Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > abscncf | Structured version Visualization version GIF version |
Description: Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.) |
Ref | Expression |
---|---|
abscncf | ⊢ abs ∈ (ℂ–cn→ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | absf 14898 | . 2 ⊢ abs:ℂ⟶ℝ | |
2 | abscn2 15157 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℝ+) → ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℂ ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((abs‘𝑤) − (abs‘𝑥))) < 𝑦)) | |
3 | 2 | rgen2 3121 | . 2 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℂ ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((abs‘𝑤) − (abs‘𝑥))) < 𝑦) |
4 | ssid 3920 | . . 3 ⊢ ℂ ⊆ ℂ | |
5 | ax-resscn 10783 | . . 3 ⊢ ℝ ⊆ ℂ | |
6 | elcncf2 23784 | . . 3 ⊢ ((ℂ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (abs ∈ (ℂ–cn→ℝ) ↔ (abs:ℂ⟶ℝ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℂ ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((abs‘𝑤) − (abs‘𝑥))) < 𝑦)))) | |
7 | 4, 5, 6 | mp2an 692 | . 2 ⊢ (abs ∈ (ℂ–cn→ℝ) ↔ (abs:ℂ⟶ℝ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℂ ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((abs‘𝑤) − (abs‘𝑥))) < 𝑦))) |
8 | 1, 3, 7 | mpbir2an 711 | 1 ⊢ abs ∈ (ℂ–cn→ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2110 ∀wral 3058 ∃wrex 3059 ⊆ wss 3863 class class class wbr 5050 ⟶wf 6373 ‘cfv 6377 (class class class)co 7210 ℂcc 10724 ℝcr 10725 < clt 10864 − cmin 11059 ℝ+crp 12583 abscabs 14794 –cn→ccncf 23770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 ax-pre-sup 10804 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rmo 3066 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-2nd 7759 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-er 8388 df-map 8507 df-en 8624 df-dom 8625 df-sdom 8626 df-sup 9055 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-div 11487 df-nn 11828 df-2 11890 df-3 11891 df-n0 12088 df-z 12174 df-uz 12436 df-rp 12584 df-seq 13572 df-exp 13633 df-cj 14659 df-re 14660 df-im 14661 df-sqrt 14795 df-abs 14796 df-cncf 23772 |
This theorem is referenced by: cniccbdd 24355 iblabslem 24722 iblabs 24723 bddmulibl 24733 logcn 25532 ftalem3 25954 ftc1cnnclem 35583 ftc2nc 35594 evthiccabs 42707 cncficcgt0 43102 |
Copyright terms: Public domain | W3C validator |