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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fuzxrpmcn | Structured version Visualization version GIF version |
Description: A function mapping from an upper set of integers to the extended reals is a partial map on the complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
fuzxrpmcn.1 | β’ π = (β€β₯βπ) |
fuzxrpmcn.2 | β’ (π β πΉ:πβΆβ*) |
Ref | Expression |
---|---|
fuzxrpmcn | β’ (π β πΉ β (β* βpm β)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 11137 | . . 3 β’ β β V | |
2 | 1 | a1i 11 | . 2 β’ (π β β β V) |
3 | xrex 12917 | . . 3 β’ β* β V | |
4 | 3 | a1i 11 | . 2 β’ (π β β* β V) |
5 | fuzxrpmcn.1 | . . . 4 β’ π = (β€β₯βπ) | |
6 | 5 | uzsscn2 43799 | . . 3 β’ π β β |
7 | 6 | a1i 11 | . 2 β’ (π β π β β) |
8 | fuzxrpmcn.2 | . 2 β’ (π β πΉ:πβΆβ*) | |
9 | 2, 4, 7, 8 | fpmd 43579 | 1 β’ (π β πΉ β (β* βpm β)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3444 β wss 3911 βΆwf 6493 βcfv 6497 (class class class)co 7358 βpm cpm 8769 βcc 11054 β*cxr 11193 β€β₯cuz 12768 |
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-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 |
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 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-pm 8771 df-xr 11198 df-neg 11393 df-z 12505 df-uz 12769 |
This theorem is referenced by: xlimconst2 44162 xlimclim2lem 44166 climxlim2 44173 xlimliminflimsup 44189 |
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