Theorem limccnp2 23462

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‘ℂfld)
limccnp2.j	⊢ 𝐽 = ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌))
limccnp2.c	⊢ (𝜑 → 𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) limℂ 𝐵))
limccnp2.d	⊢ (𝜑 → 𝐷 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑆) limℂ 𝐵))
limccnp2.h	⊢ (𝜑 → 𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩))

Assertion

Ref	Expression
limccnp2	⊢ (𝜑 → (𝐶𝐻𝐷) ∈ ((𝑥 ∈ 𝐴 ↦ (𝑅𝐻𝑆)) limℂ 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 2610	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	cnprcl 20859	. . . . . . . . . . 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‘ℂfld)
7	6	cnfldtopon 22396	. . . . . . . . . . . . . 14 ⊢ 𝐾 ∈ (TopOn‘ℂ)
8		txtopon 21204	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝐾 ∈ (TopOn‘ℂ)) → (𝐾 ×t 𝐾) ∈ (TopOn‘(ℂ × ℂ)))
9	7, 7, 8	mp2an 704	. . . . . . . . . . . . 13 ⊢ (𝐾 ×t 𝐾) ∈ (TopOn‘(ℂ × ℂ))
10		limccnp2.x	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
11		limccnp2.y	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ⊆ ℂ)
12		xpss12 5148	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑌 ⊆ ℂ) → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
13	10, 11, 12	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
14		resttopon 20775	. . . . . . . . . . . . 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 694	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)) ∈ (TopOn‘(𝑋 × 𝑌)))
16	5, 15	syl5eqel 2692	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(𝑋 × 𝑌)))
17		toponuni 20542	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑋 × 𝑌) = ∪ 𝐽)
18	16, 17	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 × 𝑌) = ∪ 𝐽)
19	4, 18	eleqtrrd 2691	. . . . . . . . 9 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
20		opelxp 5070	. . . . . . . . 9 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌) ↔ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌))
21	19, 20	sylib 207	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌))
22	21	simpld 474	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
23	22	ad2antrr 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑥 = 𝐵) → 𝐶 ∈ 𝑋)
24		simpll 786	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝜑)
25		simpr 476	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → 𝑥 ∈ (𝐴 ∪ {𝐵}))
26		elun 3715	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐵}))
27	25, 26	sylib 207	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐵}))
28	27	ord 391	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ {𝐵}))
29		elsni 4142	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
30	28, 29	syl6 34	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥 ∈ 𝐴 → 𝑥 = 𝐵))
31	30	con1d 138	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥 = 𝐵 → 𝑥 ∈ 𝐴))
32	31	imp 444	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ 𝐴)
33		limccnp2.r	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝑋)
34	24, 32, 33	syl2anc 691	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑅 ∈ 𝑋)
35	23, 34	ifclda 4070	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, 𝐶, 𝑅) ∈ 𝑋)
36	21	simprd 478	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
37	36	ad2antrr 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑥 = 𝐵) → 𝐷 ∈ 𝑌)
38		limccnp2.s	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ 𝑌)
39	24, 32, 38	syl2anc 691	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑆 ∈ 𝑌)
40	37, 39	ifclda 4070	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, 𝐷, 𝑆) ∈ 𝑌)
41		opelxpi 5072	. . . . 5 ⊢ ((if(𝑥 = 𝐵, 𝐶, 𝑅) ∈ 𝑋 ∧ if(𝑥 = 𝐵, 𝐷, 𝑆) ∈ 𝑌) → ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ ∈ (𝑋 × 𝑌))
42	35, 40, 41	syl2anc 691	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ ∈ (𝑋 × 𝑌))
43		eqidd 2611	. . . 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)⟩))
44	7	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘ℂ))
45		cnpf2 20864	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ 𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩)) → 𝐻:(𝑋 × 𝑌)⟶ℂ)
46	16, 44, 1, 45	syl3anc 1318	. . . . 5 ⊢ (𝜑 → 𝐻:(𝑋 × 𝑌)⟶ℂ)
47	46	feqmptd 6159	. . . 4 ⊢ (𝜑 → 𝐻 = (𝑦 ∈ (𝑋 × 𝑌) ↦ (𝐻‘𝑦)))
48		fveq2 6103	. . . . 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)⟩))
49		df-ov 6552	. . . . . 6 ⊢ (if(𝑥 = 𝐵, 𝐶, 𝑅)𝐻if(𝑥 = 𝐵, 𝐷, 𝑆)) = (𝐻‘⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)
50		ovif12 6637	. . . . . 6 ⊢ (if(𝑥 = 𝐵, 𝐶, 𝑅)𝐻if(𝑥 = 𝐵, 𝐷, 𝑆)) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))
51	49, 50	eqtr3i 2634	. . . . 5 ⊢ (𝐻‘⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))
52	48, 51	syl6eq 2660	. . . 4 ⊢ (𝑦 = ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ → (𝐻‘𝑦) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆)))
53	42, 43, 47, 52	fmptco 6303	. . 3 ⊢ (𝜑 → (𝐻 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))))
54		eqid 2610	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑥 ∈ 𝐴 ↦ 𝑅)
55	54, 33	dmmptd 5937	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝑅) = 𝐴)
56		limccnp2.c	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) limℂ 𝐵))
57		limcrcl 23444	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) limℂ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝑅):dom (𝑥 ∈ 𝐴 ↦ 𝑅)⟶ℂ ∧ dom (𝑥 ∈ 𝐴 ↦ 𝑅) ⊆ ℂ ∧ 𝐵 ∈ ℂ))
58	56, 57	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝑅):dom (𝑥 ∈ 𝐴 ↦ 𝑅)⟶ℂ ∧ dom (𝑥 ∈ 𝐴 ↦ 𝑅) ⊆ ℂ ∧ 𝐵 ∈ ℂ))
59	58	simp2d 1067	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝑅) ⊆ ℂ)
60	55, 59	eqsstr3d 3603	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
61	58	simp3d 1068	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ)
62	61	snssd 4281	. . . . . . . . 9 ⊢ (𝜑 → {𝐵} ⊆ ℂ)
63	60, 62	unssd 3751	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ)
64		resttopon 20775	. . . . . . . 8 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ (𝐴 ∪ {𝐵}) ⊆ ℂ) → (𝐾 ↾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
65	7, 63, 64	sylancr 694	. . . . . . 7 ⊢ (𝜑 → (𝐾 ↾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
66		ssun2 3739	. . . . . . . 8 ⊢ {𝐵} ⊆ (𝐴 ∪ {𝐵})
67		snssg 4268	. . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
68	61, 67	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
69	66, 68	mpbiri 247	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ∪ {𝐵}))
70	10	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑋 ⊆ ℂ)
71	70, 33	sseldd 3569	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ ℂ)
72		eqid 2610	. . . . . . . . 9 ⊢ (𝐾 ↾t (𝐴 ∪ {𝐵})) = (𝐾 ↾t (𝐴 ∪ {𝐵}))
73	60, 61, 71, 72, 6	limcmpt 23453	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) limℂ 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, 𝑅)) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
74	56, 73	mpbid 221	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, 𝑅)) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
75		limccnp2.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑆) limℂ 𝐵))
76	11	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ⊆ ℂ)
77	76, 38	sseldd 3569	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ ℂ)
78	60, 61, 77, 72, 6	limcmpt 23453	. . . . . . . 8 ⊢ (𝜑 → (𝐷 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑆) limℂ 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐷, 𝑆)) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
79	75, 78	mpbid 221	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐷, 𝑆)) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
80	65, 44, 44, 69, 74, 79	txcnp 21233	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP (𝐾 ×t 𝐾))‘𝐵))
81	9	topontopi 20546	. . . . . . . 8 ⊢ (𝐾 ×t 𝐾) ∈ Top
82	81	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐾 ×t 𝐾) ∈ Top)
83		eqid 2610	. . . . . . . . 9 ⊢ (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)⟩)
84	42, 83	fmptd 6292	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩):(𝐴 ∪ {𝐵})⟶(𝑋 × 𝑌))
85		toponuni 20542	. . . . . . . . . 10 ⊢ ((𝐾 ↾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})) → (𝐴 ∪ {𝐵}) = ∪ (𝐾 ↾t (𝐴 ∪ {𝐵})))
86	65, 85	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) = ∪ (𝐾 ↾t (𝐴 ∪ {𝐵})))
87	86	feq2d 5944	. . . . . . . 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)))
88	84, 87	mpbid 221	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩):∪ (𝐾 ↾t (𝐴 ∪ {𝐵}))⟶(𝑋 × 𝑌))
89		eqid 2610	. . . . . . . 8 ⊢ ∪ (𝐾 ↾t (𝐴 ∪ {𝐵})) = ∪ (𝐾 ↾t (𝐴 ∪ {𝐵}))
90	9	toponunii 20547	. . . . . . . 8 ⊢ (ℂ × ℂ) = ∪ (𝐾 ×t 𝐾)
91	89, 90	cnprest2 20904	. . . . . . 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)))
92	82, 88, 13, 91	syl3anc 1318	. . . . . 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)))
93	80, 92	mpbid 221	. . . . 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))
94	5	oveq2i 6560	. . . . . 6 ⊢ ((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐽) = ((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)))
95	94	fveq1i 6104	. . . . 5 ⊢ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵) = (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)))‘𝐵)
96	93, 95	syl6eleqr 2699	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵))
97		iftrue 4042	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐶, 𝑅) = 𝐶)
98		iftrue 4042	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐷, 𝑆) = 𝐷)
99	97, 98	opeq12d 4348	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ = ⟨𝐶, 𝐷⟩)
100		opex 4859	. . . . . . . 8 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
101	99, 83, 100	fvmpt 6191	. . . . . . 7 ⊢ (𝐵 ∈ (𝐴 ∪ {𝐵}) → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)‘𝐵) = ⟨𝐶, 𝐷⟩)
102	69, 101	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)‘𝐵) = ⟨𝐶, 𝐷⟩)
103	102	fveq2d 6107	. . . . 5 ⊢ (𝜑 → ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)‘𝐵)) = ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩))
104	1, 103	eleqtrrd 2691	. . . 4 ⊢ (𝜑 → 𝐻 ∈ ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)‘𝐵)))
105		cnpco 20881	. . . 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))
106	96, 104, 105	syl2anc 691	. . 3 ⊢ (𝜑 → (𝐻 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
107	53, 106	eqeltrrd 2689	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
108	46	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻:(𝑋 × 𝑌)⟶ℂ)
109	108, 33, 38	fovrnd 6704	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅𝐻𝑆) ∈ ℂ)
110	60, 61, 109, 72, 6	limcmpt 23453	. 2 ⊢ (𝜑 → ((𝐶𝐻𝐷) ∈ ((𝑥 ∈ 𝐴 ↦ (𝑅𝐻𝑆)) limℂ 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))) ∈ (((𝐾 ↾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
111	107, 110	mpbird 246	1 ⊢ (𝜑 → (𝐶𝐻𝐷) ∈ ((𝑥 ∈ 𝐴 ↦ (𝑅𝐻𝑆)) limℂ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∪ cun 3538   ⊆ wss 3540  ifcif 4036  {csn 4125  ⟨cop 4131  ∪ cuni 4372   ↦ cmpt 4643   × cxp 5036  dom cdm 5038   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813   ↾_t crest 15904  TopOpenctopn 15905  ℂ_fldccnfld 19567  Topctop 20517  TopOnctopon 20518   CnP ccnp 20839   ×_t ctx 21173   lim_ℂ climc 23432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-fz 12198  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-mulr 15782  df-starv 15783  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-rest 15906  df-topn 15907  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cnp 20842  df-tx 21175  df-xms 21935  df-ms 21936  df-limc 23436

This theorem is referenced by:  dvcnp2  23489  dvaddbr  23507  dvmulbr  23508  dvcobr  23515  lhop1lem  23580  taylthlem2  23932