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Mirrors > Home > MPE Home > Th. List > ulmf | Structured version Visualization version GIF version |
Description: Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.) |
Ref | Expression |
---|---|
ulmf | β’ (πΉ(βπ’βπ)πΊ β βπ β β€ πΉ:(β€β₯βπ)βΆ(β βm π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ulmscl 26309 | . . . 4 β’ (πΉ(βπ’βπ)πΊ β π β V) | |
2 | ulmval 26310 | . . . 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 | simp1 1134 | . . 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 πΉ:(β€β₯βπ)βΆ(β βm π)) | |
6 | 5 | reximi 3080 | . 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 βπ β β€ πΉ:(β€β₯βπ)βΆ(β βm π)) |
7 | 4, 6 | syl 17 | 1 β’ (πΉ(βπ’βπ)πΊ β βπ β β€ πΉ:(β€β₯βπ)βΆ(β βm π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1085 β wcel 2099 βwral 3057 βwrex 3066 Vcvv 3470 class class class wbr 5143 βΆwf 6539 βcfv 6543 (class class class)co 7415 βm cmap 8839 βcc 11131 < clt 11273 β cmin 11469 β€cz 12583 β€β₯cuz 12847 β+crp 13001 abscabs 15208 βπ’culm 26306 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 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-ov 7418 df-oprab 7419 df-mpo 7420 df-map 8841 df-pm 8842 df-neg 11472 df-z 12584 df-uz 12848 df-ulm 26307 |
This theorem is referenced by: ulmpm 26313 ulmuni 26322 ulmss 26327 |
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