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Mirrors > Home > MPE Home > Th. List > ulmcl | Structured version Visualization version GIF version |
Description: Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.) |
Ref | Expression |
---|---|
ulmcl | β’ (πΉ(βπ’βπ)πΊ β πΊ:πβΆβ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ulmscl 25882 | . . . 4 β’ (πΉ(βπ’βπ)πΊ β π β V) | |
2 | ulmval 25883 | . . . 4 β’ (π β V β (πΉ(βπ’βπ)πΊ β βπ β β€ (πΉ:(β€β₯βπ)βΆ(β βm π) β§ πΊ:πβΆβ β§ βπ₯ β β+ βπ β (β€β₯βπ)βπ β (β€β₯βπ)βπ§ β π (absβ(((πΉβπ)βπ§) β (πΊβπ§))) < π₯))) | |
3 | 1, 2 | syl 17 | . . 3 β’ (πΉ(βπ’βπ)πΊ β (πΉ(βπ’βπ)πΊ β βπ β β€ (πΉ:(β€β₯βπ)βΆ(β βm π) β§ πΊ:πβΆβ β§ βπ₯ β β+ βπ β (β€β₯βπ)βπ β (β€β₯βπ)βπ§ β π (absβ(((πΉβπ)βπ§) β (πΊβπ§))) < π₯))) |
4 | 3 | ibi 266 | . 2 β’ (πΉ(βπ’βπ)πΊ β βπ β β€ (πΉ:(β€β₯βπ)βΆ(β βm π) β§ πΊ:πβΆβ β§ βπ₯ β β+ βπ β (β€β₯βπ)βπ β (β€β₯βπ)βπ§ β π (absβ(((πΉβπ)βπ§) β (πΊβπ§))) < π₯)) |
5 | simp2 1137 | . . 3 β’ ((πΉ:(β€β₯βπ)βΆ(β βm π) β§ πΊ:πβΆβ β§ βπ₯ β β+ βπ β (β€β₯βπ)βπ β (β€β₯βπ)βπ§ β π (absβ(((πΉβπ)βπ§) β (πΊβπ§))) < π₯) β πΊ:πβΆβ) | |
6 | 5 | rexlimivw 3151 | . 2 β’ (βπ β β€ (πΉ:(β€β₯βπ)βΆ(β βm π) β§ πΊ:πβΆβ β§ βπ₯ β β+ βπ β (β€β₯βπ)βπ β (β€β₯βπ)βπ§ β π (absβ(((πΉβπ)βπ§) β (πΊβπ§))) < π₯) β πΊ:πβΆβ) |
7 | 4, 6 | syl 17 | 1 β’ (πΉ(βπ’βπ)πΊ β πΊ:πβΆβ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1087 β wcel 2106 βwral 3061 βwrex 3070 Vcvv 3474 class class class wbr 5147 βΆwf 6536 βcfv 6540 (class class class)co 7405 βm cmap 8816 βcc 11104 < clt 11244 β cmin 11440 β€cz 12554 β€β₯cuz 12818 β+crp 12970 abscabs 15177 βπ’culm 25879 |
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-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 |
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-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-map 8818 df-pm 8819 df-neg 11443 df-z 12555 df-uz 12819 df-ulm 25880 |
This theorem is referenced by: ulmi 25889 ulmclm 25890 ulmres 25891 ulmshftlem 25892 ulmuni 25895 ulmcau 25898 ulmss 25900 ulmbdd 25901 ulmcn 25902 ulmdvlem1 25903 ulmdvlem3 25905 ulmdv 25906 mbfulm 25909 iblulm 25910 itgulm 25911 itgulm2 25912 pserulm 25925 lgamgulmlem6 26527 lgamgulm2 26529 knoppcnlem9 35365 |
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