limccnp2 - Metamath Proof Explorer

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Theorem limccnp2 25859

Description: The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 28-Dec-2016.)

**Hypotheses**
Ref	Expression
limccnp2.r	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝑋)
limccnp2.s	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ 𝑌)
limccnp2.x	⊢ (𝜑 → 𝑋 ⊆ ℂ)
limccnp2.y	⊢ (𝜑 → 𝑌 ⊆ ℂ)
limccnp2.k	⊢ 𝐾 = (TopOpen‘ℂ_fld)
limccnp2.j	⊢ 𝐽 = ((𝐾 ×_t 𝐾) ↾_t (𝑋 × 𝑌))
limccnp2.c	⊢ (𝜑 → 𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) lim_ℂ 𝐵))
limccnp2.d	⊢ (𝜑 → 𝐷 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑆) lim_ℂ 𝐵))
limccnp2.h	⊢ (𝜑 → 𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩))

**Assertion**
Ref	Expression
limccnp2	⊢ (𝜑 → (𝐶𝐻𝐷) ∈ ((𝑥 ∈ 𝐴 ↦ (𝑅𝐻𝑆)) lim_ℂ 𝐵))

Distinct variable groups: 𝑥,𝐵 𝑥,𝐶 𝑥,𝐷 𝑥,𝐻 𝜑,𝑥 𝑥,𝑋 𝑥,𝐴 𝑥,𝑌 Allowed substitution hints: 𝑅(𝑥) 𝑆(𝑥) 𝐽(𝑥) 𝐾(𝑥)

Proof of Theorem limccnp2 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		limccnp2.h	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩))
2		eqid 2736	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	cnprcl 23210	. . . . . . . . . . 11 ⊢ (𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩) → ⟨𝐶, 𝐷⟩ ∈ ∪ 𝐽)
4	1, 3	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ ∪ 𝐽)
5		limccnp2.j	. . . . . . . . . . . 12 ⊢ 𝐽 = ((𝐾 ×_t 𝐾) ↾_t (𝑋 × 𝑌))
6		limccnp2.k	. . . . . . . . . . . . . . 15 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
7	6	cnfldtopon 24747	. . . . . . . . . . . . . 14 ⊢ 𝐾 ∈ (TopOn‘ℂ)
8		txtopon 23556	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝐾 ∈ (TopOn‘ℂ)) → (𝐾 ×_t 𝐾) ∈ (TopOn‘(ℂ × ℂ)))
9	7, 7, 8	mp2an 693	. . . . . . . . . . . . 13 ⊢ (𝐾 ×_t 𝐾) ∈ (TopOn‘(ℂ × ℂ))
10		limccnp2.x	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
11		limccnp2.y	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ⊆ ℂ)
12		xpss12 5646	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑌 ⊆ ℂ) → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
13	10, 11, 12	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
14		resttopon 23126	. . . . . . . . . . . . 13 ⊢ (((𝐾 ×_t 𝐾) ∈ (TopOn‘(ℂ × ℂ)) ∧ (𝑋 × 𝑌) ⊆ (ℂ × ℂ)) → ((𝐾 ×_t 𝐾) ↾_t (𝑋 × 𝑌)) ∈ (TopOn‘(𝑋 × 𝑌)))
15	9, 13, 14	sylancr 588	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾 ×_t 𝐾) ↾_t (𝑋 × 𝑌)) ∈ (TopOn‘(𝑋 × 𝑌)))
16	5, 15	eqeltrid 2840	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(𝑋 × 𝑌)))
17		toponuni 22879	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑋 × 𝑌) = ∪ 𝐽)
18	16, 17	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 × 𝑌) = ∪ 𝐽)
19	4, 18	eleqtrrd 2839	. . . . . . . . 9 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
20		opelxp 5667	. . . . . . . . 9 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌) ↔ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌))
21	19, 20	sylib 218	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌))
22	21	simpld 494	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
23	22	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑥 = 𝐵) → 𝐶 ∈ 𝑋)
24		simpll 767	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝜑)
25		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → 𝑥 ∈ (𝐴 ∪ {𝐵}))
26		elun 4093	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐵}))
27	25, 26	sylib 218	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐵}))
28	27	ord 865	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ {𝐵}))
29		elsni 4584	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
30	28, 29	syl6 35	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥 ∈ 𝐴 → 𝑥 = 𝐵))
31	30	con1d 145	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥 = 𝐵 → 𝑥 ∈ 𝐴))
32	31	imp 406	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ 𝐴)
33		limccnp2.r	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝑋)
34	24, 32, 33	syl2anc 585	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑅 ∈ 𝑋)
35	23, 34	ifclda 4502	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, 𝐶, 𝑅) ∈ 𝑋)
36	21	simprd 495	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
37	36	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑥 = 𝐵) → 𝐷 ∈ 𝑌)
38		limccnp2.s	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ 𝑌)
39	24, 32, 38	syl2anc 585	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑆 ∈ 𝑌)
40	37, 39	ifclda 4502	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, 𝐷, 𝑆) ∈ 𝑌)
41	35, 40	opelxpd 5670	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ ∈ (𝑋 × 𝑌))
42		eqidd 2737	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩))
43	7	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘ℂ))
44		cnpf2 23215	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ 𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩)) → 𝐻:(𝑋 × 𝑌)⟶ℂ)
45	16, 43, 1, 44	syl3anc 1374	. . . . 5 ⊢ (𝜑 → 𝐻:(𝑋 × 𝑌)⟶ℂ)
46	45	feqmptd 6908	. . . 4 ⊢ (𝜑 → 𝐻 = (𝑦 ∈ (𝑋 × 𝑌) ↦ (𝐻‘𝑦)))
47		fveq2 6840	. . . . 5 ⊢ (𝑦 = ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ → (𝐻‘𝑦) = (𝐻‘⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩))
48		df-ov 7370	. . . . . 6 ⊢ (if(𝑥 = 𝐵, 𝐶, 𝑅)𝐻if(𝑥 = 𝐵, 𝐷, 𝑆)) = (𝐻‘⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)
49		ovif12 7467	. . . . . 6 ⊢ (if(𝑥 = 𝐵, 𝐶, 𝑅)𝐻if(𝑥 = 𝐵, 𝐷, 𝑆)) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))
50	48, 49	eqtr3i 2761	. . . . 5 ⊢ (𝐻‘⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))
51	47, 50	eqtrdi 2787	. . . 4 ⊢ (𝑦 = ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ → (𝐻‘𝑦) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆)))
52	41, 42, 46, 51	fmptco 7082	. . 3 ⊢ (𝜑 → (𝐻 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))))
53		eqid 2736	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑥 ∈ 𝐴 ↦ 𝑅)
54	53, 33	dmmptd 6643	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝑅) = 𝐴)
55		limccnp2.c	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) lim_ℂ 𝐵))
56		limcrcl 25841	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) lim_ℂ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝑅):dom (𝑥 ∈ 𝐴 ↦ 𝑅)⟶ℂ ∧ dom (𝑥 ∈ 𝐴 ↦ 𝑅) ⊆ ℂ ∧ 𝐵 ∈ ℂ))
57	55, 56	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝑅):dom (𝑥 ∈ 𝐴 ↦ 𝑅)⟶ℂ ∧ dom (𝑥 ∈ 𝐴 ↦ 𝑅) ⊆ ℂ ∧ 𝐵 ∈ ℂ))
58	57	simp2d 1144	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝑅) ⊆ ℂ)
59	54, 58	eqsstrrd 3957	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
60	57	simp3d 1145	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ)
61	60	snssd 4730	. . . . . . . . 9 ⊢ (𝜑 → {𝐵} ⊆ ℂ)
62	59, 61	unssd 4132	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ)
63		resttopon 23126	. . . . . . . 8 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ (𝐴 ∪ {𝐵}) ⊆ ℂ) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
64	7, 62, 63	sylancr 588	. . . . . . 7 ⊢ (𝜑 → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
65		ssun2 4119	. . . . . . . 8 ⊢ {𝐵} ⊆ (𝐴 ∪ {𝐵})
66		snssg 4727	. . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
67	60, 66	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
68	65, 67	mpbiri 258	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ∪ {𝐵}))
69	10	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑋 ⊆ ℂ)
70	69, 33	sseldd 3922	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ ℂ)
71		eqid 2736	. . . . . . . . 9 ⊢ (𝐾 ↾_t (𝐴 ∪ {𝐵})) = (𝐾 ↾_t (𝐴 ∪ {𝐵}))
72	59, 60, 70, 71, 6	limcmpt 25850	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) lim_ℂ 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, 𝑅)) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
73	55, 72	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, 𝑅)) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
74		limccnp2.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑆) lim_ℂ 𝐵))
75	11	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ⊆ ℂ)
76	75, 38	sseldd 3922	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ ℂ)
77	59, 60, 76, 71, 6	limcmpt 25850	. . . . . . . 8 ⊢ (𝜑 → (𝐷 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑆) lim_ℂ 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐷, 𝑆)) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
78	74, 77	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐷, 𝑆)) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
79	64, 43, 43, 68, 73, 78	txcnp 23585	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP (𝐾 ×_t 𝐾))‘𝐵))
80	9	topontopi 22880	. . . . . . . 8 ⊢ (𝐾 ×_t 𝐾) ∈ Top
81	80	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐾 ×_t 𝐾) ∈ Top)
82	41	fmpttd 7067	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩):(𝐴 ∪ {𝐵})⟶(𝑋 × 𝑌))
83		toponuni 22879	. . . . . . . . . 10 ⊢ ((𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})) → (𝐴 ∪ {𝐵}) = ∪ (𝐾 ↾_t (𝐴 ∪ {𝐵})))
84	64, 83	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) = ∪ (𝐾 ↾_t (𝐴 ∪ {𝐵})))
85	84	feq2d 6652	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩):(𝐴 ∪ {𝐵})⟶(𝑋 × 𝑌) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩):∪ (𝐾 ↾_t (𝐴 ∪ {𝐵}))⟶(𝑋 × 𝑌)))
86	82, 85	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩):∪ (𝐾 ↾_t (𝐴 ∪ {𝐵}))⟶(𝑋 × 𝑌))
87		eqid 2736	. . . . . . . 8 ⊢ ∪ (𝐾 ↾_t (𝐴 ∪ {𝐵})) = ∪ (𝐾 ↾_t (𝐴 ∪ {𝐵}))
88	9	toponunii 22881	. . . . . . . 8 ⊢ (ℂ × ℂ) = ∪ (𝐾 ×_t 𝐾)
89	87, 88	cnprest2 23255	. . . . . . 7 ⊢ (((𝐾 ×_t 𝐾) ∈ Top ∧ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩):∪ (𝐾 ↾_t (𝐴 ∪ {𝐵}))⟶(𝑋 × 𝑌) ∧ (𝑋 × 𝑌) ⊆ (ℂ × ℂ)) → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP (𝐾 ×_t 𝐾))‘𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×_t 𝐾) ↾_t (𝑋 × 𝑌)))‘𝐵)))
90	81, 86, 13, 89	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP (𝐾 ×_t 𝐾))‘𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×_t 𝐾) ↾_t (𝑋 × 𝑌)))‘𝐵)))
91	79, 90	mpbid 232	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×_t 𝐾) ↾_t (𝑋 × 𝑌)))‘𝐵))
92	5	oveq2i 7378	. . . . . 6 ⊢ ((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽) = ((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×_t 𝐾) ↾_t (𝑋 × 𝑌)))
93	92	fveq1i 6841	. . . . 5 ⊢ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵) = (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×_t 𝐾) ↾_t (𝑋 × 𝑌)))‘𝐵)
94	91, 93	eleqtrrdi 2847	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵))
95		iftrue 4472	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐶, 𝑅) = 𝐶)
96		iftrue 4472	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐷, 𝑆) = 𝐷)
97	95, 96	opeq12d 4824	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ = ⟨𝐶, 𝐷⟩)
98		eqid 2736	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)
99		opex 5416	. . . . . . . 8 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
100	97, 98, 99	fvmpt 6947	. . . . . . 7 ⊢ (𝐵 ∈ (𝐴 ∪ {𝐵}) → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)‘𝐵) = ⟨𝐶, 𝐷⟩)
101	68, 100	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)‘𝐵) = ⟨𝐶, 𝐷⟩)
102	101	fveq2d 6844	. . . . 5 ⊢ (𝜑 → ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)‘𝐵)) = ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩))
103	1, 102	eleqtrrd 2839	. . . 4 ⊢ (𝜑 → 𝐻 ∈ ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)‘𝐵)))
104		cnpco 23232	. . . 4 ⊢ (((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵) ∧ 𝐻 ∈ ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)‘𝐵))) → (𝐻 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
105	94, 103, 104	syl2anc 585	. . 3 ⊢ (𝜑 → (𝐻 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
106	52, 105	eqeltrrd 2837	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
107	45	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻:(𝑋 × 𝑌)⟶ℂ)
108	107, 33, 38	fovcdmd 7539	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅𝐻𝑆) ∈ ℂ)
109	59, 60, 108, 71, 6	limcmpt 25850	. 2 ⊢ (𝜑 → ((𝐶𝐻𝐷) ∈ ((𝑥 ∈ 𝐴 ↦ (𝑅𝐻𝑆)) lim_ℂ 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
110	106, 109	mpbird 257	1 ⊢ (𝜑 → (𝐶𝐻𝐷) ∈ ((𝑥 ∈ 𝐴 ↦ (𝑅𝐻𝑆)) lim_ℂ 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∪ cun 3887 ⊆ wss 3889 ifcif 4466 {csn 4567 ⟨cop 4573 ∪ cuni 4850 ↦ cmpt 5166 × cxp 5629 dom cdm 5631 ∘ ccom 5635 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 ↾_t crest 17383 TopOpenctopn 17384 ℂ_fldccnfld 21352 Topctop 22858 TopOnctopon 22875 CnP ccnp 23190 ×_t ctx 23525 lim_ℂ climc 25829

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-map 8775 df-pm 8776 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fi 9324 df-sup 9355 df-inf 9356 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-fz 13462 df-seq 13964 df-exp 14024 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-plusg 17233 df-mulr 17234 df-starv 17235 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-rest 17385 df-topn 17386 df-topgen 17406 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cnp 23193 df-tx 23527 df-xms 24285 df-ms 24286 df-limc 25833

This theorem is referenced by: dvcnp2 25887 dvaddbr 25905 dvmulbr 25906 dvcobr 25913 lhop1lem 25980 taylthlem2 26339

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