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Theorem rlimcn2 15623
Description: Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 17-Sep-2014.)
Hypotheses
Ref Expression
rlimcn2.1a ((𝜑𝑧𝐴) → 𝐵𝑋)
rlimcn2.1b ((𝜑𝑧𝐴) → 𝐶𝑌)
rlimcn2.2a (𝜑𝑅𝑋)
rlimcn2.2b (𝜑𝑆𝑌)
rlimcn2.3a (𝜑 → (𝑧𝐴𝐵) ⇝𝑟 𝑅)
rlimcn2.3b (𝜑 → (𝑧𝐴𝐶) ⇝𝑟 𝑆)
rlimcn2.4 (𝜑𝐹:(𝑋 × 𝑌)⟶ℂ)
rlimcn2.5 ((𝜑𝑥 ∈ ℝ+) → ∃𝑟 ∈ ℝ+𝑠 ∈ ℝ+𝑢𝑋𝑣𝑌 (((abs‘(𝑢𝑅)) < 𝑟 ∧ (abs‘(𝑣𝑆)) < 𝑠) → (abs‘((𝑢𝐹𝑣) − (𝑅𝐹𝑆))) < 𝑥))
Assertion
Ref Expression
rlimcn2 (𝜑 → (𝑧𝐴 ↦ (𝐵𝐹𝐶)) ⇝𝑟 (𝑅𝐹𝑆))
Distinct variable groups:   𝐴,𝑟,𝑠,𝑥,𝑧   𝐹,𝑟,𝑠,𝑢,𝑣,𝑥,𝑧   𝑅,𝑟,𝑠,𝑢,𝑣,𝑥,𝑧   𝐵,𝑟,𝑠,𝑢,𝑣,𝑥   𝜑,𝑟,𝑠,𝑥,𝑧   𝑆,𝑟,𝑠,𝑢,𝑣,𝑥,𝑧   𝐶,𝑟,𝑠,𝑣,𝑥   𝑢,𝑋,𝑧   𝑢,𝑌,𝑣,𝑧
Allowed substitution hints:   𝜑(𝑣,𝑢)   𝐴(𝑣,𝑢)   𝐵(𝑧)   𝐶(𝑧,𝑢)   𝑋(𝑥,𝑣,𝑠,𝑟)   𝑌(𝑥,𝑠,𝑟)

Proof of Theorem rlimcn2
StepHypRef Expression
1 rlimcn2.1a . 2 ((𝜑𝑧𝐴) → 𝐵𝑋)
2 rlimcn2.1b . 2 ((𝜑𝑧𝐴) → 𝐶𝑌)
3 rlimcn2.4 . . . 4 (𝜑𝐹:(𝑋 × 𝑌)⟶ℂ)
43adantr 480 . . 3 ((𝜑𝑧𝐴) → 𝐹:(𝑋 × 𝑌)⟶ℂ)
54, 1, 2fovcdmd 7604 . 2 ((𝜑𝑧𝐴) → (𝐵𝐹𝐶) ∈ ℂ)
6 rlimcn2.2a . . 3 (𝜑𝑅𝑋)
7 rlimcn2.2b . . 3 (𝜑𝑆𝑌)
83, 6, 7fovcdmd 7604 . 2 (𝜑 → (𝑅𝐹𝑆) ∈ ℂ)
9 rlimcn2.3a . 2 (𝜑 → (𝑧𝐴𝐵) ⇝𝑟 𝑅)
10 rlimcn2.3b . 2 (𝜑 → (𝑧𝐴𝐶) ⇝𝑟 𝑆)
11 rlimcn2.5 . 2 ((𝜑𝑥 ∈ ℝ+) → ∃𝑟 ∈ ℝ+𝑠 ∈ ℝ+𝑢𝑋𝑣𝑌 (((abs‘(𝑢𝑅)) < 𝑟 ∧ (abs‘(𝑣𝑆)) < 𝑠) → (abs‘((𝑢𝐹𝑣) − (𝑅𝐹𝑆))) < 𝑥))
121, 2, 5, 8, 9, 10, 11rlimcn3 15622 1 (𝜑 → (𝑧𝐴 ↦ (𝐵𝐹𝐶)) ⇝𝑟 (𝑅𝐹𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2105  wral 3058  wrex 3067   class class class wbr 5147  cmpt 5230   × cxp 5686  wf 6558  cfv 6562  (class class class)co 7430  cc 11150   < clt 11292  cmin 11489  +crp 13031  abscabs 15269  𝑟 crli 15517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-pre-lttri 11226  ax-pre-lttrn 11227
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-po 5596  df-so 5597  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-er 8743  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-rlim 15521
This theorem is referenced by:  rlimsub  15676
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