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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 17839 | . 2 ⊢ ≤ ∈ TosetRel | |
2 | ledm 17836 | . . 3 ⊢ ℝ* = dom ≤ | |
3 | 2 | ordttopon 21803 | . 2 ⊢ ( ≤ ∈ TosetRel → (ordTop‘ ≤ ) ∈ (TopOn‘ℝ*)) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 ‘cfv 6357 ℝ*cxr 10676 ≤ cle 10678 ordTopcordt 16774 TosetRel ctsr 17811 TopOnctopon 21520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fi 8877 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-topgen 16719 df-ordt 16776 df-ps 17812 df-tsr 17813 df-top 21504 df-topon 21521 df-bases 21556 |
This theorem is referenced by: letop 21816 letopuni 21817 xrstopn 21818 xrstps 21819 xmetdcn 23448 metdcn2 23449 xrlimcnp 25548 xrge0pluscn 31185 xrge0mulc1cn 31186 lmlimxrge0 31193 pnfneige0 31196 lmxrge0 31197 esumcvg 31347 xlimres 42109 xlimcl 42110 xlimconst 42113 xlimbr 42115 xlimmnfvlem1 42120 xlimmnfvlem2 42121 xlimpnfvlem1 42124 xlimpnfvlem2 42125 |
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