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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 25754 | . . . 4 β’ (πΉ(βπ’βπ)πΊ β π β V) | |
2 | ulmval 25755 | . . . 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 3149 | . 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 3065 βwrex 3074 Vcvv 3448 class class class wbr 5110 βΆwf 6497 βcfv 6501 (class class class)co 7362 βm cmap 8772 βcc 11056 < clt 11196 β cmin 11392 β€cz 12506 β€β₯cuz 12770 β+crp 12922 abscabs 15126 βπ’culm 25751 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-map 8774 df-pm 8775 df-neg 11395 df-z 12507 df-uz 12771 df-ulm 25752 |
This theorem is referenced by: ulmi 25761 ulmclm 25762 ulmres 25763 ulmshftlem 25764 ulmuni 25767 ulmcau 25770 ulmss 25772 ulmbdd 25773 ulmcn 25774 ulmdvlem1 25775 ulmdvlem3 25777 ulmdv 25778 mbfulm 25781 iblulm 25782 itgulm 25783 itgulm2 25784 pserulm 25797 lgamgulmlem6 26399 lgamgulm2 26401 knoppcnlem9 34993 |
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