limccnp2 - Metamath Proof Explorer

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Theorem limccnp2 25869

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 2737	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	cnprcl 23220	. . . . . . . . . . 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 24757	. . . . . . . . . . . . . 14 ⊢ 𝐾 ∈ (TopOn‘ℂ)
8		txtopon 23566	. . . . . . . . . . . . . 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 5639	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑌 ⊆ ℂ) → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
13	10, 11, 12	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
14		resttopon 23136	. . . . . . . . . . . . 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 2841	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(𝑋 × 𝑌)))
17		toponuni 22889	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑋 × 𝑌) = ∪ 𝐽)
18	16, 17	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 × 𝑌) = ∪ 𝐽)
19	4, 18	eleqtrrd 2840	. . . . . . . . 9 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
20		opelxp 5660	. . . . . . . . 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 4094	. . . . . . . . . . . 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 4585	. . . . . . . . . 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 4503	. . . . 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 4503	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, 𝐷, 𝑆) ∈ 𝑌)
41	35, 40	opelxpd 5663	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∪ {𝐵})) → ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ ∈ (𝑋 × 𝑌))
42		eqidd 2738	. . . 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 23225	. . . . . 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 6902	. . . 4 ⊢ (𝜑 → 𝐻 = (𝑦 ∈ (𝑋 × 𝑌) ↦ (𝐻‘𝑦)))
47		fveq2 6834	. . . . 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 7363	. . . . . 6 ⊢ (if(𝑥 = 𝐵, 𝐶, 𝑅)𝐻if(𝑥 = 𝐵, 𝐷, 𝑆)) = (𝐻‘⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)
49		ovif12 7460	. . . . . 6 ⊢ (if(𝑥 = 𝐵, 𝐶, 𝑅)𝐻if(𝑥 = 𝐵, 𝐷, 𝑆)) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))
50	48, 49	eqtr3i 2762	. . . . 5 ⊢ (𝐻‘⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))
51	47, 50	eqtrdi 2788	. . . 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 7076	. . 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 2737	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 ↦ 𝑅) = (𝑥 ∈ 𝐴 ↦ 𝑅)
54	53, 33	dmmptd 6637	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝑅) = 𝐴)
55		limccnp2.c	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝑅) lim_ℂ 𝐵))
56		limcrcl 25851	. . . . . . . . . . . 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 3958	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
60	57	simp3d 1145	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ)
61	60	snssd 4753	. . . . . . . . 9 ⊢ (𝜑 → {𝐵} ⊆ ℂ)
62	59, 61	unssd 4133	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ)
63		resttopon 23136	. . . . . . . 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 4120	. . . . . . . 8 ⊢ {𝐵} ⊆ (𝐴 ∪ {𝐵})
66		snssg 4728	. . . . . . . . 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 3923	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑅 ∈ ℂ)
71		eqid 2737	. . . . . . . . 9 ⊢ (𝐾 ↾_t (𝐴 ∪ {𝐵})) = (𝐾 ↾_t (𝐴 ∪ {𝐵}))
72	59, 60, 70, 71, 6	limcmpt 25860	. . . . . . . 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 3923	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑆 ∈ ℂ)
77	59, 60, 76, 71, 6	limcmpt 25860	. . . . . . . 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 23595	. . . . . 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 22890	. . . . . . . 8 ⊢ (𝐾 ×_t 𝐾) ∈ Top
81	80	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐾 ×_t 𝐾) ∈ Top)
82	41	fmpttd 7061	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩):(𝐴 ∪ {𝐵})⟶(𝑋 × 𝑌))
83		toponuni 22889	. . . . . . . . . 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 6646	. . . . . . . 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 2737	. . . . . . . 8 ⊢ ∪ (𝐾 ↾_t (𝐴 ∪ {𝐵})) = ∪ (𝐾 ↾_t (𝐴 ∪ {𝐵}))
88	9	toponunii 22891	. . . . . . . 8 ⊢ (ℂ × ℂ) = ∪ (𝐾 ×_t 𝐾)
89	87, 88	cnprest2 23265	. . . . . . 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 7371	. . . . . 6 ⊢ ((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽) = ((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×_t 𝐾) ↾_t (𝑋 × 𝑌)))
93	92	fveq1i 6835	. . . . 5 ⊢ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵) = (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×_t 𝐾) ↾_t (𝑋 × 𝑌)))‘𝐵)
94	91, 93	eleqtrrdi 2848	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾 ↾_t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵))
95		iftrue 4473	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐶, 𝑅) = 𝐶)
96		iftrue 4473	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐷, 𝑆) = 𝐷)
97	95, 96	opeq12d 4825	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ = ⟨𝐶, 𝐷⟩)
98		eqid 2737	. . . . . . . 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 5411	. . . . . . . 8 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
100	97, 98, 99	fvmpt 6941	. . . . . . 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 6838	. . . . 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 2840	. . . 4 ⊢ (𝜑 → 𝐻 ∈ ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)‘𝐵)))
104		cnpco 23242	. . . 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 2838	. 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 7532	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅𝐻𝑆) ∈ ℂ)
109	59, 60, 108, 71, 6	limcmpt 25860	. 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 3888 ⊆ wss 3890 ifcif 4467 {csn 4568 ⟨cop 4574 ∪ cuni 4851 ↦ cmpt 5167 × cxp 5622 dom cdm 5624 ∘ ccom 5628 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 ↾_t crest 17374 TopOpenctopn 17375 ℂ_fldccnfld 21344 Topctop 22868 TopOnctopon 22885 CnP ccnp 23200 ×_t ctx 23535 lim_ℂ climc 25839

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 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-map 8768 df-pm 8769 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fi 9317 df-sup 9348 df-inf 9349 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-fz 13453 df-seq 13955 df-exp 14015 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17108 df-slot 17143 df-ndx 17155 df-base 17171 df-plusg 17224 df-mulr 17225 df-starv 17226 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-rest 17376 df-topn 17377 df-topgen 17397 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-cnfld 21345 df-top 22869 df-topon 22886 df-topsp 22908 df-bases 22921 df-cnp 23203 df-tx 23537 df-xms 24295 df-ms 24296 df-limc 25843

This theorem is referenced by: dvcnp2 25897 dvaddbr 25915 dvmulbr 25916 dvcobr 25923 lhop1lem 25990 taylthlem2 26351 taylthlem2OLD 26352

W3C validator