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Theorem limccnp2 23562
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 2621 . . . . . . . . . . . 12 𝐽 = 𝐽
32cnprcl 20959 . . . . . . . . . . 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 22496 . . . . . . . . . . . . . 14 𝐾 ∈ (TopOn‘ℂ)
8 txtopon 21304 . . . . . . . . . . . . . 14 ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝐾 ∈ (TopOn‘ℂ)) → (𝐾 ×t 𝐾) ∈ (TopOn‘(ℂ × ℂ)))
97, 7, 8mp2an 707 . . . . . . . . . . . . 13 (𝐾 ×t 𝐾) ∈ (TopOn‘(ℂ × ℂ))
10 limccnp2.x . . . . . . . . . . . . . 14 (𝜑𝑋 ⊆ ℂ)
11 limccnp2.y . . . . . . . . . . . . . 14 (𝜑𝑌 ⊆ ℂ)
12 xpss12 5186 . . . . . . . . . . . . . 14 ((𝑋 ⊆ ℂ ∧ 𝑌 ⊆ ℂ) → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
1310, 11, 12syl2anc 692 . . . . . . . . . . . . 13 (𝜑 → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
14 resttopon 20875 . . . . . . . . . . . . 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 2702 . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘(𝑋 × 𝑌)))
17 toponuni 20642 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑋 × 𝑌) = 𝐽)
1816, 17syl 17 . . . . . . . . . 10 (𝜑 → (𝑋 × 𝑌) = 𝐽)
194, 18eleqtrrd 2701 . . . . . . . . 9 (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
20 opelxp 5106 . . . . . . . . 9 (⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌) ↔ (𝐶𝑋𝐷𝑌))
2119, 20sylib 208 . . . . . . . 8 (𝜑 → (𝐶𝑋𝐷𝑌))
2221simpld 475 . . . . . . 7 (𝜑𝐶𝑋)
2322ad2antrr 761 . . . . . 6 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑥 = 𝐵) → 𝐶𝑋)
24 simpll 789 . . . . . . 7 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝜑)
25 simpr 477 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → 𝑥 ∈ (𝐴 ∪ {𝐵}))
26 elun 3731 . . . . . . . . . . . 12 (𝑥 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑥𝐴𝑥 ∈ {𝐵}))
2725, 26sylib 208 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → (𝑥𝐴𝑥 ∈ {𝐵}))
2827ord 392 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥𝐴𝑥 ∈ {𝐵}))
29 elsni 4165 . . . . . . . . . 10 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
3028, 29syl6 35 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥𝐴𝑥 = 𝐵))
3130con1d 139 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥 = 𝐵𝑥𝐴))
3231imp 445 . . . . . . 7 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑥𝐴)
33 limccnp2.r . . . . . . 7 ((𝜑𝑥𝐴) → 𝑅𝑋)
3424, 32, 33syl2anc 692 . . . . . 6 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑅𝑋)
3523, 34ifclda 4092 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, 𝐶, 𝑅) ∈ 𝑋)
3621simprd 479 . . . . . . 7 (𝜑𝐷𝑌)
3736ad2antrr 761 . . . . . 6 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑥 = 𝐵) → 𝐷𝑌)
38 limccnp2.s . . . . . . 7 ((𝜑𝑥𝐴) → 𝑆𝑌)
3924, 32, 38syl2anc 692 . . . . . 6 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑆𝑌)
4037, 39ifclda 4092 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, 𝐷, 𝑆) ∈ 𝑌)
41 opelxpi 5108 . . . . 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 692 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ ∈ (𝑋 × 𝑌))
43 eqidd 2622 . . . 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 20964 . . . . . 6 ((𝐽 ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ 𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩)) → 𝐻:(𝑋 × 𝑌)⟶ℂ)
4616, 44, 1, 45syl3anc 1323 . . . . 5 (𝜑𝐻:(𝑋 × 𝑌)⟶ℂ)
4746feqmptd 6206 . . . 4 (𝜑𝐻 = (𝑦 ∈ (𝑋 × 𝑌) ↦ (𝐻𝑦)))
48 fveq2 6148 . . . . 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 6607 . . . . . 6 (if(𝑥 = 𝐵, 𝐶, 𝑅)𝐻if(𝑥 = 𝐵, 𝐷, 𝑆)) = (𝐻‘⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)
50 ovif12 6692 . . . . . 6 (if(𝑥 = 𝐵, 𝐶, 𝑅)𝐻if(𝑥 = 𝐵, 𝐷, 𝑆)) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))
5149, 50eqtr3i 2645 . . . . 5 (𝐻‘⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))
5248, 51syl6eq 2671 . . . 4 (𝑦 = ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ → (𝐻𝑦) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆)))
5342, 43, 47, 52fmptco 6351 . . 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 2621 . . . . . . . . . . 11 (𝑥𝐴𝑅) = (𝑥𝐴𝑅)
5554, 33dmmptd 5981 . . . . . . . . . 10 (𝜑 → dom (𝑥𝐴𝑅) = 𝐴)
56 limccnp2.c . . . . . . . . . . . 12 (𝜑𝐶 ∈ ((𝑥𝐴𝑅) lim 𝐵))
57 limcrcl 23544 . . . . . . . . . . . 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 1072 . . . . . . . . . 10 (𝜑 → dom (𝑥𝐴𝑅) ⊆ ℂ)
6055, 59eqsstr3d 3619 . . . . . . . . 9 (𝜑𝐴 ⊆ ℂ)
6158simp3d 1073 . . . . . . . . . 10 (𝜑𝐵 ∈ ℂ)
6261snssd 4309 . . . . . . . . 9 (𝜑 → {𝐵} ⊆ ℂ)
6360, 62unssd 3767 . . . . . . . 8 (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ)
64 resttopon 20875 . . . . . . . 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 3755 . . . . . . . 8 {𝐵} ⊆ (𝐴 ∪ {𝐵})
67 snssg 4296 . . . . . . . . 9 (𝐵 ∈ ℂ → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
6861, 67syl 17 . . . . . . . 8 (𝜑 → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
6966, 68mpbiri 248 . . . . . . 7 (𝜑𝐵 ∈ (𝐴 ∪ {𝐵}))
7010adantr 481 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑋 ⊆ ℂ)
7170, 33sseldd 3584 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑅 ∈ ℂ)
72 eqid 2621 . . . . . . . . 9 (𝐾t (𝐴 ∪ {𝐵})) = (𝐾t (𝐴 ∪ {𝐵}))
7360, 61, 71, 72, 6limcmpt 23553 . . . . . . . 8 (𝜑 → (𝐶 ∈ ((𝑥𝐴𝑅) lim 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, 𝑅)) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
7456, 73mpbid 222 . . . . . . 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 481 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑌 ⊆ ℂ)
7776, 38sseldd 3584 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑆 ∈ ℂ)
7860, 61, 77, 72, 6limcmpt 23553 . . . . . . . 8 (𝜑 → (𝐷 ∈ ((𝑥𝐴𝑆) lim 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐷, 𝑆)) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
7975, 78mpbid 222 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐷, 𝑆)) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
8065, 44, 44, 69, 74, 79txcnp 21333 . . . . . 6 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP (𝐾 ×t 𝐾))‘𝐵))
819topontopi 20646 . . . . . . . 8 (𝐾 ×t 𝐾) ∈ Top
8281a1i 11 . . . . . . 7 (𝜑 → (𝐾 ×t 𝐾) ∈ Top)
83 eqid 2621 . . . . . . . . 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 6340 . . . . . . . 8 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩):(𝐴 ∪ {𝐵})⟶(𝑋 × 𝑌))
85 toponuni 20642 . . . . . . . . . 10 ((𝐾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})) → (𝐴 ∪ {𝐵}) = (𝐾t (𝐴 ∪ {𝐵})))
8665, 85syl 17 . . . . . . . . 9 (𝜑 → (𝐴 ∪ {𝐵}) = (𝐾t (𝐴 ∪ {𝐵})))
8786feq2d 5988 . . . . . . . 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 222 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩): (𝐾t (𝐴 ∪ {𝐵}))⟶(𝑋 × 𝑌))
89 eqid 2621 . . . . . . . 8 (𝐾t (𝐴 ∪ {𝐵})) = (𝐾t (𝐴 ∪ {𝐵}))
909toponunii 20647 . . . . . . . 8 (ℂ × ℂ) = (𝐾 ×t 𝐾)
9189, 90cnprest2 21004 . . . . . . 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 1323 . . . . . 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 222 . . . . 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 6615 . . . . . 6 ((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐽) = ((𝐾t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)))
9594fveq1i 6149 . . . . 5 (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵) = (((𝐾t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)))‘𝐵)
9693, 95syl6eleqr 2709 . . . 4 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵))
97 iftrue 4064 . . . . . . . . 9 (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐶, 𝑅) = 𝐶)
98 iftrue 4064 . . . . . . . . 9 (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐷, 𝑆) = 𝐷)
9997, 98opeq12d 4378 . . . . . . . 8 (𝑥 = 𝐵 → ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ = ⟨𝐶, 𝐷⟩)
100 opex 4893 . . . . . . . 8 𝐶, 𝐷⟩ ∈ V
10199, 83, 100fvmpt 6239 . . . . . . 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 6152 . . . . 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 2701 . . . 4 (𝜑𝐻 ∈ ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)‘𝐵)))
105 cnpco 20981 . . . 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 692 . . 3 (𝜑 → (𝐻 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
10753, 106eqeltrrd 2699 . 2 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
10846adantr 481 . . . 4 ((𝜑𝑥𝐴) → 𝐻:(𝑋 × 𝑌)⟶ℂ)
109108, 33, 38fovrnd 6759 . . 3 ((𝜑𝑥𝐴) → (𝑅𝐻𝑆) ∈ ℂ)
11060, 61, 109, 72, 6limcmpt 23553 . 2 (𝜑 → ((𝐶𝐻𝐷) ∈ ((𝑥𝐴 ↦ (𝑅𝐻𝑆)) lim 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
111107, 110mpbird 247 1 (𝜑 → (𝐶𝐻𝐷) ∈ ((𝑥𝐴 ↦ (𝑅𝐻𝑆)) lim 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  cun 3553  wss 3555  ifcif 4058  {csn 4148  cop 4154   cuni 4402  cmpt 4673   × cxp 5072  dom cdm 5074  ccom 5078  wf 5843  cfv 5847  (class class class)co 6604  cc 9878  t crest 16002  TopOpenctopn 16003  fldccnfld 19665  Topctop 20617  TopOnctopon 20618   CnP ccnp 20939   ×t ctx 21273   lim climc 23532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fi 8261  df-sup 8292  df-inf 8293  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-fz 12269  df-seq 12742  df-exp 12801  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-plusg 15875  df-mulr 15876  df-starv 15877  df-tset 15881  df-ple 15882  df-ds 15885  df-unif 15886  df-rest 16004  df-topn 16005  df-topgen 16025  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-cnfld 19666  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-cnp 20942  df-tx 21275  df-xms 22035  df-ms 22036  df-limc 23536
This theorem is referenced by:  dvcnp2  23589  dvaddbr  23607  dvmulbr  23608  dvcobr  23615  lhop1lem  23680  taylthlem2  24032
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