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Mirrors > Home > MPE Home > Th. List > txtop | Structured version Visualization version GIF version |
Description: The product of two topologies is a topology. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
txtop | ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . 3 ⊢ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) | |
2 | 1 | txval 21367 | . 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) |
3 | topbas 20776 | . . . 4 ⊢ (𝑅 ∈ Top → 𝑅 ∈ TopBases) | |
4 | topbas 20776 | . . . 4 ⊢ (𝑆 ∈ Top → 𝑆 ∈ TopBases) | |
5 | 1 | txbas 21370 | . . . 4 ⊢ ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ TopBases) |
6 | 3, 4, 5 | syl2an 494 | . . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ TopBases) |
7 | tgcl 20773 | . . 3 ⊢ (ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ TopBases → (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))) ∈ Top) | |
8 | 6, 7 | syl 17 | . 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))) ∈ Top) |
9 | 2, 8 | eqeltrd 2701 | 1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 × cxp 5112 ran crn 5115 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 topGenctg 16098 Topctop 20698 TopBasesctb 20749 ×t ctx 21363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-topgen 16104 df-top 20699 df-bases 20750 df-tx 21365 |
This theorem is referenced by: txtopi 21393 txtopon 21394 txcld 21406 neitx 21410 txlly 21439 txnlly 21440 txcmplem1 21444 txcmp 21446 hausdiag 21448 txhaus 21450 tx1stc 21453 txkgen 21455 xkococn 21463 xkoinjcn 21490 txconn 21492 imasnopn 21493 imasncls 21495 utop2nei 22054 utop3cls 22055 qtophaus 29903 txpconn 31214 |
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