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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 25891 | . . . 4 β’ (πΉ(βπ’βπ)πΊ β π β V) | |
2 | ulmval 25892 | . . . 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 267 | . 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 1138 | . . 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 3152 | . 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 1088 β wcel 2107 βwral 3062 βwrex 3071 Vcvv 3475 class class class wbr 5149 βΆwf 6540 βcfv 6544 (class class class)co 7409 βm cmap 8820 βcc 11108 < clt 11248 β cmin 11444 β€cz 12558 β€β₯cuz 12822 β+crp 12974 abscabs 15181 βπ’culm 25888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-map 8822 df-pm 8823 df-neg 11447 df-z 12559 df-uz 12823 df-ulm 25889 |
This theorem is referenced by: ulmi 25898 ulmclm 25899 ulmres 25900 ulmshftlem 25901 ulmuni 25904 ulmcau 25907 ulmss 25909 ulmbdd 25910 ulmcn 25911 ulmdvlem1 25912 ulmdvlem3 25914 ulmdv 25915 mbfulm 25918 iblulm 25919 itgulm 25920 itgulm2 25921 pserulm 25934 lgamgulmlem6 26538 lgamgulm2 26540 knoppcnlem9 35377 |
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