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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 11193 | . . 3 β’ β β V | |
2 | 1 | a1i 11 | . 2 β’ (π β β β V) |
3 | xrex 12975 | . . 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 44760 | . . 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 44540 | 1 β’ (π β πΉ β (β* βpm β)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3468 β wss 3943 βΆwf 6533 βcfv 6537 (class class class)co 7405 βpm cpm 8823 βcc 11110 β*cxr 11251 β€β₯cuz 12826 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-pm 8825 df-xr 11256 df-neg 11451 df-z 12563 df-uz 12827 |
This theorem is referenced by: xlimconst2 45123 xlimclim2lem 45127 climxlim2 45134 xlimliminflimsup 45150 |
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