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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > frexr | Structured version Visualization version GIF version |
Description: A function taking real values, is a function taking extended real values. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
frexr.1 | β’ (π β πΉ:π΄βΆβ) |
Ref | Expression |
---|---|
frexr | β’ (π β πΉ:π΄βΆβ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frexr.1 | . 2 β’ (π β πΉ:π΄βΆβ) | |
2 | ressxr 11254 | . . 3 β’ β β β* | |
3 | 2 | a1i 11 | . 2 β’ (π β β β β*) |
4 | 1, 3 | fssd 6732 | 1 β’ (π β πΉ:π΄βΆβ*) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wss 3947 βΆwf 6536 βcr 11105 β*cxr 11243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-un 3952 df-in 3954 df-ss 3964 df-f 6544 df-xr 11248 |
This theorem is referenced by: limsupubuz 44415 limsupreuz 44439 limsupvaluz2 44440 supcnvlimsup 44442 limsupgtlem 44479 liminflimsupclim 44509 climliminflimsup2 44511 climliminflimsup3 44512 climliminflimsup4 44513 xlimliminflimsup 44564 preimaioomnf 45421 incsmf 45444 issmfle 45447 decsmf 45469 smfsupdmmbllem 45546 smfinfdmmbllem 45550 |
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