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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cncfres | Structured version Visualization version GIF version |
Description: A continuous function on complex numbers restricted to a subset. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.) |
Ref | Expression |
---|---|
cncfres.1 | β’ π΄ β β |
cncfres.2 | β’ π΅ β β |
cncfres.3 | β’ πΉ = (π₯ β β β¦ πΆ) |
cncfres.4 | β’ πΊ = (π₯ β π΄ β¦ πΆ) |
cncfres.5 | β’ (π₯ β π΄ β πΆ β π΅) |
cncfres.6 | β’ πΉ β (ββcnββ) |
cncfres.7 | β’ π½ = (MetOpenβ((abs β β ) βΎ (π΄ Γ π΄))) |
cncfres.8 | β’ πΎ = (MetOpenβ((abs β β ) βΎ (π΅ Γ π΅))) |
Ref | Expression |
---|---|
cncfres | β’ πΊ β (π½ Cn πΎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncfres.4 | . . . 4 β’ πΊ = (π₯ β π΄ β¦ πΆ) | |
2 | cncfres.5 | . . . 4 β’ (π₯ β π΄ β πΆ β π΅) | |
3 | 1, 2 | fmpti 7113 | . . 3 β’ πΊ:π΄βΆπ΅ |
4 | cncfres.2 | . . . 4 β’ π΅ β β | |
5 | cncfres.1 | . . . . . . 7 β’ π΄ β β | |
6 | resmpt 6037 | . . . . . . 7 β’ (π΄ β β β ((π₯ β β β¦ πΆ) βΎ π΄) = (π₯ β π΄ β¦ πΆ)) | |
7 | 5, 6 | ax-mp 5 | . . . . . 6 β’ ((π₯ β β β¦ πΆ) βΎ π΄) = (π₯ β π΄ β¦ πΆ) |
8 | 1, 7 | eqtr4i 2763 | . . . . 5 β’ πΊ = ((π₯ β β β¦ πΆ) βΎ π΄) |
9 | cncfres.3 | . . . . . . 7 β’ πΉ = (π₯ β β β¦ πΆ) | |
10 | cncfres.6 | . . . . . . 7 β’ πΉ β (ββcnββ) | |
11 | 9, 10 | eqeltrri 2830 | . . . . . 6 β’ (π₯ β β β¦ πΆ) β (ββcnββ) |
12 | rescncf 24637 | . . . . . 6 β’ (π΄ β β β ((π₯ β β β¦ πΆ) β (ββcnββ) β ((π₯ β β β¦ πΆ) βΎ π΄) β (π΄βcnββ))) | |
13 | 5, 11, 12 | mp2 9 | . . . . 5 β’ ((π₯ β β β¦ πΆ) βΎ π΄) β (π΄βcnββ) |
14 | 8, 13 | eqeltri 2829 | . . . 4 β’ πΊ β (π΄βcnββ) |
15 | cncfcdm 24638 | . . . 4 β’ ((π΅ β β β§ πΊ β (π΄βcnββ)) β (πΊ β (π΄βcnβπ΅) β πΊ:π΄βΆπ΅)) | |
16 | 4, 14, 15 | mp2an 690 | . . 3 β’ (πΊ β (π΄βcnβπ΅) β πΊ:π΄βΆπ΅) |
17 | 3, 16 | mpbir 230 | . 2 β’ πΊ β (π΄βcnβπ΅) |
18 | eqid 2732 | . . . 4 β’ ((abs β β ) βΎ (π΄ Γ π΄)) = ((abs β β ) βΎ (π΄ Γ π΄)) | |
19 | eqid 2732 | . . . 4 β’ ((abs β β ) βΎ (π΅ Γ π΅)) = ((abs β β ) βΎ (π΅ Γ π΅)) | |
20 | cncfres.7 | . . . 4 β’ π½ = (MetOpenβ((abs β β ) βΎ (π΄ Γ π΄))) | |
21 | cncfres.8 | . . . 4 β’ πΎ = (MetOpenβ((abs β β ) βΎ (π΅ Γ π΅))) | |
22 | 18, 19, 20, 21 | cncfmet 24649 | . . 3 β’ ((π΄ β β β§ π΅ β β) β (π΄βcnβπ΅) = (π½ Cn πΎ)) |
23 | 5, 4, 22 | mp2an 690 | . 2 β’ (π΄βcnβπ΅) = (π½ Cn πΎ) |
24 | 17, 23 | eleqtri 2831 | 1 β’ πΊ β (π½ Cn πΎ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1541 β wcel 2106 β wss 3948 β¦ cmpt 5231 Γ cxp 5674 βΎ cres 5678 β ccom 5680 βΆwf 6539 βcfv 6543 (class class class)co 7411 βcc 11110 β cmin 11448 abscabs 15185 MetOpencmopn 21134 Cn ccn 22948 βcnβccncf 24616 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-seq 13971 df-exp 14032 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-topgen 17393 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-top 22616 df-topon 22633 df-bases 22669 df-cn 22951 df-cnp 22952 df-cncf 24618 |
This theorem is referenced by: (None) |
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