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Mirrors > Home > MPE Home > Th. List > xrsbas | Structured version Visualization version GIF version |
Description: The base set of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
xrsbas | ⊢ ℝ* = (Base‘ℝ*𝑠) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrex 12068 | . 2 ⊢ ℝ* ∈ V | |
2 | df-xrs 16474 | . . 3 ⊢ ℝ*𝑠 = ({〈(Base‘ndx), ℝ*〉, 〈(+g‘ndx), +𝑒 〉, 〈(.r‘ndx), ·e 〉} ∪ {〈(TopSet‘ndx), (ordTop‘ ≤ )〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))〉}) | |
3 | 2 | odrngbas 16379 | . 2 ⊢ (ℝ* ∈ V → ℝ* = (Base‘ℝ*𝑠)) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ ℝ* = (Base‘ℝ*𝑠) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ∈ wcel 2157 Vcvv 3384 ifcif 4276 class class class wbr 4842 ‘cfv 6100 (class class class)co 6877 ↦ cmpt2 6879 ℝ*cxr 10361 ≤ cle 10363 -𝑒cxne 12187 +𝑒 cxad 12188 ·e cxmu 12189 Basecbs 16181 ordTopcordt 16471 ℝ*𝑠cxrs 16472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 ax-cnex 10279 ax-resscn 10280 ax-1cn 10281 ax-icn 10282 ax-addcl 10283 ax-addrcl 10284 ax-mulcl 10285 ax-mulrcl 10286 ax-mulcom 10287 ax-addass 10288 ax-mulass 10289 ax-distr 10290 ax-i2m1 10291 ax-1ne0 10292 ax-1rid 10293 ax-rnegex 10294 ax-rrecex 10295 ax-cnre 10296 ax-pre-lttri 10297 ax-pre-lttrn 10298 ax-pre-ltadd 10299 ax-pre-mulgt0 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-pss 3784 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-tp 4372 df-op 4374 df-uni 4628 df-int 4667 df-iun 4711 df-br 4843 df-opab 4905 df-mpt 4922 df-tr 4945 df-id 5219 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-we 5272 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-pred 5897 df-ord 5943 df-on 5944 df-lim 5945 df-suc 5946 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-riota 6838 df-ov 6880 df-oprab 6881 df-mpt2 6882 df-om 7299 df-1st 7400 df-2nd 7401 df-wrecs 7644 df-recs 7706 df-rdg 7744 df-1o 7798 df-oadd 7802 df-er 7981 df-en 8195 df-dom 8196 df-sdom 8197 df-fin 8198 df-pnf 10364 df-mnf 10365 df-xr 10366 df-ltxr 10367 df-le 10368 df-sub 10557 df-neg 10558 df-nn 11312 df-2 11373 df-3 11374 df-4 11375 df-5 11376 df-6 11377 df-7 11378 df-8 11379 df-9 11380 df-n0 11578 df-z 11664 df-dec 11781 df-uz 11928 df-fz 12578 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-plusg 16277 df-mulr 16278 df-tset 16283 df-ple 16284 df-ds 16286 df-xrs 16474 |
This theorem is referenced by: xrsmcmn 20088 xrsmgm 20100 xrsnsgrp 20101 xrs1mnd 20103 xrs10 20104 xrs1cmn 20105 xrge0subm 20106 xrge0cmn 20107 xrstopn 21338 xrstps 21339 imasdsf1olem 22503 xrge0gsumle 22961 xrge0tsms 22962 xrs0 30184 xrsinvgval 30186 xrsmulgzz 30187 xrstos 30188 xrsclat 30189 xrsp0 30190 xrsp1 30191 xrge0base 30194 xrge00 30195 xrge0mulgnn0 30198 pnfinf 30246 xrnarchi 30247 xrge0tsmsd 30294 xrge0tmdOLD 30500 esumpfinvallem 30645 gsumge0cl 41320 sge0tsms 41329 |
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