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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 480 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐹:(𝑋 × 𝑌)⟶ℂ) |
5 | 4, 1, 2 | fovcdmd 7604 | . 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 7604 | . 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 15622 | 1 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ (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 ℂcc 11150 < clt 11292 − cmin 11489 ℝ+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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