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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 44697 | . 2 β’ (π β πΉ:πβΆ(0[,)+β)) |
4 | rge0ssre 13379 | . . 3 β’ (0[,)+β) β β | |
5 | 4 | a1i 11 | . 2 β’ (π β (0[,)+β) β β) |
6 | 3, 5 | fssd 6687 | 1 β’ (π β πΉ:πβΆβ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wcel 2107 β wss 3911 ran crn 5635 βΆwf 6493 (class class class)co 7358 βcr 11055 0cc0 11056 +βcpnf 11191 [,)cico 13272 [,]cicc 13273 |
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 ax-1cn 11114 ax-addrcl 11117 ax-rnegex 11127 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 |
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-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 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-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 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-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-ico 13276 df-icc 13277 |
This theorem is referenced by: sge0fsum 44714 |
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