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Mirrors > Home > MPE Home > Th. List > txtopon | Structured version Visualization version GIF version |
Description: The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 22-Aug-2015.) (Revised by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
txtopon | ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topontop 21205 | . . 3 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top) | |
2 | topontop 21205 | . . 3 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top) | |
3 | txtop 21861 | . . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top) | |
4 | 1, 2, 3 | syl2an 595 | . 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ Top) |
5 | eqid 2795 | . . . . 5 ⊢ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) | |
6 | eqid 2795 | . . . . 5 ⊢ ∪ 𝑅 = ∪ 𝑅 | |
7 | eqid 2795 | . . . . 5 ⊢ ∪ 𝑆 = ∪ 𝑆 | |
8 | 5, 6, 7 | txuni2 21857 | . . . 4 ⊢ (∪ 𝑅 × ∪ 𝑆) = ∪ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) |
9 | toponuni 21206 | . . . . 5 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑅) | |
10 | toponuni 21206 | . . . . 5 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝑆) | |
11 | xpeq12 5468 | . . . . 5 ⊢ ((𝑋 = ∪ 𝑅 ∧ 𝑌 = ∪ 𝑆) → (𝑋 × 𝑌) = (∪ 𝑅 × ∪ 𝑆)) | |
12 | 9, 10, 11 | syl2an 595 | . . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = (∪ 𝑅 × ∪ 𝑆)) |
13 | 5 | txbasex 21858 | . . . . 5 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ V) |
14 | unitg 21259 | . . . . 5 ⊢ (ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ V → ∪ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))) = ∪ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∪ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))) = ∪ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))) |
16 | 8, 12, 15 | 3eqtr4a 2857 | . . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = ∪ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) |
17 | 5 | txval 21856 | . . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) |
18 | 17 | unieqd 4755 | . . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∪ (𝑅 ×t 𝑆) = ∪ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) |
19 | 16, 18 | eqtr4d 2834 | . 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆)) |
20 | istopon 21204 | . 2 ⊢ ((𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆))) | |
21 | 4, 19, 20 | sylanbrc 583 | 1 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 Vcvv 3437 ∪ cuni 4745 × cxp 5441 ran crn 5444 ‘cfv 6225 (class class class)co 7016 ∈ cmpo 7018 topGenctg 16540 Topctop 21185 TopOnctopon 21202 ×t ctx 21852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-fv 6233 df-ov 7019 df-oprab 7020 df-mpo 7021 df-1st 7545 df-2nd 7546 df-topgen 16546 df-top 21186 df-topon 21203 df-bases 21238 df-tx 21854 |
This theorem is referenced by: txuni 21884 txcls 21896 tx1cn 21901 tx2cn 21902 txcnp 21912 txcnmpt 21916 txindis 21926 txdis1cn 21927 txlm 21940 lmcn2 21941 xkococn 21952 cnmpt12 21959 cnmpt2c 21962 cnmpt21 21963 cnmpt2t 21965 cnmpt22 21966 cnmpt22f 21967 cnmpt2res 21969 cnmptcom 21970 cnmpt2k 21980 ptunhmeo 22100 xpstopnlem1 22101 xkocnv 22106 xkohmeo 22107 txflf 22298 flfcnp2 22299 cnmpt2plusg 22380 tmdcn2 22381 indistgp 22392 clssubg 22400 qustgplem 22412 prdstmdd 22415 tsmsadd 22438 cnmpt2vsca 22486 txmetcn 22841 cnmpt2ds 23134 fsum2cn 23162 cnmpopc 23215 htpyco2 23266 phtpyco2 23277 cnmpt2ip 23534 limccnp2 24173 dvcnp2 24200 dvaddbr 24218 dvmulbr 24219 dvcobr 24226 lhop1lem 24293 taylthlem2 24645 cxpcn3 25010 tpr2tp 30764 txsconnlem 32095 txsconn 32096 cvmlift2lem11 32168 cvmlift2lem12 32169 |
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