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Mirrors > Home > MPE Home > Th. List > Mathboxes > istoprelowl | Structured version Visualization version GIF version |
Description: The set of all closed-below, open-above intervals of reals generate a topology on the reals. (Contributed by ML, 27-Jul-2020.) |
Ref | Expression |
---|---|
istoprelowl.1 | ⊢ 𝐼 = ([,) “ (ℝ × ℝ)) |
Ref | Expression |
---|---|
istoprelowl | ⊢ (topGen‘𝐼) ∈ (TopOn‘ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istoprelowl.1 | . . 3 ⊢ 𝐼 = ([,) “ (ℝ × ℝ)) | |
2 | 1 | isbasisrelowl 33704 | . 2 ⊢ 𝐼 ∈ TopBases |
3 | tgtopon 21104 | . . 3 ⊢ (𝐼 ∈ TopBases → (topGen‘𝐼) ∈ (TopOn‘∪ 𝐼)) | |
4 | 1 | icoreunrn 33705 | . . . . 5 ⊢ ℝ = ∪ 𝐼 |
5 | 4 | eqcomi 2808 | . . . 4 ⊢ ∪ 𝐼 = ℝ |
6 | 5 | fveq2i 6414 | . . 3 ⊢ (TopOn‘∪ 𝐼) = (TopOn‘ℝ) |
7 | 3, 6 | syl6eleq 2888 | . 2 ⊢ (𝐼 ∈ TopBases → (topGen‘𝐼) ∈ (TopOn‘ℝ)) |
8 | 2, 7 | ax-mp 5 | 1 ⊢ (topGen‘𝐼) ∈ (TopOn‘ℝ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ∈ wcel 2157 ∪ cuni 4628 × cxp 5310 “ cima 5315 ‘cfv 6101 ℝcr 10223 [,)cico 12426 topGenctg 16413 TopOnctopon 21043 TopBasesctb 21078 |
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 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
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 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-po 5233 df-so 5234 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-1st 7401 df-2nd 7402 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-ico 12430 df-topgen 16419 df-top 21027 df-topon 21044 df-bases 21079 |
This theorem is referenced by: (None) |
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