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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 7605 | . 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 7605 | . 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 15626 | 1 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ (𝐵𝐹𝐶)) ⇝𝑟 (𝑅𝐹𝑆)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 class class class wbr 5143 ↦ cmpt 5225 × cxp 5683 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 < clt 11295 − cmin 11492 ℝ+crp 13034 abscabs 15273 ⇝𝑟 crli 15521 | 
| 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-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 | 
| 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-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 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-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-rlim 15525 | 
| This theorem is referenced by: rlimsub 15680 | 
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