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Mirrors > Home > MPE Home > Th. List > Mathboxes > tpr2tp | Structured version Visualization version GIF version |
Description: The usual topology on (ℝ × ℝ) is the product topology of the usual topology on ℝ. (Contributed by Thierry Arnoux, 21-Sep-2017.) |
Ref | Expression |
---|---|
tpr2tp.0 | ⊢ 𝐽 = (topGen‘ran (,)) |
Ref | Expression |
---|---|
tpr2tp | ⊢ (𝐽 ×t 𝐽) ∈ (TopOn‘(ℝ × ℝ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tpr2tp.0 | . . 3 ⊢ 𝐽 = (topGen‘ran (,)) | |
2 | retopon 23301 | . . 3 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) | |
3 | 1, 2 | eqeltri 2909 | . 2 ⊢ 𝐽 ∈ (TopOn‘ℝ) |
4 | txtopon 22129 | . 2 ⊢ ((𝐽 ∈ (TopOn‘ℝ) ∧ 𝐽 ∈ (TopOn‘ℝ)) → (𝐽 ×t 𝐽) ∈ (TopOn‘(ℝ × ℝ))) | |
5 | 3, 3, 4 | mp2an 688 | 1 ⊢ (𝐽 ×t 𝐽) ∈ (TopOn‘(ℝ × ℝ)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 × cxp 5547 ran crn 5550 ‘cfv 6349 (class class class)co 7145 ℝcr 10525 (,)cioo 12728 topGenctg 16701 TopOnctopon 21448 ×t ctx 22098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7680 df-2nd 7681 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-ioo 12732 df-topgen 16707 df-top 21432 df-topon 21449 df-bases 21484 df-tx 22100 |
This theorem is referenced by: tpr2uni 31048 sxbrsigalem4 31445 sxbrsiga 31448 |
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