![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cnmpt2nd | Structured version Visualization version GIF version |
Description: The projection onto the second coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
cnmpt21.j | β’ (π β π½ β (TopOnβπ)) |
cnmpt21.k | β’ (π β πΎ β (TopOnβπ)) |
Ref | Expression |
---|---|
cnmpt2nd | β’ (π β (π₯ β π, π¦ β π β¦ π¦) β ((π½ Γt πΎ) Cn πΎ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fo2nd 7995 | . . . . . 6 β’ 2nd :VβontoβV | |
2 | fofn 6801 | . . . . . 6 β’ (2nd :VβontoβV β 2nd Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 β’ 2nd Fn V |
4 | ssv 4001 | . . . . 5 β’ (π Γ π) β V | |
5 | fnssres 6667 | . . . . 5 β’ ((2nd Fn V β§ (π Γ π) β V) β (2nd βΎ (π Γ π)) Fn (π Γ π)) | |
6 | 3, 4, 5 | mp2an 689 | . . . 4 β’ (2nd βΎ (π Γ π)) Fn (π Γ π) |
7 | dffn5 6944 | . . . 4 β’ ((2nd βΎ (π Γ π)) Fn (π Γ π) β (2nd βΎ (π Γ π)) = (π§ β (π Γ π) β¦ ((2nd βΎ (π Γ π))βπ§))) | |
8 | 6, 7 | mpbi 229 | . . 3 β’ (2nd βΎ (π Γ π)) = (π§ β (π Γ π) β¦ ((2nd βΎ (π Γ π))βπ§)) |
9 | fvres 6904 | . . . 4 β’ (π§ β (π Γ π) β ((2nd βΎ (π Γ π))βπ§) = (2nd βπ§)) | |
10 | 9 | mpteq2ia 5244 | . . 3 β’ (π§ β (π Γ π) β¦ ((2nd βΎ (π Γ π))βπ§)) = (π§ β (π Γ π) β¦ (2nd βπ§)) |
11 | vex 3472 | . . . . 5 β’ π₯ β V | |
12 | vex 3472 | . . . . 5 β’ π¦ β V | |
13 | 11, 12 | op2ndd 7985 | . . . 4 β’ (π§ = β¨π₯, π¦β© β (2nd βπ§) = π¦) |
14 | 13 | mpompt 7518 | . . 3 β’ (π§ β (π Γ π) β¦ (2nd βπ§)) = (π₯ β π, π¦ β π β¦ π¦) |
15 | 8, 10, 14 | 3eqtri 2758 | . 2 β’ (2nd βΎ (π Γ π)) = (π₯ β π, π¦ β π β¦ π¦) |
16 | cnmpt21.j | . . 3 β’ (π β π½ β (TopOnβπ)) | |
17 | cnmpt21.k | . . 3 β’ (π β πΎ β (TopOnβπ)) | |
18 | tx2cn 23469 | . . 3 β’ ((π½ β (TopOnβπ) β§ πΎ β (TopOnβπ)) β (2nd βΎ (π Γ π)) β ((π½ Γt πΎ) Cn πΎ)) | |
19 | 16, 17, 18 | syl2anc 583 | . 2 β’ (π β (2nd βΎ (π Γ π)) β ((π½ Γt πΎ) Cn πΎ)) |
20 | 15, 19 | eqeltrrid 2832 | 1 β’ (π β (π₯ β π, π¦ β π β¦ π¦) β ((π½ Γt πΎ) Cn πΎ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3468 β wss 3943 β¦ cmpt 5224 Γ cxp 5667 βΎ cres 5671 Fn wfn 6532 βontoβwfo 6535 βcfv 6537 (class class class)co 7405 β cmpo 7407 2nd c2nd 7973 TopOnctopon 22767 Cn ccn 23083 Γt ctx 23419 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fo 6543 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-map 8824 df-topgen 17398 df-top 22751 df-topon 22768 df-bases 22804 df-cn 23086 df-tx 23421 |
This theorem is referenced by: cnmptcom 23537 xkofvcn 23543 cnmptk2 23545 txhmeo 23662 txswaphmeo 23664 ptunhmeo 23667 xkohmeo 23674 tgpsubcn 23949 istgp2 23950 oppgtmd 23956 prdstmdd 23983 dvrcn 24043 divcnOLD 24739 divcn 24741 cnrehmeo 24833 cnrehmeoOLD 24834 htpycom 24857 htpyco1 24859 htpycc 24861 reparphti 24878 reparphtiOLD 24879 pcohtpylem 24901 pcorevlem 24908 cxpcn 26634 cxpcnOLD 26635 vmcn 30461 dipcn 30482 mndpluscn 33436 cvxsconn 34762 cvmlift2lem6 34827 |
Copyright terms: Public domain | W3C validator |