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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 2728 | . . 3 β’ (π§ = β¨π₯, π¦β© β π = π) | |
2 | 1 | mpompt 7538 | . 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 23513 | . . . 4 β’ ((π½ β (TopOnβπ) β§ πΎ β (TopOnβπ)) β (π½ Γt πΎ) β (TopOnβ(π Γ π))) | |
6 | 3, 4, 5 | syl2anc 582 | . . 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 23584 | . 2 β’ (π β (π§ β (π Γ π) β¦ π) β ((π½ Γt πΎ) Cn πΏ)) |
10 | 2, 9 | eqeltrrid 2833 | 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 4636 β¦ cmpt 5233 Γ cxp 5678 βcfv 6551 (class class class)co 7424 β cmpo 7426 TopOnctopon 22830 Cn ccn 23146 Γt ctx 23482 |
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 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 7997 df-2nd 7998 df-map 8851 df-topgen 17430 df-top 22814 df-topon 22831 df-bases 22867 df-cn 23149 df-cnp 23150 df-tx 23484 |
This theorem is referenced by: cnrehmeo 24896 cnrehmeoOLD 24897 pcopt 24967 pcopt2 24968 vmcn 30527 dipcn 30548 cvxsconn 34858 |
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