![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cnmpt2c | Structured version Visualization version GIF version |
Description: A constant function is continuous. (Contributed by Mario Carneiro, 5-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)) |
cnmpt2c.l | β’ (π β πΏ β (TopOnβπ)) |
cnmpt2c.p | β’ (π β π β π) |
Ref | Expression |
---|---|
cnmpt2c | β’ (π β (π₯ β π, π¦ β π β¦ π) β ((π½ Γt πΎ) Cn πΏ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2727 | . . 3 β’ (π§ = β¨π₯, π¦β© β π = π) | |
2 | 1 | mpompt 7517 | . 2 β’ (π§ β (π Γ π) β¦ π) = (π₯ β π, π¦ β π β¦ π) |
3 | cnmpt21.j | . . . 4 β’ (π β π½ β (TopOnβπ)) | |
4 | cnmpt21.k | . . . 4 β’ (π β πΎ β (TopOnβπ)) | |
5 | txtopon 23446 | . . . 4 β’ ((π½ β (TopOnβπ) β§ πΎ β (TopOnβπ)) β (π½ Γt πΎ) β (TopOnβ(π Γ π))) | |
6 | 3, 4, 5 | syl2anc 583 | . . 3 β’ (π β (π½ Γt πΎ) β (TopOnβ(π Γ π))) |
7 | cnmpt2c.l | . . 3 β’ (π β πΏ β (TopOnβπ)) | |
8 | cnmpt2c.p | . . 3 β’ (π β π β π) | |
9 | 6, 7, 8 | cnmptc 23517 | . 2 β’ (π β (π§ β (π Γ π) β¦ π) β ((π½ Γt πΎ) Cn πΏ)) |
10 | 2, 9 | eqeltrrid 2832 | 1 β’ (π β (π₯ β π, π¦ β π β¦ π) β ((π½ Γt πΎ) Cn πΏ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β¨cop 4629 β¦ cmpt 5224 Γ cxp 5667 βcfv 6536 (class class class)co 7404 β cmpo 7406 TopOnctopon 22763 Cn ccn 23079 Γt ctx 23415 |
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 7721 |
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 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-map 8821 df-topgen 17396 df-top 22747 df-topon 22764 df-bases 22800 df-cn 23082 df-cnp 23083 df-tx 23417 |
This theorem is referenced by: cnrehmeo 24829 cnrehmeoOLD 24830 pcopt 24900 pcopt2 24901 vmcn 30457 dipcn 30478 cvxsconn 34762 |
Copyright terms: Public domain | W3C validator |