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Mirrors > Home > MPE Home > Th. List > ulmi | Structured version Visualization version GIF version |
Description: The uniform limit property. (Contributed by Mario Carneiro, 27-Feb-2015.) |
Ref | Expression |
---|---|
ulm2.z | β’ π = (β€β₯βπ) |
ulm2.m | β’ (π β π β β€) |
ulm2.f | β’ (π β πΉ:πβΆ(β βm π)) |
ulm2.b | β’ ((π β§ (π β π β§ π§ β π)) β ((πΉβπ)βπ§) = π΅) |
ulm2.a | β’ ((π β§ π§ β π) β (πΊβπ§) = π΄) |
ulmi.u | β’ (π β πΉ(βπ’βπ)πΊ) |
ulmi.c | β’ (π β πΆ β β+) |
Ref | Expression |
---|---|
ulmi | β’ (π β βπ β π βπ β (β€β₯βπ)βπ§ β π (absβ(π΅ β π΄)) < πΆ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 5114 | . . . 4 β’ (π₯ = πΆ β ((absβ(π΅ β π΄)) < π₯ β (absβ(π΅ β π΄)) < πΆ)) | |
2 | 1 | ralbidv 3175 | . . 3 β’ (π₯ = πΆ β (βπ§ β π (absβ(π΅ β π΄)) < π₯ β βπ§ β π (absβ(π΅ β π΄)) < πΆ)) |
3 | 2 | rexralbidv 3215 | . 2 β’ (π₯ = πΆ β (βπ β π βπ β (β€β₯βπ)βπ§ β π (absβ(π΅ β π΄)) < π₯ β βπ β π βπ β (β€β₯βπ)βπ§ β π (absβ(π΅ β π΄)) < πΆ)) |
4 | ulmi.u | . . 3 β’ (π β πΉ(βπ’βπ)πΊ) | |
5 | ulm2.z | . . . 4 β’ π = (β€β₯βπ) | |
6 | ulm2.m | . . . 4 β’ (π β π β β€) | |
7 | ulm2.f | . . . 4 β’ (π β πΉ:πβΆ(β βm π)) | |
8 | ulm2.b | . . . 4 β’ ((π β§ (π β π β§ π§ β π)) β ((πΉβπ)βπ§) = π΅) | |
9 | ulm2.a | . . . 4 β’ ((π β§ π§ β π) β (πΊβπ§) = π΄) | |
10 | ulmcl 25756 | . . . . 5 β’ (πΉ(βπ’βπ)πΊ β πΊ:πβΆβ) | |
11 | 4, 10 | syl 17 | . . . 4 β’ (π β πΊ:πβΆβ) |
12 | ulmscl 25754 | . . . . 5 β’ (πΉ(βπ’βπ)πΊ β π β V) | |
13 | 4, 12 | syl 17 | . . . 4 β’ (π β π β V) |
14 | 5, 6, 7, 8, 9, 11, 13 | ulm2 25760 | . . 3 β’ (π β (πΉ(βπ’βπ)πΊ β βπ₯ β β+ βπ β π βπ β (β€β₯βπ)βπ§ β π (absβ(π΅ β π΄)) < π₯)) |
15 | 4, 14 | mpbid 231 | . 2 β’ (π β βπ₯ β β+ βπ β π βπ β (β€β₯βπ)βπ§ β π (absβ(π΅ β π΄)) < π₯) |
16 | ulmi.c | . 2 β’ (π β πΆ β β+) | |
17 | 3, 15, 16 | rspcdva 3585 | 1 β’ (π β βπ β π βπ β (β€β₯βπ)βπ§ β π (absβ(π΅ β π΄)) < πΆ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 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 ax-pre-lttri 11132 ax-pre-lttrn 11133 |
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-nel 3051 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-po 5550 df-so 5551 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-er 8655 df-map 8774 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-neg 11395 df-z 12507 df-uz 12771 df-ulm 25752 |
This theorem is referenced by: ulmshftlem 25764 ulmcau 25770 ulmbdd 25773 ulmcn 25774 iblulm 25782 itgulm 25783 |
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