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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fge0iccre | Structured version Visualization version GIF version |
Description: A range of nonnegative extended reals without plus infinity. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
fge0iccre.1 | β’ (π β πΉ:πβΆ(0[,]+β)) |
fge0iccre.2 | β’ (π β Β¬ +β β ran πΉ) |
Ref | Expression |
---|---|
fge0iccre | β’ (π β πΉ:πβΆβ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fge0iccre.1 | . . 3 β’ (π β πΉ:πβΆ(0[,]+β)) | |
2 | fge0iccre.2 | . . 3 β’ (π β Β¬ +β β ran πΉ) | |
3 | 1, 2 | fge0iccico 45793 | . 2 β’ (π β πΉ:πβΆ(0[,)+β)) |
4 | rge0ssre 13463 | . . 3 β’ (0[,)+β) β β | |
5 | 4 | a1i 11 | . 2 β’ (π β (0[,)+β) β β) |
6 | 3, 5 | fssd 6733 | 1 β’ (π β πΉ:πβΆβ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wcel 2098 β wss 3939 ran crn 5671 βΆwf 6537 (class class class)co 7414 βcr 11135 0cc0 11136 +βcpnf 11273 [,)cico 13356 [,]cicc 13357 |
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 2166 ax-ext 2696 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-addrcl 11197 ax-rnegex 11207 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5568 df-po 5582 df-so 5583 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7989 df-2nd 7990 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-ico 13360 df-icc 13361 |
This theorem is referenced by: sge0fsum 45810 |
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