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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 37273 | . 2 ⊢ 𝐼 ∈ TopBases |
3 | tgtopon 22992 | . . 3 ⊢ (𝐼 ∈ TopBases → (topGen‘𝐼) ∈ (TopOn‘∪ 𝐼)) | |
4 | 1 | icoreunrn 37274 | . . . . 5 ⊢ ℝ = ∪ 𝐼 |
5 | 4 | eqcomi 2743 | . . . 4 ⊢ ∪ 𝐼 = ℝ |
6 | 5 | fveq2i 6922 | . . 3 ⊢ (TopOn‘∪ 𝐼) = (TopOn‘ℝ) |
7 | 3, 6 | eleqtrdi 2848 | . 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 1537 ∈ wcel 2103 ∪ cuni 4931 × cxp 5697 “ cima 5702 ‘cfv 6572 ℝcr 11179 [,)cico 13405 topGenctg 17492 TopOnctopon 22930 TopBasesctb 22966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-po 5611 df-so 5612 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-1st 8026 df-2nd 8027 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-ico 13409 df-topgen 17498 df-top 22914 df-topon 22931 df-bases 22967 |
This theorem is referenced by: (None) |
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