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Theorem limccnp2 23997
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 2799 . . . . . . . . . . . 12 𝐽 = 𝐽
32cnprcl 21378 . . . . . . . . . . 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 22914 . . . . . . . . . . . . . 14 𝐾 ∈ (TopOn‘ℂ)
8 txtopon 21723 . . . . . . . . . . . . . 14 ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝐾 ∈ (TopOn‘ℂ)) → (𝐾 ×t 𝐾) ∈ (TopOn‘(ℂ × ℂ)))
97, 7, 8mp2an 684 . . . . . . . . . . . . 13 (𝐾 ×t 𝐾) ∈ (TopOn‘(ℂ × ℂ))
10 limccnp2.x . . . . . . . . . . . . . 14 (𝜑𝑋 ⊆ ℂ)
11 limccnp2.y . . . . . . . . . . . . . 14 (𝜑𝑌 ⊆ ℂ)
12 xpss12 5327 . . . . . . . . . . . . . 14 ((𝑋 ⊆ ℂ ∧ 𝑌 ⊆ ℂ) → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
1310, 11, 12syl2anc 580 . . . . . . . . . . . . 13 (𝜑 → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
14 resttopon 21294 . . . . . . . . . . . . 13 (((𝐾 ×t 𝐾) ∈ (TopOn‘(ℂ × ℂ)) ∧ (𝑋 × 𝑌) ⊆ (ℂ × ℂ)) → ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)) ∈ (TopOn‘(𝑋 × 𝑌)))
159, 13, 14sylancr 582 . . . . . . . . . . . 12 (𝜑 → ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)) ∈ (TopOn‘(𝑋 × 𝑌)))
165, 15syl5eqel 2882 . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘(𝑋 × 𝑌)))
17 toponuni 21047 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑋 × 𝑌) = 𝐽)
1816, 17syl 17 . . . . . . . . . 10 (𝜑 → (𝑋 × 𝑌) = 𝐽)
194, 18eleqtrrd 2881 . . . . . . . . 9 (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
20 opelxp 5348 . . . . . . . . 9 (⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌) ↔ (𝐶𝑋𝐷𝑌))
2119, 20sylib 210 . . . . . . . 8 (𝜑 → (𝐶𝑋𝐷𝑌))
2221simpld 489 . . . . . . 7 (𝜑𝐶𝑋)
2322ad2antrr 718 . . . . . 6 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑥 = 𝐵) → 𝐶𝑋)
24 simpll 784 . . . . . . 7 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝜑)
25 simpr 478 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → 𝑥 ∈ (𝐴 ∪ {𝐵}))
26 elun 3951 . . . . . . . . . . . 12 (𝑥 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑥𝐴𝑥 ∈ {𝐵}))
2725, 26sylib 210 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → (𝑥𝐴𝑥 ∈ {𝐵}))
2827ord 891 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥𝐴𝑥 ∈ {𝐵}))
29 elsni 4385 . . . . . . . . . 10 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
3028, 29syl6 35 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥𝐴𝑥 = 𝐵))
3130con1d 142 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥 = 𝐵𝑥𝐴))
3231imp 396 . . . . . . 7 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑥𝐴)
33 limccnp2.r . . . . . . 7 ((𝜑𝑥𝐴) → 𝑅𝑋)
3424, 32, 33syl2anc 580 . . . . . 6 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑅𝑋)
3523, 34ifclda 4311 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, 𝐶, 𝑅) ∈ 𝑋)
3621simprd 490 . . . . . . 7 (𝜑𝐷𝑌)
3736ad2antrr 718 . . . . . 6 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑥 = 𝐵) → 𝐷𝑌)
38 limccnp2.s . . . . . . 7 ((𝜑𝑥𝐴) → 𝑆𝑌)
3924, 32, 38syl2anc 580 . . . . . 6 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑆𝑌)
4037, 39ifclda 4311 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, 𝐷, 𝑆) ∈ 𝑌)
41 opelxpi 5349 . . . . 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 580 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ ∈ (𝑋 × 𝑌))
43 eqidd 2800 . . . 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 21383 . . . . . 6 ((𝐽 ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ 𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩)) → 𝐻:(𝑋 × 𝑌)⟶ℂ)
4616, 44, 1, 45syl3anc 1491 . . . . 5 (𝜑𝐻:(𝑋 × 𝑌)⟶ℂ)
4746feqmptd 6474 . . . 4 (𝜑𝐻 = (𝑦 ∈ (𝑋 × 𝑌) ↦ (𝐻𝑦)))
48 fveq2 6411 . . . . 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 6881 . . . . . 6 (if(𝑥 = 𝐵, 𝐶, 𝑅)𝐻if(𝑥 = 𝐵, 𝐷, 𝑆)) = (𝐻‘⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)
50 ovif12 6973 . . . . . 6 (if(𝑥 = 𝐵, 𝐶, 𝑅)𝐻if(𝑥 = 𝐵, 𝐷, 𝑆)) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))
5149, 50eqtr3i 2823 . . . . 5 (𝐻‘⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))
5248, 51syl6eq 2849 . . . 4 (𝑦 = ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ → (𝐻𝑦) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆)))
5342, 43, 47, 52fmptco 6623 . . 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 2799 . . . . . . . . . . 11 (𝑥𝐴𝑅) = (𝑥𝐴𝑅)
5554, 33dmmptd 6235 . . . . . . . . . 10 (𝜑 → dom (𝑥𝐴𝑅) = 𝐴)
56 limccnp2.c . . . . . . . . . . . 12 (𝜑𝐶 ∈ ((𝑥𝐴𝑅) lim 𝐵))
57 limcrcl 23979 . . . . . . . . . . . 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 1174 . . . . . . . . . 10 (𝜑 → dom (𝑥𝐴𝑅) ⊆ ℂ)
6055, 59eqsstr3d 3836 . . . . . . . . 9 (𝜑𝐴 ⊆ ℂ)
6158simp3d 1175 . . . . . . . . . 10 (𝜑𝐵 ∈ ℂ)
6261snssd 4528 . . . . . . . . 9 (𝜑 → {𝐵} ⊆ ℂ)
6360, 62unssd 3987 . . . . . . . 8 (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ)
64 resttopon 21294 . . . . . . . 8 ((𝐾 ∈ (TopOn‘ℂ) ∧ (𝐴 ∪ {𝐵}) ⊆ ℂ) → (𝐾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
657, 63, 64sylancr 582 . . . . . . 7 (𝜑 → (𝐾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
66 ssun2 3975 . . . . . . . 8 {𝐵} ⊆ (𝐴 ∪ {𝐵})
67 snssg 4503 . . . . . . . . 9 (𝐵 ∈ ℂ → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
6861, 67syl 17 . . . . . . . 8 (𝜑 → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
6966, 68mpbiri 250 . . . . . . 7 (𝜑𝐵 ∈ (𝐴 ∪ {𝐵}))
7010adantr 473 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑋 ⊆ ℂ)
7170, 33sseldd 3799 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑅 ∈ ℂ)
72 eqid 2799 . . . . . . . . 9 (𝐾t (𝐴 ∪ {𝐵})) = (𝐾t (𝐴 ∪ {𝐵}))
7360, 61, 71, 72, 6limcmpt 23988 . . . . . . . 8 (𝜑 → (𝐶 ∈ ((𝑥𝐴𝑅) lim 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, 𝑅)) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
7456, 73mpbid 224 . . . . . . 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 473 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑌 ⊆ ℂ)
7776, 38sseldd 3799 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑆 ∈ ℂ)
7860, 61, 77, 72, 6limcmpt 23988 . . . . . . . 8 (𝜑 → (𝐷 ∈ ((𝑥𝐴𝑆) lim 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐷, 𝑆)) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
7975, 78mpbid 224 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐷, 𝑆)) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
8065, 44, 44, 69, 74, 79txcnp 21752 . . . . . 6 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP (𝐾 ×t 𝐾))‘𝐵))
819topontopi 21048 . . . . . . . 8 (𝐾 ×t 𝐾) ∈ Top
8281a1i 11 . . . . . . 7 (𝜑 → (𝐾 ×t 𝐾) ∈ Top)
8342fmpttd 6611 . . . . . . . 8 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩):(𝐴 ∪ {𝐵})⟶(𝑋 × 𝑌))
84 toponuni 21047 . . . . . . . . . 10 ((𝐾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})) → (𝐴 ∪ {𝐵}) = (𝐾t (𝐴 ∪ {𝐵})))
8565, 84syl 17 . . . . . . . . 9 (𝜑 → (𝐴 ∪ {𝐵}) = (𝐾t (𝐴 ∪ {𝐵})))
8685feq2d 6242 . . . . . . . 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)))
8783, 86mpbid 224 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩): (𝐾t (𝐴 ∪ {𝐵}))⟶(𝑋 × 𝑌))
88 eqid 2799 . . . . . . . 8 (𝐾t (𝐴 ∪ {𝐵})) = (𝐾t (𝐴 ∪ {𝐵}))
899toponunii 21049 . . . . . . . 8 (ℂ × ℂ) = (𝐾 ×t 𝐾)
9088, 89cnprest2 21423 . . . . . . 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)))
9182, 87, 13, 90syl3anc 1491 . . . . . 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)))
9280, 91mpbid 224 . . . . 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))
935oveq2i 6889 . . . . . 6 ((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐽) = ((𝐾t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)))
9493fveq1i 6412 . . . . 5 (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵) = (((𝐾t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)))‘𝐵)
9592, 94syl6eleqr 2889 . . . 4 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵))
96 iftrue 4283 . . . . . . . . 9 (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐶, 𝑅) = 𝐶)
97 iftrue 4283 . . . . . . . . 9 (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐷, 𝑆) = 𝐷)
9896, 97opeq12d 4601 . . . . . . . 8 (𝑥 = 𝐵 → ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ = ⟨𝐶, 𝐷⟩)
99 eqid 2799 . . . . . . . 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)⟩)
100 opex 5123 . . . . . . . 8 𝐶, 𝐷⟩ ∈ V
10198, 99, 100fvmpt 6507 . . . . . . 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 6415 . . . . 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 2881 . . . 4 (𝜑𝐻 ∈ ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)‘𝐵)))
105 cnpco 21400 . . . 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))
10695, 104, 105syl2anc 580 . . 3 (𝜑 → (𝐻 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
10753, 106eqeltrrd 2879 . 2 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
10846adantr 473 . . . 4 ((𝜑𝑥𝐴) → 𝐻:(𝑋 × 𝑌)⟶ℂ)
109108, 33, 38fovrnd 7040 . . 3 ((𝜑𝑥𝐴) → (𝑅𝐻𝑆) ∈ ℂ)
11060, 61, 109, 72, 6limcmpt 23988 . 2 (𝜑 → ((𝐶𝐻𝐷) ∈ ((𝑥𝐴 ↦ (𝑅𝐻𝑆)) lim 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
111107, 110mpbird 249 1 (𝜑 → (𝐶𝐻𝐷) ∈ ((𝑥𝐴 ↦ (𝑅𝐻𝑆)) lim 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  wo 874  w3a 1108   = wceq 1653  wcel 2157  cun 3767  wss 3769  ifcif 4277  {csn 4368  cop 4374   cuni 4628  cmpt 4922   × cxp 5310  dom cdm 5312  ccom 5316  wf 6097  cfv 6101  (class class class)co 6878  cc 10222  t crest 16396  TopOpenctopn 16397  fldccnfld 20068  Topctop 21026  TopOnctopon 21043   CnP ccnp 21358   ×t ctx 21692   lim climc 23967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301  ax-pre-sup 10302
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-map 8097  df-pm 8098  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-fi 8559  df-sup 8590  df-inf 8591  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-div 10977  df-nn 11313  df-2 11376  df-3 11377  df-4 11378  df-5 11379  df-6 11380  df-7 11381  df-8 11382  df-9 11383  df-n0 11581  df-z 11667  df-dec 11784  df-uz 11931  df-q 12034  df-rp 12075  df-xneg 12193  df-xadd 12194  df-xmul 12195  df-fz 12581  df-seq 13056  df-exp 13115  df-cj 14180  df-re 14181  df-im 14182  df-sqrt 14316  df-abs 14317  df-struct 16186  df-ndx 16187  df-slot 16188  df-base 16190  df-plusg 16280  df-mulr 16281  df-starv 16282  df-tset 16286  df-ple 16287  df-ds 16289  df-unif 16290  df-rest 16398  df-topn 16399  df-topgen 16419  df-psmet 20060  df-xmet 20061  df-met 20062  df-bl 20063  df-mopn 20064  df-cnfld 20069  df-top 21027  df-topon 21044  df-topsp 21066  df-bases 21079  df-cnp 21361  df-tx 21694  df-xms 22453  df-ms 22454  df-limc 23971
This theorem is referenced by:  dvcnp2  24024  dvaddbr  24042  dvmulbr  24043  dvcobr  24050  lhop1lem  24117  taylthlem2  24469
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