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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 23908 | . . 3 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) | |
3 | 1, 2 | eqeltri 2836 | . 2 ⊢ 𝐽 ∈ (TopOn‘ℝ) |
4 | txtopon 22723 | . 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 1541 ∈ wcel 2109 × cxp 5586 ran crn 5589 ‘cfv 6430 (class class class)co 7268 ℝcr 10854 (,)cioo 13061 topGenctg 17129 TopOnctopon 22040 ×t ctx 22692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-pre-lttri 10929 ax-pre-lttrn 10930 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-po 5502 df-so 5503 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-ov 7271 df-oprab 7272 df-mpo 7273 df-1st 7817 df-2nd 7818 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-ioo 13065 df-topgen 17135 df-top 22024 df-topon 22041 df-bases 22077 df-tx 22694 |
This theorem is referenced by: tpr2uni 31834 sxbrsigalem4 32233 sxbrsiga 32236 |
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