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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 37065 | . 2 ⊢ 𝐼 ∈ TopBases |
3 | tgtopon 22965 | . . 3 ⊢ (𝐼 ∈ TopBases → (topGen‘𝐼) ∈ (TopOn‘∪ 𝐼)) | |
4 | 1 | icoreunrn 37066 | . . . . 5 ⊢ ℝ = ∪ 𝐼 |
5 | 4 | eqcomi 2735 | . . . 4 ⊢ ∪ 𝐼 = ℝ |
6 | 5 | fveq2i 6904 | . . 3 ⊢ (TopOn‘∪ 𝐼) = (TopOn‘ℝ) |
7 | 3, 6 | eleqtrdi 2836 | . 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 1534 ∈ wcel 2099 ∪ cuni 4913 × cxp 5680 “ cima 5685 ‘cfv 6554 ℝcr 11157 [,)cico 13380 topGenctg 17452 TopOnctopon 22903 TopBasesctb 22939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 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-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-ico 13384 df-topgen 17458 df-top 22887 df-topon 22904 df-bases 22940 |
This theorem is referenced by: (None) |
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