Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  limccnp2 Structured version   Visualization version   GIF version

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 ((𝜑𝑥𝐴) → 𝑅𝑋)
limccnp2.s ((𝜑𝑥𝐴) → 𝑆𝑌)
limccnp2.x (𝜑𝑋 ⊆ ℂ)
limccnp2.y (𝜑𝑌 ⊆ ℂ)
limccnp2.k 𝐾 = (TopOpen‘ℂfld)
limccnp2.j 𝐽 = ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌))
limccnp2.c (𝜑𝐶 ∈ ((𝑥𝐴𝑅) lim 𝐵))
limccnp2.d (𝜑𝐷 ∈ ((𝑥𝐴𝑆) lim 𝐵))
limccnp2.h (𝜑𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩))
Assertion
Ref Expression
limccnp2 (𝜑 → (𝐶𝐻𝐷) ∈ ((𝑥𝐴 ↦ (𝑅𝐻𝑆)) lim 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.
StepHypRef Expression
1 limccnp2.h . . . . . . . . . . 11 (𝜑𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩))
2 eqid 2610 . . . . . . . . . . . 12 𝐽 = 𝐽
32cnprcl 20859 . . . . . . . . . . 11 (𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩) → ⟨𝐶, 𝐷⟩ ∈ 𝐽)
41, 3syl 17 . . . . . . . . . 10 (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝐽)
5 limccnp2.j . . . . . . . . . . . 12 𝐽 = ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌))
6 limccnp2.k . . . . . . . . . . . . . . 15 𝐾 = (TopOpen‘ℂfld)
76cnfldtopon 22396 . . . . . . . . . . . . . 14 𝐾 ∈ (TopOn‘ℂ)
8 txtopon 21204 . . . . . . . . . . . . . 14 ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝐾 ∈ (TopOn‘ℂ)) → (𝐾 ×t 𝐾) ∈ (TopOn‘(ℂ × ℂ)))
97, 7, 8mp2an 704 . . . . . . . . . . . . 13 (𝐾 ×t 𝐾) ∈ (TopOn‘(ℂ × ℂ))
10 limccnp2.x . . . . . . . . . . . . . 14 (𝜑𝑋 ⊆ ℂ)
11 limccnp2.y . . . . . . . . . . . . . 14 (𝜑𝑌 ⊆ ℂ)
12 xpss12 5148 . . . . . . . . . . . . . 14 ((𝑋 ⊆ ℂ ∧ 𝑌 ⊆ ℂ) → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
1310, 11, 12syl2anc 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)))
159, 13, 14sylancr 694 . . . . . . . . . . . 12 (𝜑 → ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)) ∈ (TopOn‘(𝑋 × 𝑌)))
165, 15syl5eqel 2692 . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘(𝑋 × 𝑌)))
17 toponuni 20542 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑋 × 𝑌) = 𝐽)
1816, 17syl 17 . . . . . . . . . 10 (𝜑 → (𝑋 × 𝑌) = 𝐽)
194, 18eleqtrrd 2691 . . . . . . . . 9 (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
20 opelxp 5070 . . . . . . . . 9 (⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌) ↔ (𝐶𝑋𝐷𝑌))
2119, 20sylib 207 . . . . . . . 8 (𝜑 → (𝐶𝑋𝐷𝑌))
2221simpld 474 . . . . . . 7 (𝜑𝐶𝑋)
2322ad2antrr 758 . . . . . 6 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑥 = 𝐵) → 𝐶𝑋)
24 simpll 786 . . . . . . 7 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝜑)
25 simpr 476 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → 𝑥 ∈ (𝐴 ∪ {𝐵}))
26 elun 3715 . . . . . . . . . . . 12 (𝑥 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑥𝐴𝑥 ∈ {𝐵}))
2725, 26sylib 207 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → (𝑥𝐴𝑥 ∈ {𝐵}))
2827ord 391 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥𝐴𝑥 ∈ {𝐵}))
29 elsni 4142 . . . . . . . . . 10 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
3028, 29syl6 34 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥𝐴𝑥 = 𝐵))
3130con1d 138 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥 = 𝐵𝑥𝐴))
3231imp 444 . . . . . . 7 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑥𝐴)
33 limccnp2.r . . . . . . 7 ((𝜑𝑥𝐴) → 𝑅𝑋)
3424, 32, 33syl2anc 691 . . . . . 6 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑅𝑋)
3523, 34ifclda 4070 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, 𝐶, 𝑅) ∈ 𝑋)
3621simprd 478 . . . . . . 7 (𝜑𝐷𝑌)
3736ad2antrr 758 . . . . . 6 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑥 = 𝐵) → 𝐷𝑌)
38 limccnp2.s . . . . . . 7 ((𝜑𝑥𝐴) → 𝑆𝑌)
3924, 32, 38syl2anc 691 . . . . . 6 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑆𝑌)
4037, 39ifclda 4070 . . . . 5 ((𝜑𝑥 ∈ (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))
4235, 40, 41syl2anc 691 . . . 4 ((𝜑𝑥 ∈ (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)⟩))
447a1i 11 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘ℂ))
45 cnpf2 20864 . . . . . 6 ((𝐽 ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ 𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩)) → 𝐻:(𝑋 × 𝑌)⟶ℂ)
4616, 44, 1, 45syl3anc 1318 . . . . 5 (𝜑𝐻:(𝑋 × 𝑌)⟶ℂ)
4746feqmptd 6159 . . . 4 (𝜑𝐻 = (𝑦 ∈ (𝑋 × 𝑌) ↦ (𝐻𝑦)))
48 fveq2 6103 . . . . 5 (𝑦 = ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ → (𝐻𝑦) = (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))
5149, 50eqtr3i 2634 . . . . 5 (𝐻‘⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))
5248, 51syl6eq 2660 . . . 4 (𝑦 = ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ → (𝐻𝑦) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆)))
5342, 43, 47, 52fmptco 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 (𝑥𝐴𝑅) = (𝑥𝐴𝑅)
5554, 33dmmptd 5937 . . . . . . . . . 10 (𝜑 → dom (𝑥𝐴𝑅) = 𝐴)
56 limccnp2.c . . . . . . . . . . . 12 (𝜑𝐶 ∈ ((𝑥𝐴𝑅) lim 𝐵))
57 limcrcl 23444 . . . . . . . . . . . 12 (𝐶 ∈ ((𝑥𝐴𝑅) lim 𝐵) → ((𝑥𝐴𝑅):dom (𝑥𝐴𝑅)⟶ℂ ∧ dom (𝑥𝐴𝑅) ⊆ ℂ ∧ 𝐵 ∈ ℂ))
5856, 57syl 17 . . . . . . . . . . 11 (𝜑 → ((𝑥𝐴𝑅):dom (𝑥𝐴𝑅)⟶ℂ ∧ dom (𝑥𝐴𝑅) ⊆ ℂ ∧ 𝐵 ∈ ℂ))
5958simp2d 1067 . . . . . . . . . 10 (𝜑 → dom (𝑥𝐴𝑅) ⊆ ℂ)
6055, 59eqsstr3d 3603 . . . . . . . . 9 (𝜑𝐴 ⊆ ℂ)
6158simp3d 1068 . . . . . . . . . 10 (𝜑𝐵 ∈ ℂ)
6261snssd 4281 . . . . . . . . 9 (𝜑 → {𝐵} ⊆ ℂ)
6360, 62unssd 3751 . . . . . . . 8 (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ)
64 resttopon 20775 . . . . . . . 8 ((𝐾 ∈ (TopOn‘ℂ) ∧ (𝐴 ∪ {𝐵}) ⊆ ℂ) → (𝐾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
657, 63, 64sylancr 694 . . . . . . 7 (𝜑 → (𝐾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
66 ssun2 3739 . . . . . . . 8 {𝐵} ⊆ (𝐴 ∪ {𝐵})
67 snssg 4268 . . . . . . . . 9 (𝐵 ∈ ℂ → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
6861, 67syl 17 . . . . . . . 8 (𝜑 → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
6966, 68mpbiri 247 . . . . . . 7 (𝜑𝐵 ∈ (𝐴 ∪ {𝐵}))
7010adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑋 ⊆ ℂ)
7170, 33sseldd 3569 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑅 ∈ ℂ)
72 eqid 2610 . . . . . . . . 9 (𝐾t (𝐴 ∪ {𝐵})) = (𝐾t (𝐴 ∪ {𝐵}))
7360, 61, 71, 72, 6limcmpt 23453 . . . . . . . 8 (𝜑 → (𝐶 ∈ ((𝑥𝐴𝑅) lim 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, 𝑅)) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
7456, 73mpbid 221 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, 𝑅)) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
75 limccnp2.d . . . . . . . 8 (𝜑𝐷 ∈ ((𝑥𝐴𝑆) lim 𝐵))
7611adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑌 ⊆ ℂ)
7776, 38sseldd 3569 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑆 ∈ ℂ)
7860, 61, 77, 72, 6limcmpt 23453 . . . . . . . 8 (𝜑 → (𝐷 ∈ ((𝑥𝐴𝑆) lim 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐷, 𝑆)) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
7975, 78mpbid 221 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐷, 𝑆)) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
8065, 44, 44, 69, 74, 79txcnp 21233 . . . . . 6 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP (𝐾 ×t 𝐾))‘𝐵))
819topontopi 20546 . . . . . . . 8 (𝐾 ×t 𝐾) ∈ Top
8281a1i 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)⟩)
8442, 83fmptd 6292 . . . . . . . 8 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩):(𝐴 ∪ {𝐵})⟶(𝑋 × 𝑌))
85 toponuni 20542 . . . . . . . . . 10 ((𝐾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})) → (𝐴 ∪ {𝐵}) = (𝐾t (𝐴 ∪ {𝐵})))
8665, 85syl 17 . . . . . . . . 9 (𝜑 → (𝐴 ∪ {𝐵}) = (𝐾t (𝐴 ∪ {𝐵})))
8786feq2d 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)⟩): (𝐾t (𝐴 ∪ {𝐵}))⟶(𝑋 × 𝑌)))
8884, 87mpbid 221 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩): (𝐾t (𝐴 ∪ {𝐵}))⟶(𝑋 × 𝑌))
89 eqid 2610 . . . . . . . 8 (𝐾t (𝐴 ∪ {𝐵})) = (𝐾t (𝐴 ∪ {𝐵}))
909toponunii 20547 . . . . . . . 8 (ℂ × ℂ) = (𝐾 ×t 𝐾)
9189, 90cnprest2 20904 . . . . . . 7 (((𝐾 ×t 𝐾) ∈ Top ∧ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩): (𝐾t (𝐴 ∪ {𝐵}))⟶(𝑋 × 𝑌) ∧ (𝑋 × 𝑌) ⊆ (ℂ × ℂ)) → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP (𝐾 ×t 𝐾))‘𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)))‘𝐵)))
9282, 88, 13, 91syl3anc 1318 . . . . . 6 (𝜑 → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP (𝐾 ×t 𝐾))‘𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)))‘𝐵)))
9380, 92mpbid 221 . . . . 5 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)))‘𝐵))
945oveq2i 6560 . . . . . 6 ((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐽) = ((𝐾t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)))
9594fveq1i 6104 . . . . 5 (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵) = (((𝐾t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)))‘𝐵)
9693, 95syl6eleqr 2699 . . . 4 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾t (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)
9997, 98opeq12d 4348 . . . . . . . 8 (𝑥 = 𝐵 → ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ = ⟨𝐶, 𝐷⟩)
100 opex 4859 . . . . . . . 8 𝐶, 𝐷⟩ ∈ V
10199, 83, 100fvmpt 6191 . . . . . . 7 (𝐵 ∈ (𝐴 ∪ {𝐵}) → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)‘𝐵) = ⟨𝐶, 𝐷⟩)
10269, 101syl 17 . . . . . 6 (𝜑 → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)‘𝐵) = ⟨𝐶, 𝐷⟩)
103102fveq2d 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⟩))
1041, 103eleqtrrd 2691 . . . 4 (𝜑𝐻 ∈ ((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)⟩) ∈ (((𝐾t (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)⟩)) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
10696, 104, 105syl2anc 691 . . 3 (𝜑 → (𝐻 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
10753, 106eqeltrrd 2689 . 2 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
10846adantr 480 . . . 4 ((𝜑𝑥𝐴) → 𝐻:(𝑋 × 𝑌)⟶ℂ)
109108, 33, 38fovrnd 6704 . . 3 ((𝜑𝑥𝐴) → (𝑅𝐻𝑆) ∈ ℂ)
11060, 61, 109, 72, 6limcmpt 23453 . 2 (𝜑 → ((𝐶𝐻𝐷) ∈ ((𝑥𝐴 ↦ (𝑅𝐻𝑆)) lim 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
111107, 110mpbird 246 1 (𝜑 → (𝐶𝐻𝐷) ∈ ((𝑥𝐴 ↦ (𝑅𝐻𝑆)) lim 𝐵))
 Colors of variables: wff setvar class 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  ℂfldccnfld 19567  Topctop 20517  TopOnctopon 20518   CnP ccnp 20839   ×t ctx 21173   limℂ 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
 Copyright terms: Public domain W3C validator