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Mirrors > Home > ILE Home > Th. List > cncfcncntop | GIF version |
Description: Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.) |
Ref | Expression |
---|---|
cncfcn.2 | β’ π½ = (MetOpenβ(abs β β )) |
cncfcn.3 | β’ πΎ = (π½ βΎt π΄) |
cncfcn.4 | β’ πΏ = (π½ βΎt π΅) |
Ref | Expression |
---|---|
cncfcncntop | β’ ((π΄ β β β§ π΅ β β) β (π΄βcnβπ΅) = (πΎ Cn πΏ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2177 | . . 3 β’ ((abs β β ) βΎ (π΄ Γ π΄)) = ((abs β β ) βΎ (π΄ Γ π΄)) | |
2 | eqid 2177 | . . 3 β’ ((abs β β ) βΎ (π΅ Γ π΅)) = ((abs β β ) βΎ (π΅ Γ π΅)) | |
3 | eqid 2177 | . . 3 β’ (MetOpenβ((abs β β ) βΎ (π΄ Γ π΄))) = (MetOpenβ((abs β β ) βΎ (π΄ Γ π΄))) | |
4 | eqid 2177 | . . 3 β’ (MetOpenβ((abs β β ) βΎ (π΅ Γ π΅))) = (MetOpenβ((abs β β ) βΎ (π΅ Γ π΅))) | |
5 | 1, 2, 3, 4 | cncfmet 14015 | . 2 β’ ((π΄ β β β§ π΅ β β) β (π΄βcnβπ΅) = ((MetOpenβ((abs β β ) βΎ (π΄ Γ π΄))) Cn (MetOpenβ((abs β β ) βΎ (π΅ Γ π΅))))) |
6 | cncfcn.3 | . . . 4 β’ πΎ = (π½ βΎt π΄) | |
7 | cnxmet 13967 | . . . . 5 β’ (abs β β ) β (βMetββ) | |
8 | simpl 109 | . . . . 5 β’ ((π΄ β β β§ π΅ β β) β π΄ β β) | |
9 | cncfcn.2 | . . . . . 6 β’ π½ = (MetOpenβ(abs β β )) | |
10 | 1, 9, 3 | metrest 13942 | . . . . 5 β’ (((abs β β ) β (βMetββ) β§ π΄ β β) β (π½ βΎt π΄) = (MetOpenβ((abs β β ) βΎ (π΄ Γ π΄)))) |
11 | 7, 8, 10 | sylancr 414 | . . . 4 β’ ((π΄ β β β§ π΅ β β) β (π½ βΎt π΄) = (MetOpenβ((abs β β ) βΎ (π΄ Γ π΄)))) |
12 | 6, 11 | eqtrid 2222 | . . 3 β’ ((π΄ β β β§ π΅ β β) β πΎ = (MetOpenβ((abs β β ) βΎ (π΄ Γ π΄)))) |
13 | cncfcn.4 | . . . 4 β’ πΏ = (π½ βΎt π΅) | |
14 | simpr 110 | . . . . 5 β’ ((π΄ β β β§ π΅ β β) β π΅ β β) | |
15 | 2, 9, 4 | metrest 13942 | . . . . 5 β’ (((abs β β ) β (βMetββ) β§ π΅ β β) β (π½ βΎt π΅) = (MetOpenβ((abs β β ) βΎ (π΅ Γ π΅)))) |
16 | 7, 14, 15 | sylancr 414 | . . . 4 β’ ((π΄ β β β§ π΅ β β) β (π½ βΎt π΅) = (MetOpenβ((abs β β ) βΎ (π΅ Γ π΅)))) |
17 | 13, 16 | eqtrid 2222 | . . 3 β’ ((π΄ β β β§ π΅ β β) β πΏ = (MetOpenβ((abs β β ) βΎ (π΅ Γ π΅)))) |
18 | 12, 17 | oveq12d 5892 | . 2 β’ ((π΄ β β β§ π΅ β β) β (πΎ Cn πΏ) = ((MetOpenβ((abs β β ) βΎ (π΄ Γ π΄))) Cn (MetOpenβ((abs β β ) βΎ (π΅ Γ π΅))))) |
19 | 5, 18 | eqtr4d 2213 | 1 β’ ((π΄ β β β§ π΅ β β) β (π΄βcnβπ΅) = (πΎ Cn πΏ)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1353 β wcel 2148 β wss 3129 Γ cxp 4624 βΎ cres 4628 β ccom 4630 βcfv 5216 (class class class)co 5874 βcc 7808 β cmin 8127 abscabs 11005 βΎt crest 12687 βMetcxmet 13376 MetOpencmopn 13381 Cn ccn 13621 βcnβccncf 13993 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-isom 5225 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-frec 6391 df-map 6649 df-sup 6982 df-inf 6983 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-sub 8129 df-neg 8130 df-reap 8531 df-ap 8538 df-div 8629 df-inn 8919 df-2 8977 df-3 8978 df-4 8979 df-n0 9176 df-z 9253 df-uz 9528 df-q 9619 df-rp 9653 df-xneg 9771 df-xadd 9772 df-seqfrec 10445 df-exp 10519 df-cj 10850 df-re 10851 df-im 10852 df-rsqrt 11006 df-abs 11007 df-rest 12689 df-topgen 12708 df-psmet 13383 df-xmet 13384 df-met 13385 df-bl 13386 df-mopn 13387 df-top 13434 df-topon 13447 df-bases 13479 df-cn 13624 df-cnp 13625 df-cncf 13994 |
This theorem is referenced by: cncfcn1cntop 14017 cncfmpt2fcntop 14021 cnrehmeocntop 14029 cnlimcim 14076 cnlimc 14077 dvcn 14100 |
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