14 ⊢ (0(,)+∞) = ℝ⁺
63		iooretop 24502	. . . . . . . . . . . . . 14 ⊢ (0(,)+∞) ∈ (topGen‘ran (,))
64	62, 63	eqeltrri 2828	. . . . . . . . . . . . 13 ⊢ ℝ⁺ ∈ (topGen‘ran (,))
65		retop 24498	. . . . . . . . . . . . . . 15 ⊢ (topGen‘ran (,)) ∈ Top
66		ovex 7444	. . . . . . . . . . . . . . 15 ⊢ (0[,)+∞) ∈ V
67		restopnb 22899	. . . . . . . . . . . . . . 15 ⊢ ((((topGen‘ran (,)) ∈ Top ∧ (0[,)+∞) ∈ V) ∧ (ℝ⁺ ∈ (topGen‘ran (,)) ∧ ℝ⁺ ⊆ (0[,)+∞) ∧ ℝ⁺ ⊆ ℝ⁺)) → (ℝ⁺ ∈ (topGen‘ran (,)) ↔ ℝ⁺ ∈ ((topGen‘ran (,)) ↾_t (0[,)+∞))))
68	65, 66, 67	mpanl12 698	. . . . . . . . . . . . . 14 ⊢ ((ℝ⁺ ∈ (topGen‘ran (,)) ∧ ℝ⁺ ⊆ (0[,)+∞) ∧ ℝ⁺ ⊆ ℝ⁺) → (ℝ⁺ ∈ (topGen‘ran (,)) ↔ ℝ⁺ ∈ ((topGen‘ran (,)) ↾_t (0[,)+∞))))
69	64, 24, 30, 68	mp3an 1459	. . . . . . . . . . . . 13 ⊢ (ℝ⁺ ∈ (topGen‘ran (,)) ↔ ℝ⁺ ∈ ((topGen‘ran (,)) ↾_t (0[,)+∞)))
70	64, 69	mpbi 229	. . . . . . . . . . . 12 ⊢ ℝ⁺ ∈ ((topGen‘ran (,)) ↾_t (0[,)+∞))
71		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
72	18, 71	rerest 24540	. . . . . . . . . . . . . 14 ⊢ ((0[,)+∞) ⊆ ℝ → (𝐽 ↾_t (0[,)+∞)) = ((topGen‘ran (,)) ↾_t (0[,)+∞)))
73	1, 72	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝐽 ↾_t (0[,)+∞)) = ((topGen‘ran (,)) ↾_t (0[,)+∞))
74	58, 73	eqtri 2758	. . . . . . . . . . . 12 ⊢ 𝐾 = ((topGen‘ran (,)) ↾_t (0[,)+∞))
75	70, 74	eleqtrri 2830	. . . . . . . . . . 11 ⊢ ℝ⁺ ∈ 𝐾
76		toponmax 22648	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ (TopOn‘𝐷) → 𝐷 ∈ 𝐿)
77	49, 76	ax-mp 5	. . . . . . . . . . 11 ⊢ 𝐷 ∈ 𝐿
78		txrest 23355	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ (TopOn‘(0[,)+∞)) ∧ 𝐿 ∈ (TopOn‘𝐷)) ∧ (ℝ⁺ ∈ 𝐾 ∧ 𝐷 ∈ 𝐿)) → ((𝐾 ×_t 𝐿) ↾_t (ℝ⁺ × 𝐷)) = ((𝐾 ↾_t ℝ⁺) ×_t (𝐿 ↾_t 𝐷)))
79	61, 49, 75, 77, 78	mp4an 689	. . . . . . . . . 10 ⊢ ((𝐾 ×_t 𝐿) ↾_t (ℝ⁺ × 𝐷)) = ((𝐾 ↾_t ℝ⁺) ×_t (𝐿 ↾_t 𝐷))
80	58	oveq1i 7421	. . . . . . . . . . . 12 ⊢ (𝐾 ↾_t ℝ⁺) = ((𝐽 ↾_t (0[,)+∞)) ↾_t ℝ⁺)
81		restabs 22889	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ ℝ⁺ ⊆ (0[,)+∞) ∧ (0[,)+∞) ∈ V) → ((𝐽 ↾_t (0[,)+∞)) ↾_t ℝ⁺) = (𝐽 ↾_t ℝ⁺))
82	19, 24, 66, 81	mp3an 1459	. . . . . . . . . . . 12 ⊢ ((𝐽 ↾_t (0[,)+∞)) ↾_t ℝ⁺) = (𝐽 ↾_t ℝ⁺)
83	80, 82	eqtri 2758	. . . . . . . . . . 11 ⊢ (𝐾 ↾_t ℝ⁺) = (𝐽 ↾_t ℝ⁺)
84	49	toponunii 22638	. . . . . . . . . . . . 13 ⊢ 𝐷 = ∪ 𝐿
85	84	restid 17383	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ (TopOn‘𝐷) → (𝐿 ↾_t 𝐷) = 𝐿)
86	49, 85	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐿 ↾_t 𝐷) = 𝐿
87	83, 86	oveq12i 7423	. . . . . . . . . 10 ⊢ ((𝐾 ↾_t ℝ⁺) ×_t (𝐿 ↾_t 𝐷)) = ((𝐽 ↾_t ℝ⁺) ×_t 𝐿)
88	79, 87	eqtri 2758	. . . . . . . . 9 ⊢ ((𝐾 ×_t 𝐿) ↾_t (ℝ⁺ × 𝐷)) = ((𝐽 ↾_t ℝ⁺) ×_t 𝐿)
89	88	oveq1i 7421	. . . . . . . 8 ⊢ (((𝐾 ×_t 𝐿) ↾_t (ℝ⁺ × 𝐷)) CnP 𝐽) = (((𝐽 ↾_t ℝ⁺) ×_t 𝐿) CnP 𝐽)
90	89	fveq1i 6891	. . . . . . 7 ⊢ ((((𝐾 ×_t 𝐿) ↾_t (ℝ⁺ × 𝐷)) CnP 𝐽)‘⟨𝑢, 𝑣⟩) = ((((𝐽 ↾_t ℝ⁺) ×_t 𝐿) CnP 𝐽)‘⟨𝑢, 𝑣⟩)
91	54, 57, 90	3eltr4g 2848	. . . . . 6 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 < 𝑢) → ((𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ↾ (ℝ⁺ × 𝐷)) ∈ ((((𝐾 ×_t 𝐿) ↾_t (ℝ⁺ × 𝐷)) CnP 𝐽)‘⟨𝑢, 𝑣⟩))
92		txtopon 23315	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘(0[,)+∞)) ∧ 𝐿 ∈ (TopOn‘𝐷)) → (𝐾 ×_t 𝐿) ∈ (TopOn‘((0[,)+∞) × 𝐷)))
93	61, 49, 92	mp2an 688	. . . . . . . . 9 ⊢ (𝐾 ×_t 𝐿) ∈ (TopOn‘((0[,)+∞) × 𝐷))
94	93	topontopi 22637	. . . . . . . 8 ⊢ (𝐾 ×_t 𝐿) ∈ Top
95	94	a1i 11	. . . . . . 7 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 < 𝑢) → (𝐾 ×_t 𝐿) ∈ Top)
96		xpss1 5694	. . . . . . . 8 ⊢ (ℝ⁺ ⊆ (0[,)+∞) → (ℝ⁺ × 𝐷) ⊆ ((0[,)+∞) × 𝐷))
97	24, 96	mp1i 13	. . . . . . 7 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 < 𝑢) → (ℝ⁺ × 𝐷) ⊆ ((0[,)+∞) × 𝐷))
98		txopn 23326	. . . . . . . . . 10 ⊢ (((𝐾 ∈ (TopOn‘(0[,)+∞)) ∧ 𝐿 ∈ (TopOn‘𝐷)) ∧ (ℝ⁺ ∈ 𝐾 ∧ 𝐷 ∈ 𝐿)) → (ℝ⁺ × 𝐷) ∈ (𝐾 ×_t 𝐿))
99	61, 49, 75, 77, 98	mp4an 689	. . . . . . . . 9 ⊢ (ℝ⁺ × 𝐷) ∈ (𝐾 ×_t 𝐿)
100		isopn3i 22806	. . . . . . . . 9 ⊢ (((𝐾 ×_t 𝐿) ∈ Top ∧ (ℝ⁺ × 𝐷) ∈ (𝐾 ×_t 𝐿)) → ((int‘(𝐾 ×_t 𝐿))‘(ℝ⁺ × 𝐷)) = (ℝ⁺ × 𝐷))
101	94, 99, 100	mp2an 688	. . . . . . . 8 ⊢ ((int‘(𝐾 ×_t 𝐿))‘(ℝ⁺ × 𝐷)) = (ℝ⁺ × 𝐷)
102	46, 101	eleqtrrdi 2842	. . . . . . 7 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 < 𝑢) → ⟨𝑢, 𝑣⟩ ∈ ((int‘(𝐾 ×_t 𝐿))‘(ℝ⁺ × 𝐷)))
103	17	a1i 11	. . . . . . 7 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 < 𝑢) → (𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)):((0[,)+∞) × 𝐷)⟶ℂ)
104	61	topontopi 22637	. . . . . . . . 9 ⊢ 𝐾 ∈ Top
105	49	topontopi 22637	. . . . . . . . 9 ⊢ 𝐿 ∈ Top
106	61	toponunii 22638	. . . . . . . . 9 ⊢ (0[,)+∞) = ∪ 𝐾
107	104, 105, 106, 84	txunii 23317	. . . . . . . 8 ⊢ ((0[,)+∞) × 𝐷) = ∪ (𝐾 ×_t 𝐿)
108	19	toponunii 22638	. . . . . . . 8 ⊢ ℂ = ∪ 𝐽
109	107, 108	cnprest 23013	. . . . . . 7 ⊢ ((((𝐾 ×_t 𝐿) ∈ Top ∧ (ℝ⁺ × 𝐷) ⊆ ((0[,)+∞) × 𝐷)) ∧ (⟨𝑢, 𝑣⟩ ∈ ((int‘(𝐾 ×_t 𝐿))‘(ℝ⁺ × 𝐷)) ∧ (𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)):((0[,)+∞) × 𝐷)⟶ℂ)) → ((𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐾 ×_t 𝐿) CnP 𝐽)‘⟨𝑢, 𝑣⟩) ↔ ((𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ↾ (ℝ⁺ × 𝐷)) ∈ ((((𝐾 ×_t 𝐿) ↾_t (ℝ⁺ × 𝐷)) CnP 𝐽)‘⟨𝑢, 𝑣⟩)))
110	95, 97, 102, 103, 109	syl22anc 835	. . . . . 6 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 < 𝑢) → ((𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐾 ×_t 𝐿) CnP 𝐽)‘⟨𝑢, 𝑣⟩) ↔ ((𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ↾ (ℝ⁺ × 𝐷)) ∈ ((((𝐾 ×_t 𝐿) ↾_t (ℝ⁺ × 𝐷)) CnP 𝐽)‘⟨𝑢, 𝑣⟩)))
111	91, 110	mpbird 256	. . . . 5 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 < 𝑢) → (𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐾 ×_t 𝐿) CnP 𝐽)‘⟨𝑢, 𝑣⟩))
112	17	a1i 11	. . . . . . . 8 ⊢ (𝑣 ∈ 𝐷 → (𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)):((0[,)+∞) × 𝐷)⟶ℂ)
113		eqid 2730	. . . . . . . . . . 11 ⊢ (if((ℜ‘𝑣) ≤ 1, (ℜ‘𝑣), 1) / 2) = (if((ℜ‘𝑣) ≤ 1, (ℜ‘𝑣), 1) / 2)
114		eqid 2730	. . . . . . . . . . 11 ⊢ if((if((ℜ‘𝑣) ≤ 1, (ℜ‘𝑣), 1) / 2) ≤ (𝑒↑_𝑐(1 / (if((ℜ‘𝑣) ≤ 1, (ℜ‘𝑣), 1) / 2))), (if((ℜ‘𝑣) ≤ 1, (ℜ‘𝑣), 1) / 2), (𝑒↑_𝑐(1 / (if((ℜ‘𝑣) ≤ 1, (ℜ‘𝑣), 1) / 2)))) = if((if((ℜ‘𝑣) ≤ 1, (ℜ‘𝑣), 1) / 2) ≤ (𝑒↑_𝑐(1 / (if((ℜ‘𝑣) ≤ 1, (ℜ‘𝑣), 1) / 2))), (if((ℜ‘𝑣) ≤ 1, (ℜ‘𝑣), 1) / 2), (𝑒↑_𝑐(1 / (if((ℜ‘𝑣) ≤ 1, (ℜ‘𝑣), 1) / 2))))
115	5, 18, 58, 32, 113, 114	cxpcn3lem 26491	. . . . . . . . . 10 ⊢ ((𝑣 ∈ 𝐷 ∧ 𝑒 ∈ ℝ⁺) → ∃𝑑 ∈ ℝ⁺ ∀𝑎 ∈ (0[,)+∞)∀𝑏 ∈ 𝐷 (((abs‘𝑎) < 𝑑 ∧ (abs‘(𝑣 − 𝑏)) < 𝑑) → (abs‘(𝑎↑_𝑐𝑏)) < 𝑒))
116	115	ralrimiva 3144	. . . . . . . . 9 ⊢ (𝑣 ∈ 𝐷 → ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑎 ∈ (0[,)+∞)∀𝑏 ∈ 𝐷 (((abs‘𝑎) < 𝑑 ∧ (abs‘(𝑣 − 𝑏)) < 𝑑) → (abs‘(𝑎↑_𝑐𝑏)) < 𝑒))
117		0e0icopnf 13439	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ (0[,)+∞)
118	117	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → 0 ∈ (0[,)+∞))
119		simprl 767	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → 𝑎 ∈ (0[,)+∞))
120	118, 119	ovresd 7576	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → (0((abs ∘ − ) ↾ ((0[,)+∞) × (0[,)+∞)))𝑎) = (0(abs ∘ − )𝑎))
121		0cn 11210	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℂ
122	3, 119	sselid 3979	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → 𝑎 ∈ ℂ)
123		eqid 2730	. . . . . . . . . . . . . . . . . 18 ⊢ (abs ∘ − ) = (abs ∘ − )
124	123	cnmetdval 24507	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (0(abs ∘ − )𝑎) = (abs‘(0 − 𝑎)))
125	121, 122, 124	sylancr 585	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → (0(abs ∘ − )𝑎) = (abs‘(0 − 𝑎)))
126		df-neg 11451	. . . . . . . . . . . . . . . . . 18 ⊢ -𝑎 = (0 − 𝑎)
127	126	fveq2i 6893	. . . . . . . . . . . . . . . . 17 ⊢ (abs‘-𝑎) = (abs‘(0 − 𝑎))
128	122	absnegd 15400	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → (abs‘-𝑎) = (abs‘𝑎))
129	127, 128	eqtr3id 2784	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → (abs‘(0 − 𝑎)) = (abs‘𝑎))
130	120, 125, 129	3eqtrd 2774	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → (0((abs ∘ − ) ↾ ((0[,)+∞) × (0[,)+∞)))𝑎) = (abs‘𝑎))
131	130	breq1d 5157	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → ((0((abs ∘ − ) ↾ ((0[,)+∞) × (0[,)+∞)))𝑎) < 𝑑 ↔ (abs‘𝑎) < 𝑑))
132		simpl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → 𝑣 ∈ 𝐷)
133		simprr 769	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → 𝑏 ∈ 𝐷)
134	132, 133	ovresd 7576	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → (𝑣((abs ∘ − ) ↾ (𝐷 × 𝐷))𝑏) = (𝑣(abs ∘ − )𝑏))
135	10, 132	sselid 3979	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → 𝑣 ∈ ℂ)
136	10, 133	sselid 3979	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → 𝑏 ∈ ℂ)
137	123	cnmetdval 24507	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑣(abs ∘ − )𝑏) = (abs‘(𝑣 − 𝑏)))
138	135, 136, 137	syl2anc 582	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → (𝑣(abs ∘ − )𝑏) = (abs‘(𝑣 − 𝑏)))
139	134, 138	eqtrd 2770	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → (𝑣((abs ∘ − ) ↾ (𝐷 × 𝐷))𝑏) = (abs‘(𝑣 − 𝑏)))
140	139	breq1d 5157	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → ((𝑣((abs ∘ − ) ↾ (𝐷 × 𝐷))𝑏) < 𝑑 ↔ (abs‘(𝑣 − 𝑏)) < 𝑑))
141	131, 140	anbi12d 629	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → (((0((abs ∘ − ) ↾ ((0[,)+∞) × (0[,)+∞)))𝑎) < 𝑑 ∧ (𝑣((abs ∘ − ) ↾ (𝐷 × 𝐷))𝑏) < 𝑑) ↔ ((abs‘𝑎) < 𝑑 ∧ (abs‘(𝑣 − 𝑏)) < 𝑑)))
142		oveq12 7420	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑣) → (𝑥↑_𝑐𝑦) = (0↑_𝑐𝑣))
143		ovex 7444	. . . . . . . . . . . . . . . . . . 19 ⊢ (0↑_𝑐𝑣) ∈ V
144	142, 15, 143	ovmpoa 7565	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) → (0(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑣) = (0↑_𝑐𝑣))
145	117, 132, 144	sylancr 585	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → (0(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑣) = (0↑_𝑐𝑣))
146	5	eleq2i 2823	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ 𝐷 ↔ 𝑣 ∈ (^◡ℜ “ ℝ⁺))
147		ffn 6716	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℜ:ℂ⟶ℝ → ℜ Fn ℂ)
148		elpreima 7058	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℜ Fn ℂ → (𝑣 ∈ (^◡ℜ “ ℝ⁺) ↔ (𝑣 ∈ ℂ ∧ (ℜ‘𝑣) ∈ ℝ⁺)))
149	7, 147, 148	mp2b 10	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ (^◡ℜ “ ℝ⁺) ↔ (𝑣 ∈ ℂ ∧ (ℜ‘𝑣) ∈ ℝ⁺))
150	146, 149	bitri 274	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 ∈ 𝐷 ↔ (𝑣 ∈ ℂ ∧ (ℜ‘𝑣) ∈ ℝ⁺))
151	150	simplbi 496	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ∈ 𝐷 → 𝑣 ∈ ℂ)
152	150	simprbi 495	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ 𝐷 → (ℜ‘𝑣) ∈ ℝ⁺)
153	152	rpne0d 13025	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 ∈ 𝐷 → (ℜ‘𝑣) ≠ 0)
154		fveq2 6890	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = 0 → (ℜ‘𝑣) = (ℜ‘0))
155		re0 15103	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℜ‘0) = 0
156	154, 155	eqtrdi 2786	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = 0 → (ℜ‘𝑣) = 0)
157	156	necon3i 2971	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℜ‘𝑣) ≠ 0 → 𝑣 ≠ 0)
158	153, 157	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ∈ 𝐷 → 𝑣 ≠ 0)
159	151, 158	0cxpd 26454	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ 𝐷 → (0↑_𝑐𝑣) = 0)
160	159	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → (0↑_𝑐𝑣) = 0)
161	145, 160	eqtrd 2770	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → (0(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑣) = 0)
162		oveq12 7420	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥↑_𝑐𝑦) = (𝑎↑_𝑐𝑏))
163		ovex 7444	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎↑_𝑐𝑏) ∈ V
164	162, 15, 163	ovmpoa 7565	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷) → (𝑎(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑏) = (𝑎↑_𝑐𝑏))
165	164	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → (𝑎(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑏) = (𝑎↑_𝑐𝑏))
166	161, 165	oveq12d 7429	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → ((0(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑣)(abs ∘ − )(𝑎(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑏)) = (0(abs ∘ − )(𝑎↑_𝑐𝑏)))
167	122, 136	cxpcld 26452	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → (𝑎↑_𝑐𝑏) ∈ ℂ)
168	123	cnmetdval 24507	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℂ ∧ (𝑎↑_𝑐𝑏) ∈ ℂ) → (0(abs ∘ − )(𝑎↑_𝑐𝑏)) = (abs‘(0 − (𝑎↑_𝑐𝑏))))
169	121, 167, 168	sylancr 585	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → (0(abs ∘ − )(𝑎↑_𝑐𝑏)) = (abs‘(0 − (𝑎↑_𝑐𝑏))))
170		df-neg 11451	. . . . . . . . . . . . . . . . 17 ⊢ -(𝑎↑_𝑐𝑏) = (0 − (𝑎↑_𝑐𝑏))
171	170	fveq2i 6893	. . . . . . . . . . . . . . . 16 ⊢ (abs‘-(𝑎↑_𝑐𝑏)) = (abs‘(0 − (𝑎↑_𝑐𝑏)))
172	167	absnegd 15400	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → (abs‘-(𝑎↑_𝑐𝑏)) = (abs‘(𝑎↑_𝑐𝑏)))
173	171, 172	eqtr3id 2784	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → (abs‘(0 − (𝑎↑_𝑐𝑏))) = (abs‘(𝑎↑_𝑐𝑏)))
174	166, 169, 173	3eqtrd 2774	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → ((0(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑣)(abs ∘ − )(𝑎(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑏)) = (abs‘(𝑎↑_𝑐𝑏)))
175	174	breq1d 5157	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → (((0(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑣)(abs ∘ − )(𝑎(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑏)) < 𝑒 ↔ (abs‘(𝑎↑_𝑐𝑏)) < 𝑒))
176	141, 175	imbi12d 343	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ 𝐷 ∧ (𝑎 ∈ (0[,)+∞) ∧ 𝑏 ∈ 𝐷)) → ((((0((abs ∘ − ) ↾ ((0[,)+∞) × (0[,)+∞)))𝑎) < 𝑑 ∧ (𝑣((abs ∘ − ) ↾ (𝐷 × 𝐷))𝑏) < 𝑑) → ((0(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑣)(abs ∘ − )(𝑎(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑏)) < 𝑒) ↔ (((abs‘𝑎) < 𝑑 ∧ (abs‘(𝑣 − 𝑏)) < 𝑑) → (abs‘(𝑎↑_𝑐𝑏)) < 𝑒)))
177	176	2ralbidva 3214	. . . . . . . . . . 11 ⊢ (𝑣 ∈ 𝐷 → (∀𝑎 ∈ (0[,)+∞)∀𝑏 ∈ 𝐷 (((0((abs ∘ − ) ↾ ((0[,)+∞) × (0[,)+∞)))𝑎) < 𝑑 ∧ (𝑣((abs ∘ − ) ↾ (𝐷 × 𝐷))𝑏) < 𝑑) → ((0(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑣)(abs ∘ − )(𝑎(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑏)) < 𝑒) ↔ ∀𝑎 ∈ (0[,)+∞)∀𝑏 ∈ 𝐷 (((abs‘𝑎) < 𝑑 ∧ (abs‘(𝑣 − 𝑏)) < 𝑑) → (abs‘(𝑎↑_𝑐𝑏)) < 𝑒)))
178	177	rexbidv 3176	. . . . . . . . . 10 ⊢ (𝑣 ∈ 𝐷 → (∃𝑑 ∈ ℝ⁺ ∀𝑎 ∈ (0[,)+∞)∀𝑏 ∈ 𝐷 (((0((abs ∘ − ) ↾ ((0[,)+∞) × (0[,)+∞)))𝑎) < 𝑑 ∧ (𝑣((abs ∘ − ) ↾ (𝐷 × 𝐷))𝑏) < 𝑑) → ((0(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑣)(abs ∘ − )(𝑎(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑏)) < 𝑒) ↔ ∃𝑑 ∈ ℝ⁺ ∀𝑎 ∈ (0[,)+∞)∀𝑏 ∈ 𝐷 (((abs‘𝑎) < 𝑑 ∧ (abs‘(𝑣 − 𝑏)) < 𝑑) → (abs‘(𝑎↑_𝑐𝑏)) < 𝑒)))
179	178	ralbidv 3175	. . . . . . . . 9 ⊢ (𝑣 ∈ 𝐷 → (∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑎 ∈ (0[,)+∞)∀𝑏 ∈ 𝐷 (((0((abs ∘ − ) ↾ ((0[,)+∞) × (0[,)+∞)))𝑎) < 𝑑 ∧ (𝑣((abs ∘ − ) ↾ (𝐷 × 𝐷))𝑏) < 𝑑) → ((0(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑣)(abs ∘ − )(𝑎(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑏)) < 𝑒) ↔ ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑎 ∈ (0[,)+∞)∀𝑏 ∈ 𝐷 (((abs‘𝑎) < 𝑑 ∧ (abs‘(𝑣 − 𝑏)) < 𝑑) → (abs‘(𝑎↑_𝑐𝑏)) < 𝑒)))
180	116, 179	mpbird 256	. . . . . . . 8 ⊢ (𝑣 ∈ 𝐷 → ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑎 ∈ (0[,)+∞)∀𝑏 ∈ 𝐷 (((0((abs ∘ − ) ↾ ((0[,)+∞) × (0[,)+∞)))𝑎) < 𝑑 ∧ (𝑣((abs ∘ − ) ↾ (𝐷 × 𝐷))𝑏) < 𝑑) → ((0(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑣)(abs ∘ − )(𝑎(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑏)) < 𝑒))
181		cnxmet 24509	. . . . . . . . . . 11 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
182	181	a1i 11	. . . . . . . . . 10 ⊢ (𝑣 ∈ 𝐷 → (abs ∘ − ) ∈ (∞Met‘ℂ))
183		xmetres2 24087	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (0[,)+∞) ⊆ ℂ) → ((abs ∘ − ) ↾ ((0[,)+∞) × (0[,)+∞))) ∈ (∞Met‘(0[,)+∞)))
184	182, 3, 183	sylancl 584	. . . . . . . . 9 ⊢ (𝑣 ∈ 𝐷 → ((abs ∘ − ) ↾ ((0[,)+∞) × (0[,)+∞))) ∈ (∞Met‘(0[,)+∞)))
185		xmetres2 24087	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐷 ⊆ ℂ) → ((abs ∘ − ) ↾ (𝐷 × 𝐷)) ∈ (∞Met‘𝐷))
186	182, 10, 185	sylancl 584	. . . . . . . . 9 ⊢ (𝑣 ∈ 𝐷 → ((abs ∘ − ) ↾ (𝐷 × 𝐷)) ∈ (∞Met‘𝐷))
187	117	a1i 11	. . . . . . . . 9 ⊢ (𝑣 ∈ 𝐷 → 0 ∈ (0[,)+∞))
188		id 22	. . . . . . . . 9 ⊢ (𝑣 ∈ 𝐷 → 𝑣 ∈ 𝐷)
189		eqid 2730	. . . . . . . . . . . . 13 ⊢ ((abs ∘ − ) ↾ ((0[,)+∞) × (0[,)+∞))) = ((abs ∘ − ) ↾ ((0[,)+∞) × (0[,)+∞)))
190	18	cnfldtopn 24518	. . . . . . . . . . . . 13 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
191		eqid 2730	. . . . . . . . . . . . 13 ⊢ (MetOpen‘((abs ∘ − ) ↾ ((0[,)+∞) × (0[,)+∞)))) = (MetOpen‘((abs ∘ − ) ↾ ((0[,)+∞) × (0[,)+∞))))
192	189, 190, 191	metrest 24253	. . . . . . . . . . . 12 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (0[,)+∞) ⊆ ℂ) → (𝐽 ↾_t (0[,)+∞)) = (MetOpen‘((abs ∘ − ) ↾ ((0[,)+∞) × (0[,)+∞)))))
193	181, 3, 192	mp2an 688	. . . . . . . . . . 11 ⊢ (𝐽 ↾_t (0[,)+∞)) = (MetOpen‘((abs ∘ − ) ↾ ((0[,)+∞) × (0[,)+∞))))
194	58, 193	eqtri 2758	. . . . . . . . . 10 ⊢ 𝐾 = (MetOpen‘((abs ∘ − ) ↾ ((0[,)+∞) × (0[,)+∞))))
195		eqid 2730	. . . . . . . . . . . . 13 ⊢ ((abs ∘ − ) ↾ (𝐷 × 𝐷)) = ((abs ∘ − ) ↾ (𝐷 × 𝐷))
196		eqid 2730	. . . . . . . . . . . . 13 ⊢ (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) = (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷)))
197	195, 190, 196	metrest 24253	. . . . . . . . . . . 12 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐷 ⊆ ℂ) → (𝐽 ↾_t 𝐷) = (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))))
198	181, 10, 197	mp2an 688	. . . . . . . . . . 11 ⊢ (𝐽 ↾_t 𝐷) = (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷)))
199	32, 198	eqtri 2758	. . . . . . . . . 10 ⊢ 𝐿 = (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷)))
200	194, 199, 190	txmetcnp 24276	. . . . . . . . 9 ⊢ (((((abs ∘ − ) ↾ ((0[,)+∞) × (0[,)+∞))) ∈ (∞Met‘(0[,)+∞)) ∧ ((abs ∘ − ) ↾ (𝐷 × 𝐷)) ∈ (∞Met‘𝐷) ∧ (abs ∘ − ) ∈ (∞Met‘ℂ)) ∧ (0 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷)) → ((𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐾 ×_t 𝐿) CnP 𝐽)‘⟨0, 𝑣⟩) ↔ ((𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)):((0[,)+∞) × 𝐷)⟶ℂ ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑎 ∈ (0[,)+∞)∀𝑏 ∈ 𝐷 (((0((abs ∘ − ) ↾ ((0[,)+∞) × (0[,)+∞)))𝑎) < 𝑑 ∧ (𝑣((abs ∘ − ) ↾ (𝐷 × 𝐷))𝑏) < 𝑑) → ((0(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑣)(abs ∘ − )(𝑎(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑏)) < 𝑒))))
201	184, 186, 182, 187, 188, 200	syl32anc 1376	. . . . . . . 8 ⊢ (𝑣 ∈ 𝐷 → ((𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐾 ×_t 𝐿) CnP 𝐽)‘⟨0, 𝑣⟩) ↔ ((𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)):((0[,)+∞) × 𝐷)⟶ℂ ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑎 ∈ (0[,)+∞)∀𝑏 ∈ 𝐷 (((0((abs ∘ − ) ↾ ((0[,)+∞) × (0[,)+∞)))𝑎) < 𝑑 ∧ (𝑣((abs ∘ − ) ↾ (𝐷 × 𝐷))𝑏) < 𝑑) → ((0(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑣)(abs ∘ − )(𝑎(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦))𝑏)) < 𝑒))))
202	112, 180, 201	mpbir2and 709	. . . . . . 7 ⊢ (𝑣 ∈ 𝐷 → (𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐾 ×_t 𝐿) CnP 𝐽)‘⟨0, 𝑣⟩))
203	202	ad2antlr 723	. . . . . 6 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 = 𝑢) → (𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐾 ×_t 𝐿) CnP 𝐽)‘⟨0, 𝑣⟩))
204		simpr 483	. . . . . . . 8 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 = 𝑢) → 0 = 𝑢)
205	204	opeq1d 4878	. . . . . . 7 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 = 𝑢) → ⟨0, 𝑣⟩ = ⟨𝑢, 𝑣⟩)
206	205	fveq2d 6894	. . . . . 6 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 = 𝑢) → (((𝐾 ×_t 𝐿) CnP 𝐽)‘⟨0, 𝑣⟩) = (((𝐾 ×_t 𝐿) CnP 𝐽)‘⟨𝑢, 𝑣⟩))
207	203, 206	eleqtrd 2833	. . . . 5 ⊢ (((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) ∧ 0 = 𝑢) → (𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐾 ×_t 𝐿) CnP 𝐽)‘⟨𝑢, 𝑣⟩))
208	39	simprbi 495	. . . . . . 7 ⊢ (𝑢 ∈ (0[,)+∞) → 0 ≤ 𝑢)
209	208	adantr 479	. . . . . 6 ⊢ ((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) → 0 ≤ 𝑢)
210		0re 11220	. . . . . . 7 ⊢ 0 ∈ ℝ
211		leloe 11304	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (0 ≤ 𝑢 ↔ (0 < 𝑢 ∨ 0 = 𝑢)))
212	210, 41, 211	sylancr 585	. . . . . 6 ⊢ ((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) → (0 ≤ 𝑢 ↔ (0 < 𝑢 ∨ 0 = 𝑢)))
213	209, 212	mpbid 231	. . . . 5 ⊢ ((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) → (0 < 𝑢 ∨ 0 = 𝑢))
214	111, 207, 213	mpjaodan 955	. . . 4 ⊢ ((𝑢 ∈ (0[,)+∞) ∧ 𝑣 ∈ 𝐷) → (𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐾 ×_t 𝐿) CnP 𝐽)‘⟨𝑢, 𝑣⟩))
215	214	rgen2 3195	. . 3 ⊢ ∀𝑢 ∈ (0[,)+∞)∀𝑣 ∈ 𝐷 (𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐾 ×_t 𝐿) CnP 𝐽)‘⟨𝑢, 𝑣⟩)
216		fveq2 6890	. . . . 5 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (((𝐾 ×_t 𝐿) CnP 𝐽)‘𝑧) = (((𝐾 ×_t 𝐿) CnP 𝐽)‘⟨𝑢, 𝑣⟩))
217	216	eleq2d 2817	. . . 4 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐾 ×_t 𝐿) CnP 𝐽)‘𝑧) ↔ (𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐾 ×_t 𝐿) CnP 𝐽)‘⟨𝑢, 𝑣⟩)))
218	217	ralxp 5840	. . 3 ⊢ (∀𝑧 ∈ ((0[,)+∞) × 𝐷)(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐾 ×_t 𝐿) CnP 𝐽)‘𝑧) ↔ ∀𝑢 ∈ (0[,)+∞)∀𝑣 ∈ 𝐷 (𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐾 ×_t 𝐿) CnP 𝐽)‘⟨𝑢, 𝑣⟩))
219	215, 218	mpbir 230	. 2 ⊢ ∀𝑧 ∈ ((0[,)+∞) × 𝐷)(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐾 ×_t 𝐿) CnP 𝐽)‘𝑧)
220		cncnp 23004	. . 3 ⊢ (((𝐾 ×_t 𝐿) ∈ (TopOn‘((0[,)+∞) × 𝐷)) ∧ 𝐽 ∈ (TopOn‘ℂ)) → ((𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ ((𝐾 ×_t 𝐿) Cn 𝐽) ↔ ((𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)):((0[,)+∞) × 𝐷)⟶ℂ ∧ ∀𝑧 ∈ ((0[,)+∞) × 𝐷)(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐾 ×_t 𝐿) CnP 𝐽)‘𝑧))))
221	93, 19, 220	mp2an 688	. 2 ⊢ ((𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ ((𝐾 ×_t 𝐿) Cn 𝐽) ↔ ((𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)):((0[,)+∞) × 𝐷)⟶ℂ ∧ ∀𝑧 ∈ ((0[,)+∞) × 𝐷)(𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ (((𝐾 ×_t 𝐿) CnP 𝐽)‘𝑧)))
222	17, 219, 221	mpbir2an 707	1 ⊢ (𝑥 ∈ (0[,)+∞), 𝑦 ∈ 𝐷 ↦ (𝑥↑_𝑐𝑦)) ∈ ((𝐾 ×_t 𝐿) Cn 𝐽)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 843 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 Vcvv 3472 ⊆ wss 3947 ifcif 4527 ⟨cop 4633 class class class wbr 5147 × cxp 5673 ^◡ccnv 5674 dom cdm 5675 ran crn 5676 ↾ cres 5677 “ cima 5678 ∘ ccom 5679 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ∈ cmpo 7413 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 +∞cpnf 11249 < clt 11252 ≤ cle 11253 − cmin 11448 -cneg 11449 / cdiv 11875 2c2 12271 ℝ⁺crp 12978 (,)cioo 13328 [,)cico 13330 ℜcre 15048 abscabs 15185 ↾_t crest 17370 TopOpenctopn 17371 topGenctg 17387 ∞Metcxmet 21129 MetOpencmopn 21134 ℂ_fldccnfld 21144 Topctop 22615 TopOnctopon 22632 intcnt 22741 Cn ccn 22948 CnP ccnp 22949 ×_t ctx 23284 ↑_𝑐ccxp 26300

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-ioc 13333 df-ico 13334 df-icc 13335 df-fz 13489 df-fzo 13632 df-fl 13761 df-mod 13839 df-seq 13971 df-exp 14032 df-fac 14238 df-bc 14267 df-hash 14295 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ef 16015 df-sin 16017 df-cos 16018 df-tan 16019 df-pi 16020 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-xrs 17452 df-qtop 17457 df-imas 17458 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-mulg 18987 df-cntz 19222 df-cmn 19691 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-fbas 21141 df-fg 21142 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cld 22743 df-ntr 22744 df-cls 22745 df-nei 22822 df-lp 22860 df-perf 22861 df-cn 22951 df-cnp 22952 df-haus 23039 df-cmp 23111 df-tx 23286 df-hmeo 23479 df-fil 23570 df-fm 23662 df-flim 23663 df-flf 23664 df-xms 24046 df-ms 24047 df-tms 24048 df-cncf 24618 df-limc 25615 df-dv 25616 df-log 26301 df-cxp 26302

This theorem is referenced by: resqrtcn 26493

W3C validator