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Theorem limccnp2 25927
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 2737 . . . . . . . . . . . 12 𝐽 = 𝐽
32cnprcl 23253 . . . . . . . . . . 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 24803 . . . . . . . . . . . . . 14 𝐾 ∈ (TopOn‘ℂ)
8 txtopon 23599 . . . . . . . . . . . . . 14 ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝐾 ∈ (TopOn‘ℂ)) → (𝐾 ×t 𝐾) ∈ (TopOn‘(ℂ × ℂ)))
97, 7, 8mp2an 692 . . . . . . . . . . . . 13 (𝐾 ×t 𝐾) ∈ (TopOn‘(ℂ × ℂ))
10 limccnp2.x . . . . . . . . . . . . . 14 (𝜑𝑋 ⊆ ℂ)
11 limccnp2.y . . . . . . . . . . . . . 14 (𝜑𝑌 ⊆ ℂ)
12 xpss12 5700 . . . . . . . . . . . . . 14 ((𝑋 ⊆ ℂ ∧ 𝑌 ⊆ ℂ) → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
1310, 11, 12syl2anc 584 . . . . . . . . . . . . 13 (𝜑 → (𝑋 × 𝑌) ⊆ (ℂ × ℂ))
14 resttopon 23169 . . . . . . . . . . . . 13 (((𝐾 ×t 𝐾) ∈ (TopOn‘(ℂ × ℂ)) ∧ (𝑋 × 𝑌) ⊆ (ℂ × ℂ)) → ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)) ∈ (TopOn‘(𝑋 × 𝑌)))
159, 13, 14sylancr 587 . . . . . . . . . . . 12 (𝜑 → ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)) ∈ (TopOn‘(𝑋 × 𝑌)))
165, 15eqeltrid 2845 . . . . . . . . . . 11 (𝜑𝐽 ∈ (TopOn‘(𝑋 × 𝑌)))
17 toponuni 22920 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑋 × 𝑌) = 𝐽)
1816, 17syl 17 . . . . . . . . . 10 (𝜑 → (𝑋 × 𝑌) = 𝐽)
194, 18eleqtrrd 2844 . . . . . . . . 9 (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
20 opelxp 5721 . . . . . . . . 9 (⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌) ↔ (𝐶𝑋𝐷𝑌))
2119, 20sylib 218 . . . . . . . 8 (𝜑 → (𝐶𝑋𝐷𝑌))
2221simpld 494 . . . . . . 7 (𝜑𝐶𝑋)
2322ad2antrr 726 . . . . . 6 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑥 = 𝐵) → 𝐶𝑋)
24 simpll 767 . . . . . . 7 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝜑)
25 simpr 484 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → 𝑥 ∈ (𝐴 ∪ {𝐵}))
26 elun 4153 . . . . . . . . . . . 12 (𝑥 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑥𝐴𝑥 ∈ {𝐵}))
2725, 26sylib 218 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → (𝑥𝐴𝑥 ∈ {𝐵}))
2827ord 865 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥𝐴𝑥 ∈ {𝐵}))
29 elsni 4643 . . . . . . . . . 10 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
3028, 29syl6 35 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥𝐴𝑥 = 𝐵))
3130con1d 145 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → (¬ 𝑥 = 𝐵𝑥𝐴))
3231imp 406 . . . . . . 7 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑥𝐴)
33 limccnp2.r . . . . . . 7 ((𝜑𝑥𝐴) → 𝑅𝑋)
3424, 32, 33syl2anc 584 . . . . . 6 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑅𝑋)
3523, 34ifclda 4561 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, 𝐶, 𝑅) ∈ 𝑋)
3621simprd 495 . . . . . . 7 (𝜑𝐷𝑌)
3736ad2antrr 726 . . . . . 6 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ 𝑥 = 𝐵) → 𝐷𝑌)
38 limccnp2.s . . . . . . 7 ((𝜑𝑥𝐴) → 𝑆𝑌)
3924, 32, 38syl2anc 584 . . . . . 6 (((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑆𝑌)
4037, 39ifclda 4561 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∪ {𝐵})) → if(𝑥 = 𝐵, 𝐷, 𝑆) ∈ 𝑌)
4135, 40opelxpd 5724 . . . 4 ((𝜑𝑥 ∈ (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)⟩))
437a1i 11 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘ℂ))
44 cnpf2 23258 . . . . . 6 ((𝐽 ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ 𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩)) → 𝐻:(𝑋 × 𝑌)⟶ℂ)
4516, 43, 1, 44syl3anc 1373 . . . . 5 (𝜑𝐻:(𝑋 × 𝑌)⟶ℂ)
4645feqmptd 6977 . . . 4 (𝜑𝐻 = (𝑦 ∈ (𝑋 × 𝑌) ↦ (𝐻𝑦)))
47 fveq2 6906 . . . . 5 (𝑦 = ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ → (𝐻𝑦) = (𝐻‘⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩))
48 df-ov 7434 . . . . . 6 (if(𝑥 = 𝐵, 𝐶, 𝑅)𝐻if(𝑥 = 𝐵, 𝐷, 𝑆)) = (𝐻‘⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)
49 ovif12 7533 . . . . . 6 (if(𝑥 = 𝐵, 𝐶, 𝑅)𝐻if(𝑥 = 𝐵, 𝐷, 𝑆)) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))
5048, 49eqtr3i 2767 . . . . 5 (𝐻‘⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))
5147, 50eqtrdi 2793 . . . 4 (𝑦 = ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩ → (𝐻𝑦) = if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆)))
5241, 42, 46, 51fmptco 7149 . . 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 (𝑥𝐴𝑅) = (𝑥𝐴𝑅)
5453, 33dmmptd 6713 . . . . . . . . . 10 (𝜑 → dom (𝑥𝐴𝑅) = 𝐴)
55 limccnp2.c . . . . . . . . . . . 12 (𝜑𝐶 ∈ ((𝑥𝐴𝑅) lim 𝐵))
56 limcrcl 25909 . . . . . . . . . . . 12 (𝐶 ∈ ((𝑥𝐴𝑅) lim 𝐵) → ((𝑥𝐴𝑅):dom (𝑥𝐴𝑅)⟶ℂ ∧ dom (𝑥𝐴𝑅) ⊆ ℂ ∧ 𝐵 ∈ ℂ))
5755, 56syl 17 . . . . . . . . . . 11 (𝜑 → ((𝑥𝐴𝑅):dom (𝑥𝐴𝑅)⟶ℂ ∧ dom (𝑥𝐴𝑅) ⊆ ℂ ∧ 𝐵 ∈ ℂ))
5857simp2d 1144 . . . . . . . . . 10 (𝜑 → dom (𝑥𝐴𝑅) ⊆ ℂ)
5954, 58eqsstrrd 4019 . . . . . . . . 9 (𝜑𝐴 ⊆ ℂ)
6057simp3d 1145 . . . . . . . . . 10 (𝜑𝐵 ∈ ℂ)
6160snssd 4809 . . . . . . . . 9 (𝜑 → {𝐵} ⊆ ℂ)
6259, 61unssd 4192 . . . . . . . 8 (𝜑 → (𝐴 ∪ {𝐵}) ⊆ ℂ)
63 resttopon 23169 . . . . . . . 8 ((𝐾 ∈ (TopOn‘ℂ) ∧ (𝐴 ∪ {𝐵}) ⊆ ℂ) → (𝐾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
647, 62, 63sylancr 587 . . . . . . 7 (𝜑 → (𝐾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})))
65 ssun2 4179 . . . . . . . 8 {𝐵} ⊆ (𝐴 ∪ {𝐵})
66 snssg 4783 . . . . . . . . 9 (𝐵 ∈ ℂ → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
6760, 66syl 17 . . . . . . . 8 (𝜑 → (𝐵 ∈ (𝐴 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝐴 ∪ {𝐵})))
6865, 67mpbiri 258 . . . . . . 7 (𝜑𝐵 ∈ (𝐴 ∪ {𝐵}))
6910adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑋 ⊆ ℂ)
7069, 33sseldd 3984 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑅 ∈ ℂ)
71 eqid 2737 . . . . . . . . 9 (𝐾t (𝐴 ∪ {𝐵})) = (𝐾t (𝐴 ∪ {𝐵}))
7259, 60, 70, 71, 6limcmpt 25918 . . . . . . . 8 (𝜑 → (𝐶 ∈ ((𝑥𝐴𝑅) lim 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, 𝑅)) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
7355, 72mpbid 232 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐶, 𝑅)) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
74 limccnp2.d . . . . . . . 8 (𝜑𝐷 ∈ ((𝑥𝐴𝑆) lim 𝐵))
7511adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑌 ⊆ ℂ)
7675, 38sseldd 3984 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑆 ∈ ℂ)
7759, 60, 76, 71, 6limcmpt 25918 . . . . . . . 8 (𝜑 → (𝐷 ∈ ((𝑥𝐴𝑆) lim 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐷, 𝑆)) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
7874, 77mpbid 232 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, 𝐷, 𝑆)) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
7964, 43, 43, 68, 73, 78txcnp 23628 . . . . . 6 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP (𝐾 ×t 𝐾))‘𝐵))
809topontopi 22921 . . . . . . . 8 (𝐾 ×t 𝐾) ∈ Top
8180a1i 11 . . . . . . 7 (𝜑 → (𝐾 ×t 𝐾) ∈ Top)
8241fmpttd 7135 . . . . . . . 8 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩):(𝐴 ∪ {𝐵})⟶(𝑋 × 𝑌))
83 toponuni 22920 . . . . . . . . . 10 ((𝐾t (𝐴 ∪ {𝐵})) ∈ (TopOn‘(𝐴 ∪ {𝐵})) → (𝐴 ∪ {𝐵}) = (𝐾t (𝐴 ∪ {𝐵})))
8464, 83syl 17 . . . . . . . . 9 (𝜑 → (𝐴 ∪ {𝐵}) = (𝐾t (𝐴 ∪ {𝐵})))
8584feq2d 6722 . . . . . . . 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)))
8682, 85mpbid 232 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩): (𝐾t (𝐴 ∪ {𝐵}))⟶(𝑋 × 𝑌))
87 eqid 2737 . . . . . . . 8 (𝐾t (𝐴 ∪ {𝐵})) = (𝐾t (𝐴 ∪ {𝐵}))
889toponunii 22922 . . . . . . . 8 (ℂ × ℂ) = (𝐾 ×t 𝐾)
8987, 88cnprest2 23298 . . . . . . 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)))
9081, 86, 13, 89syl3anc 1373 . . . . . 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)))
9179, 90mpbid 232 . . . . 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))
925oveq2i 7442 . . . . . 6 ((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐽) = ((𝐾t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)))
9392fveq1i 6907 . . . . 5 (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵) = (((𝐾t (𝐴 ∪ {𝐵})) CnP ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌)))‘𝐵)
9491, 93eleqtrrdi 2852 . . . 4 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐽)‘𝐵))
95 iftrue 4531 . . . . . . . . 9 (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐶, 𝑅) = 𝐶)
96 iftrue 4531 . . . . . . . . 9 (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐷, 𝑆) = 𝐷)
9795, 96opeq12d 4881 . . . . . . . 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 5469 . . . . . . . 8 𝐶, 𝐷⟩ ∈ V
10097, 98, 99fvmpt 7016 . . . . . . 7 (𝐵 ∈ (𝐴 ∪ {𝐵}) → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)‘𝐵) = ⟨𝐶, 𝐷⟩)
10168, 100syl 17 . . . . . 6 (𝜑 → ((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)‘𝐵) = ⟨𝐶, 𝐷⟩)
102101fveq2d 6910 . . . . 5 (𝜑 → ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)‘𝐵)) = ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩))
1031, 102eleqtrrd 2844 . . . 4 (𝜑𝐻 ∈ ((𝐽 CnP 𝐾)‘((𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)‘𝐵)))
104 cnpco 23275 . . . 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))
10594, 103, 104syl2anc 584 . . 3 (𝜑 → (𝐻 ∘ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ ⟨if(𝑥 = 𝐵, 𝐶, 𝑅), if(𝑥 = 𝐵, 𝐷, 𝑆)⟩)) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
10652, 105eqeltrrd 2842 . 2 (𝜑 → (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵))
10745adantr 480 . . . 4 ((𝜑𝑥𝐴) → 𝐻:(𝑋 × 𝑌)⟶ℂ)
108107, 33, 38fovcdmd 7605 . . 3 ((𝜑𝑥𝐴) → (𝑅𝐻𝑆) ∈ ℂ)
10959, 60, 108, 71, 6limcmpt 25918 . 2 (𝜑 → ((𝐶𝐻𝐷) ∈ ((𝑥𝐴 ↦ (𝑅𝐻𝑆)) lim 𝐵) ↔ (𝑥 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑥 = 𝐵, (𝐶𝐻𝐷), (𝑅𝐻𝑆))) ∈ (((𝐾t (𝐴 ∪ {𝐵})) CnP 𝐾)‘𝐵)))
110106, 109mpbird 257 1 (𝜑 → (𝐶𝐻𝐷) ∈ ((𝑥𝐴 ↦ (𝑅𝐻𝑆)) lim 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  cun 3949  wss 3951  ifcif 4525  {csn 4626  cop 4632   cuni 4907  cmpt 5225   × cxp 5683  dom cdm 5685  ccom 5689  wf 6557  cfv 6561  (class class class)co 7431  cc 11153  t crest 17465  TopOpenctopn 17466  fldccnfld 21364  Topctop 22899  TopOnctopon 22916   CnP ccnp 23233   ×t ctx 23568   lim climc 25897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-fz 13548  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-plusg 17310  df-mulr 17311  df-starv 17312  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-rest 17467  df-topn 17468  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-cnfld 21365  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cnp 23236  df-tx 23570  df-xms 24330  df-ms 24331  df-limc 25901
This theorem is referenced by:  dvcnp2  25955  dvcnp2OLD  25956  dvaddbr  25974  dvmulbr  25975  dvmulbrOLD  25976  dvcobr  25983  dvcobrOLD  25984  lhop1lem  26052  taylthlem2  26416  taylthlem2OLD  26417
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