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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 2737 | . . 3 ⊢ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) | |
| 2 | 1 | txval 23572 | . 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) |
| 3 | topbas 22979 | . . . 4 ⊢ (𝑅 ∈ Top → 𝑅 ∈ TopBases) | |
| 4 | topbas 22979 | . . . 4 ⊢ (𝑆 ∈ Top → 𝑆 ∈ TopBases) | |
| 5 | 1 | txbas 23575 | . . . 4 ⊢ ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ TopBases) |
| 6 | 3, 4, 5 | syl2an 596 | . . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ TopBases) |
| 7 | tgcl 22976 | . . 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 2841 | 1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 × cxp 5683 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 topGenctg 17482 Topctop 22899 TopBasesctb 22952 ×t ctx 23568 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-topgen 17488 df-top 22900 df-bases 22953 df-tx 23570 |
| This theorem is referenced by: txtopi 23598 txtopon 23599 txcld 23611 neitx 23615 txlly 23644 txnlly 23645 txcmplem1 23649 txcmp 23651 hausdiag 23653 txhaus 23655 tx1stc 23658 txkgen 23660 xkococn 23668 xkoinjcn 23695 txconn 23697 imasnopn 23698 imasncls 23700 utop2nei 24259 utop3cls 24260 qtophaus 33835 txpconn 35237 |
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