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Mirrors > Home > MPE Home > Th. List > letopon | Structured version Visualization version GIF version |
Description: The topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
letopon | ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | letsr 17580 | . 2 ⊢ ≤ ∈ TosetRel | |
2 | ledm 17577 | . . 3 ⊢ ℝ* = dom ≤ | |
3 | 2 | ordttopon 21368 | . 2 ⊢ ( ≤ ∈ TosetRel → (ordTop‘ ≤ ) ∈ (TopOn‘ℝ*)) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2166 ‘cfv 6123 ℝ*cxr 10390 ≤ cle 10392 ordTopcordt 16512 TosetRel ctsr 17552 TopOnctopon 21085 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-pre-lttri 10326 ax-pre-lttrn 10327 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fi 8586 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-topgen 16457 df-ordt 16514 df-ps 17553 df-tsr 17554 df-top 21069 df-topon 21086 df-bases 21121 |
This theorem is referenced by: letop 21381 letopuni 21382 xrstopn 21383 xrstps 21384 xmetdcn 23011 metdcn2 23012 xrlimcnp 25108 xrge0pluscn 30531 xrge0mulc1cn 30532 lmlimxrge0 30539 pnfneige0 30542 lmxrge0 30543 esumcvg 30693 xlimres 40842 xlimcl 40843 xlimconst 40846 xlimbr 40848 xlimmnfvlem1 40853 xlimmnfvlem2 40854 xlimpnfvlem1 40857 xlimpnfvlem2 40858 |
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