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Mirrors > Home > MPE Home > Th. List > rlimcn2 | Structured version Visualization version GIF version |
Description: Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 17-Sep-2014.) |
Ref | Expression |
---|---|
rlimcn2.1a | ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐵 ∈ 𝑋) |
rlimcn2.1b | ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐶 ∈ 𝑌) |
rlimcn2.2a | ⊢ (𝜑 → 𝑅 ∈ 𝑋) |
rlimcn2.2b | ⊢ (𝜑 → 𝑆 ∈ 𝑌) |
rlimcn2.3a | ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝑅) |
rlimcn2.3b | ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝑆) |
rlimcn2.4 | ⊢ (𝜑 → 𝐹:(𝑋 × 𝑌)⟶ℂ) |
rlimcn2.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((abs‘(𝑢 − 𝑅)) < 𝑟 ∧ (abs‘(𝑣 − 𝑆)) < 𝑠) → (abs‘((𝑢𝐹𝑣) − (𝑅𝐹𝑆))) < 𝑥)) |
Ref | Expression |
---|---|
rlimcn2 | ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ (𝐵𝐹𝐶)) ⇝𝑟 (𝑅𝐹𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlimcn2.1a | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐵 ∈ 𝑋) | |
2 | rlimcn2.1b | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐶 ∈ 𝑌) | |
3 | rlimcn2.4 | . . . 4 ⊢ (𝜑 → 𝐹:(𝑋 × 𝑌)⟶ℂ) | |
4 | 3 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐹:(𝑋 × 𝑌)⟶ℂ) |
5 | 4, 1, 2 | fovcdmd 7523 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐵𝐹𝐶) ∈ ℂ) |
6 | rlimcn2.2a | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑋) | |
7 | rlimcn2.2b | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑌) | |
8 | 3, 6, 7 | fovcdmd 7523 | . 2 ⊢ (𝜑 → (𝑅𝐹𝑆) ∈ ℂ) |
9 | rlimcn2.3a | . 2 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝑅) | |
10 | rlimcn2.3b | . 2 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝑆) | |
11 | rlimcn2.5 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((abs‘(𝑢 − 𝑅)) < 𝑟 ∧ (abs‘(𝑣 − 𝑆)) < 𝑠) → (abs‘((𝑢𝐹𝑣) − (𝑅𝐹𝑆))) < 𝑥)) | |
12 | 1, 2, 5, 8, 9, 10, 11 | rlimcn3 15469 | 1 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ (𝐵𝐹𝐶)) ⇝𝑟 (𝑅𝐹𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ∀wral 3063 ∃wrex 3072 class class class wbr 5104 ↦ cmpt 5187 × cxp 5630 ⟶wf 6490 ‘cfv 6494 (class class class)co 7354 ℂcc 11046 < clt 11186 − cmin 11382 ℝ+crp 12912 abscabs 15116 ⇝𝑟 crli 15364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-pre-lttri 11122 ax-pre-lttrn 11123 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-po 5544 df-so 5545 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-ov 7357 df-oprab 7358 df-mpo 7359 df-er 8645 df-pm 8765 df-en 8881 df-dom 8882 df-sdom 8883 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-rlim 15368 |
This theorem is referenced by: rlimaddOLD 15523 rlimsub 15524 rlimmulOLD 15526 |
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