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| Mirrors > Home > MPE Home > Th. List > cncfi | Structured version Visualization version GIF version | ||
| Description: Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.) |
| Ref | Expression |
|---|---|
| cncfi | ⊢ ((𝐹 ∈ (𝐴–cn→𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝑅 ∈ ℝ+) → ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝐶))) < 𝑅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cncfrss 24917 | . . . . . 6 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ) | |
| 2 | cncfrss2 24918 | . . . . . 6 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ) | |
| 3 | elcncf2 24916 | . . . . . 6 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴–cn→𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑦)))) | |
| 4 | 1, 2, 3 | syl2anc 584 | . . . . 5 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → (𝐹 ∈ (𝐴–cn→𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑦)))) |
| 5 | 4 | ibi 267 | . . . 4 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑦))) |
| 6 | 5 | simprd 495 | . . 3 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑦)) |
| 7 | oveq2 7439 | . . . . . . . 8 ⊢ (𝑥 = 𝐶 → (𝑤 − 𝑥) = (𝑤 − 𝐶)) | |
| 8 | 7 | fveq2d 6910 | . . . . . . 7 ⊢ (𝑥 = 𝐶 → (abs‘(𝑤 − 𝑥)) = (abs‘(𝑤 − 𝐶))) |
| 9 | 8 | breq1d 5153 | . . . . . 6 ⊢ (𝑥 = 𝐶 → ((abs‘(𝑤 − 𝑥)) < 𝑧 ↔ (abs‘(𝑤 − 𝐶)) < 𝑧)) |
| 10 | fveq2 6906 | . . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝐹‘𝑥) = (𝐹‘𝐶)) | |
| 11 | 10 | oveq2d 7447 | . . . . . . . 8 ⊢ (𝑥 = 𝐶 → ((𝐹‘𝑤) − (𝐹‘𝑥)) = ((𝐹‘𝑤) − (𝐹‘𝐶))) |
| 12 | 11 | fveq2d 6910 | . . . . . . 7 ⊢ (𝑥 = 𝐶 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) = (abs‘((𝐹‘𝑤) − (𝐹‘𝐶)))) |
| 13 | 12 | breq1d 5153 | . . . . . 6 ⊢ (𝑥 = 𝐶 → ((abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑦 ↔ (abs‘((𝐹‘𝑤) − (𝐹‘𝐶))) < 𝑦)) |
| 14 | 9, 13 | imbi12d 344 | . . . . 5 ⊢ (𝑥 = 𝐶 → (((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑦) ↔ ((abs‘(𝑤 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝐶))) < 𝑦))) |
| 15 | 14 | rexralbidv 3223 | . . . 4 ⊢ (𝑥 = 𝐶 → (∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑦) ↔ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝐶))) < 𝑦))) |
| 16 | breq2 5147 | . . . . . 6 ⊢ (𝑦 = 𝑅 → ((abs‘((𝐹‘𝑤) − (𝐹‘𝐶))) < 𝑦 ↔ (abs‘((𝐹‘𝑤) − (𝐹‘𝐶))) < 𝑅)) | |
| 17 | 16 | imbi2d 340 | . . . . 5 ⊢ (𝑦 = 𝑅 → (((abs‘(𝑤 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝐶))) < 𝑦) ↔ ((abs‘(𝑤 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝐶))) < 𝑅))) |
| 18 | 17 | rexralbidv 3223 | . . . 4 ⊢ (𝑦 = 𝑅 → (∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝐶))) < 𝑦) ↔ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝐶))) < 𝑅))) |
| 19 | 15, 18 | rspc2v 3633 | . . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑅 ∈ ℝ+) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑦) → ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝐶))) < 𝑅))) |
| 20 | 6, 19 | mpan9 506 | . 2 ⊢ ((𝐹 ∈ (𝐴–cn→𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝑅 ∈ ℝ+)) → ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝐶))) < 𝑅)) |
| 21 | 20 | 3impb 1115 | 1 ⊢ ((𝐹 ∈ (𝐴–cn→𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝑅 ∈ ℝ+) → ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝐶))) < 𝑅)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 class class class wbr 5143 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 < clt 11295 − cmin 11492 ℝ+crp 13034 abscabs 15273 –cn→ccncf 24902 |
| 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-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 |
| 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-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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-map 8868 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-sub 11494 df-neg 11495 df-div 11921 df-2 12329 df-cj 15138 df-re 15139 df-im 15140 df-abs 15275 df-cncf 24904 |
| This theorem is referenced by: cncfcdm 24924 climcncf 24926 cncfco 24933 ivthlem2 25487 ivthlem3 25488 ulmcn 26442 pntlem3 27653 sinccvglem 35677 itg2gt0cn 37682 |
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