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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 18592 | . 2 ⊢ ≤ ∈ TosetRel | |
2 | ledm 18589 | . . 3 ⊢ ℝ* = dom ≤ | |
3 | 2 | ordttopon 23117 | . 2 ⊢ ( ≤ ∈ TosetRel → (ordTop‘ ≤ ) ∈ (TopOn‘ℝ*)) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ‘cfv 6553 ℝ*cxr 11285 ≤ cle 11287 ordTopcordt 17488 TosetRel ctsr 18564 TopOnctopon 22832 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-pre-lttri 11220 ax-pre-lttrn 11221 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-om 7877 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fi 9442 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-topgen 17432 df-ordt 17490 df-ps 18565 df-tsr 18566 df-top 22816 df-topon 22833 df-bases 22869 |
This theorem is referenced by: letop 23130 letopuni 23131 xrstopn 23132 xrstps 23133 xmetdcn 24774 metdcn2 24775 xrlimcnp 26920 xrge0pluscn 33574 xrge0mulc1cn 33575 lmlimxrge0 33582 pnfneige0 33585 lmxrge0 33586 esumcvg 33738 xlimres 45238 xlimcl 45239 xlimconst 45242 xlimbr 45244 xlimmnfvlem1 45249 xlimmnfvlem2 45250 xlimpnfvlem1 45253 xlimpnfvlem2 45254 |
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