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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 2734 | . . 3 β’ (π§ = β¨π₯, π¦β© β π = π) | |
2 | 1 | mpompt 7474 | . 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 22965 | . . . 4 β’ ((π½ β (TopOnβπ) β§ πΎ β (TopOnβπ)) β (π½ Γt πΎ) β (TopOnβ(π Γ π))) | |
6 | 3, 4, 5 | syl2anc 585 | . . 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 23036 | . 2 β’ (π β (π§ β (π Γ π) β¦ π) β ((π½ Γt πΎ) Cn πΏ)) |
10 | 2, 9 | eqeltrrid 2839 | 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 1542 β wcel 2107 β¨cop 4596 β¦ cmpt 5192 Γ cxp 5635 βcfv 6500 (class class class)co 7361 β cmpo 7363 TopOnctopon 22282 Cn ccn 22598 Γt ctx 22934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 df-map 8773 df-topgen 17333 df-top 22266 df-topon 22283 df-bases 22319 df-cn 22601 df-cnp 22602 df-tx 22936 |
This theorem is referenced by: cnrehmeo 24339 pcopt 24408 pcopt2 24409 vmcn 29690 dipcn 29711 cvxsconn 33901 |
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