limccnp2 - Metamath Proof Explorer

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Theorem limccnp2 26012

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 2765	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	cnprcl 23363	. . . . . . . . . . 11 ⊢ (𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩) → ⟨𝐶, 𝐷⟩ ∈ ∪ 𝐽)
4	1, 3	syl 18	. . . . . . . . . 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 24900	. . . . . . . . . . . . . 14 ⊢ 𝐾 ∈ (TopOn‘ℂ)
8		txtopon 23709	. . . . . . . . . . . . . 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 5667	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑌 ⊆ ℂ) → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
13	10, 11, 12	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
14		resttopon 23279	. . . . . . . . . . . . 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 598	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾 ×_t 𝐾) ↾_t (𝑋 × 𝑌)) ∈ (TopOn‘(𝑋 × 𝑌)))
16	5, 15	eqeltrid 2869	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(𝑋 × 𝑌)))
17		toponuni 23032	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑋 × 𝑌) = ∪ 𝐽)
18	16, 17	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 × 𝑌) = ∪ 𝐽)
19	4, 18	eleqtrrd 2868	. . . . . . . . 9 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
20		opelxp 5688	. . . . . . . . 9 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌) ↔ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌))
21	19, 20	sylib 221	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌))
22	21	simpld 499	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
23	22	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑥 = 𝐵) → 𝐶 ∈ 𝑋)
24		simpll 778	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝜑)
25		elun 4109	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐵}))
26	25	bilani 509	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐵}))
27	26	ord 877	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥 ∈ 𝐴 → 𝑥 ∈ {𝐵}))
28		elsni 4602	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
29	27, 28	syl6 36	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥 ∈ 𝐴 → 𝑥 = 𝐵))
30	29	con1d 146	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥 = 𝐵 → 𝑥 ∈ 𝐴))
31	30	imp 411	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ 𝐴)
32		limccnp2.r	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ 𝑋)
33	24, 31, 32	syl2anc 595	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑅 ∈ 𝑋)
34	23, 33	ifclda 4519	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, 𝐶, 𝑅) ∈ 𝑋)
35	21	simprd 500	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
36	35	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑥 = 𝐵) → 𝐷 ∈ 𝑌)
37		limccnp2.s	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ 𝑌)
38	24, 31, 37	syl2anc 595	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑆 ∈ 𝑌)
39	36, 38	ifclda 4519	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, 𝐷, 𝑆) ∈ 𝑌)
40	34, 39	opelxpd 5691	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ ∈ (𝑋 × 𝑌))
41		eqidd 2766	. . . 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)⟩))
42	7	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘ℂ))
43		cnpf2 23368	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ 𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩)) → 𝐻:(𝑋 × 𝑌)⟶ℂ)
44	16, 42, 1, 43	syl3anc 1394	. . . . 5 ⊢ (𝜑 → 𝐻:(𝑋 × 𝑌)⟶ℂ)
45	44	feqmptd 6939	. . . 4 ⊢ (𝜑 → 𝐻 = (𝑦 ∈ (𝑋 × 𝑌) ↦ (𝐻‘𝑦)))
46		fveq2 6871	. . . . 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)⟩))
47		df-ov 7403	. . . . . 6 ⊢ (if(𝑥 = 𝐵, 𝐶, 𝑅)𝐻if(𝑥 = 𝐵, 𝐷, 𝑆)) = (𝐻‘⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)
48		ovif12 7500	. . . . . 6 ⊢ (if(𝑥 = 𝐵, 𝐶, 𝑅)𝐻if(𝑥 = 𝐵, 𝐷, 𝑆)) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))
49	47, 48	eqtr3i 2790	. . . . 5 ⊢ (𝐻‘⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))
50	46, 49	eqtrdi 2816	. . . 4 ⊢ (𝑦 = ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ → (𝐻‘𝑦) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆)))
51	40, 41, 45, 50	fmptco 7115	. . 3 ⊢ (𝜑 → (𝐻 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)) = (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))))
52		eqid 2765	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑥 ∈ 𝐴 ↦ 𝑅)
53	52, 32	dmmptd 6670	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝑅) = 𝐴)
54		limccnp2.c	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) lim_ℂ 𝐵))
55		limcrcl 25994	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) lim_ℂ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝑅):dom (𝑥 ∈ 𝐴 ↦ 𝑅)⟶ℂ ∧ dom (𝑥 ∈ 𝐴 ↦ 𝑅) ⊆ ℂ ∧ 𝐵 ∈ ℂ))
56	54, 55	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝑅):dom (𝑥 ∈ 𝐴 ↦ 𝑅)⟶ℂ ∧ dom (𝑥 ∈ 𝐴 ↦ 𝑅) ⊆ ℂ ∧ 𝐵 ∈ ℂ))
57	56	simp2d 1159	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝑅) ⊆ ℂ)
58	53, 57	eqsstrrd 3974	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
59	56	simp3d 1160	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ)
60	59	snssd 4748	. . . . . . . . 9 ⊢ (𝜑 → {𝐵} ⊆ ℂ)
61	58, 60	unssd 4147	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ)
62		resttopon 23279	. . . . . . . 8 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ (𝐴 ∪ {𝐵}) ⊆ ℂ) → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
63	7, 61, 62	sylancr 598	. . . . . . 7 ⊢ (𝜑 → (𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
64		ssun2 4134	. . . . . . . 8 ⊢ {𝐵} ⊆ (𝐴 ∪ {𝐵})
65		snssg 4745	. . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
66	59, 65	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
67	64, 66	mpbiri 261	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ∪ {𝐵}))
68	10	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑋 ⊆ ℂ)
69	68, 32	sseldd 3940	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ ℂ)
70		eqid 2765	. . . . . . . . 9 ⊢ (𝐾 ↾_t (𝐴 ∪ {𝐵})) = (𝐾 ↾_t (𝐴 ∪ {𝐵}))
71	58, 59, 69, 70, 6	limcmpt 26003	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) lim_ℂ 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, 𝑅)) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
72	54, 71	mpbid 235	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, 𝑅)) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
73		limccnp2.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑆) lim_ℂ 𝐵))
74	11	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ⊆ ℂ)
75	74, 37	sseldd 3940	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ ℂ)
76	58, 59, 75, 70, 6	limcmpt 26003	. . . . . . . 8 ⊢ (𝜑 → (𝐷 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑆) lim_ℂ 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐷, 𝑆)) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
77	73, 76	mpbid 235	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐷, 𝑆)) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
78	63, 42, 42, 67, 72, 77	txcnp 23738	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP (𝐾 ×_t 𝐾))‘𝐵))
79	9	topontopi 23033	. . . . . . . 8 ⊢ (𝐾 ×_t 𝐾) ∈ Top
80	79	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐾 ×_t 𝐾) ∈ Top)
81	40	fmpttd 7100	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩):(𝐴 ∪ {𝐵})⟶(𝑋 × 𝑌))
82		toponuni 23032	. . . . . . . . . 10 ⊢ ((𝐾 ↾_t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})) → (𝐴 ∪ {𝐵}) = ∪ (𝐾 ↾_t (𝐴 ∪ {𝐵})))
83	63, 82	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) = ∪ (𝐾 ↾_t (𝐴 ∪ {𝐵})))
84	83	feq2d 6679	. . . . . . . 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)))
85	81, 84	mpbid 235	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩):∪ (𝐾 ↾_t (𝐴 ∪ {𝐵}))⟶(𝑋 × 𝑌))
86		eqid 2765	. . . . . . . 8 ⊢ ∪ (𝐾 ↾_t (𝐴 ∪ {𝐵})) = ∪ (𝐾 ↾_t (𝐴 ∪ {𝐵}))
87	9	toponunii 23034	. . . . . . . 8 ⊢ (ℂ × ℂ) = ∪ (𝐾 ×_t 𝐾)
88	86, 87	cnprest2 23408	. . . . . . 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)))
89	80, 85, 13, 88	syl3anc 1394	. . . . . 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)))
90	78, 89	mpbid 235	. . . . 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))
91	5	oveq2i 7411	. . . . . 6 ⊢ ((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽) = ((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×_t 𝐾) ↾_t (𝑋 × 𝑌)))
92	91	fveq1i 6872	. . . . 5 ⊢ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵) = (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×_t 𝐾) ↾_t (𝑋 × 𝑌)))‘𝐵)
93	90, 92	eleqtrrdi 2876	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵))
94		iftrue 4489	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐶, 𝑅) = 𝐶)
95		iftrue 4489	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐷, 𝑆) = 𝐷)
96	94, 95	opeq12d 4842	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ = ⟨𝐶, 𝐷⟩)
97		eqid 2765	. . . . . . . 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)⟩)
98		opex 5436	. . . . . . . 8 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
99	96, 97, 98	fvmpt 6979	. . . . . . 7 ⊢ (𝐵 ∈ (𝐴 ∪ {𝐵}) → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)‘𝐵) = ⟨𝐶, 𝐷⟩)
100	67, 99	syl 18	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)‘𝐵) = ⟨𝐶, 𝐷⟩)
101	100	fveq2d 6875	. . . . 5 ⊢ (𝜑 → ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)‘𝐵)) = ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩))
102	1, 101	eleqtrrd 2868	. . . 4 ⊢ (𝜑 → 𝐻 ∈ ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)‘𝐵)))
103		cnpco 23385	. . . 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))
104	93, 102, 103	syl2anc 595	. . 3 ⊢ (𝜑 → (𝐻 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
105	51, 104	eqeltrrd 2866	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
106	44	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐻:(𝑋 × 𝑌)⟶ℂ)
107	106, 32, 37	fovcdmd 7572	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅𝐻𝑆) ∈ ℂ)
108	58, 59, 107, 70, 6	limcmpt 26003	. 2 ⊢ (𝜑 → ((𝐶𝐻𝐷) ∈ ((𝑥 ∈ 𝐴 ↦ (𝑅𝐻𝑆)) lim_ℂ 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
109	105, 108	mpbird 260	1 ⊢ (𝜑 → (𝐶𝐻𝐷) ∈ ((𝑥 ∈ 𝐴 ↦ (𝑅𝐻𝑆)) lim_ℂ 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 ∪ cun 3905 ⊆ wss 3907 ifcif 4483 {csn 4585 ⟨cop 4591 ∪ cuni 4868 ↦ cmpt 5186 × cxp 5650 dom cdm 5652 ∘ ccom 5656 ⟶wf 6521 ‘cfv 6525 (class class class)co 7400 ℂcc 11086 ↾_t crest 17463 TopOpenctopn 17464 ℂ_fldccnfld 21482 Topctop 23011 TopOnctopon 23028 CnP ccnp 23343 ×_t ctx 23678 lim_ℂ climc 25982

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-er 8682 df-map 8814 df-pm 8815 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fi 9359 df-sup 9390 df-inf 9391 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-fz 13527 df-seq 14029 df-exp 14089 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277 df-struct 17197 df-slot 17232 df-ndx 17244 df-base 17260 df-plusg 17313 df-mulr 17314 df-starv 17315 df-tset 17319 df-ple 17320 df-ds 17322 df-unif 17323 df-rest 17465 df-topn 17466 df-topgen 17486 df-psmet 21474 df-xmet 21475 df-met 21476 df-bl 21477 df-mopn 21478 df-cnfld 21483 df-top 23012 df-topon 23029 df-topsp 23051 df-bases 23064 df-cnp 23346 df-tx 23680 df-xms 24438 df-ms 24439 df-limc 25986

This theorem is referenced by: dvcnp2 26040 dvaddbr 26058 dvmulbr 26059 dvcobr 26066 lhop1lem 26133 taylthlem2 26495

W3C validator