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Mirrors > Home > MPE Home > Th. List > imcncf | Structured version Visualization version GIF version |
Description: Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.) |
Ref | Expression |
---|---|
imcncf | ⊢ ℑ ∈ (ℂ–cn→ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imf 15084 | . 2 ⊢ ℑ:ℂ⟶ℝ | |
2 | imcn2 15570 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℝ+) → ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℂ ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((ℑ‘𝑤) − (ℑ‘𝑥))) < 𝑦)) | |
3 | 2 | rgen2 3192 | . 2 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℂ ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((ℑ‘𝑤) − (ℑ‘𝑥))) < 𝑦) |
4 | ssid 4000 | . . 3 ⊢ ℂ ⊆ ℂ | |
5 | ax-resscn 11187 | . . 3 ⊢ ℝ ⊆ ℂ | |
6 | elcncf2 24797 | . . 3 ⊢ ((ℂ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℑ ∈ (ℂ–cn→ℝ) ↔ (ℑ:ℂ⟶ℝ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℂ ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((ℑ‘𝑤) − (ℑ‘𝑥))) < 𝑦)))) | |
7 | 4, 5, 6 | mp2an 691 | . 2 ⊢ (ℑ ∈ (ℂ–cn→ℝ) ↔ (ℑ:ℂ⟶ℝ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℂ ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((ℑ‘𝑤) − (ℑ‘𝑥))) < 𝑦))) |
8 | 1, 3, 7 | mpbir2an 710 | 1 ⊢ ℑ ∈ (ℂ–cn→ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2099 ∀wral 3056 ∃wrex 3065 ⊆ wss 3944 class class class wbr 5142 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ℂcc 11128 ℝcr 11129 < clt 11270 − cmin 11466 ℝ+crp 12998 ℑcim 15069 abscabs 15205 –cn→ccncf 24783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-sup 9457 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-n0 12495 df-z 12581 df-uz 12845 df-rp 12999 df-seq 13991 df-exp 14051 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-cncf 24785 |
This theorem is referenced by: cnrehmeo 24865 cnrehmeoOLD 24866 cncombf 25574 cnmbf 25575 dvloglem 26569 mbfresfi 37074 ftc1anclem8 37108 |
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