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Mirrors > Home > MPE Home > Th. List > cnmptc | 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 |
---|---|
cnmptid.j | β’ (π β π½ β (TopOnβπ)) |
cnmptc.k | β’ (π β πΎ β (TopOnβπ)) |
cnmptc.p | β’ (π β π β π) |
Ref | Expression |
---|---|
cnmptc | β’ (π β (π₯ β π β¦ π) β (π½ Cn πΎ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fconstmpt 5698 | . 2 β’ (π Γ {π}) = (π₯ β π β¦ π) | |
2 | cnmptid.j | . . 3 β’ (π β π½ β (TopOnβπ)) | |
3 | cnmptc.k | . . 3 β’ (π β πΎ β (TopOnβπ)) | |
4 | cnmptc.p | . . 3 β’ (π β π β π) | |
5 | cnconst2 22657 | . . 3 β’ ((π½ β (TopOnβπ) β§ πΎ β (TopOnβπ) β§ π β π) β (π Γ {π}) β (π½ Cn πΎ)) | |
6 | 2, 3, 4, 5 | syl3anc 1372 | . 2 β’ (π β (π Γ {π}) β (π½ Cn πΎ)) |
7 | 1, 6 | eqeltrrid 2839 | 1 β’ (π β (π₯ β π β¦ π) β (π½ Cn πΎ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2107 {csn 4590 β¦ cmpt 5192 Γ cxp 5635 βcfv 6500 (class class class)co 7361 TopOnctopon 22282 Cn ccn 22598 |
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-cn 22601 df-cnp 22602 |
This theorem is referenced by: cnmpt2c 23044 xkoinjcn 23061 txconn 23063 imasnopn 23064 imasncld 23065 imasncls 23066 istgp2 23465 tmdmulg 23466 tmdgsum 23469 tmdlactcn 23476 clsnsg 23484 tgpt0 23493 tlmtgp 23570 nmcn 24230 fsumcn 24256 expcn 24258 divccn 24259 cncfmptc 24298 cdivcncf 24307 iirevcn 24316 iihalf1cn 24318 iihalf2cn 24320 icchmeo 24327 evth 24345 evth2 24346 pcocn 24403 pcopt 24408 pcopt2 24409 pcoass 24410 csscld 24636 clsocv 24637 dvcnvlem 25363 plycn 25645 psercn2 25805 resqrtcn 26125 sqrtcn 26126 atansopn 26305 efrlim 26342 ipasslem7 29827 occllem 30294 rmulccn 32573 cxpcncf1 33272 txsconnlem 33898 cvxpconn 33900 cvmlift2lem2 33962 cvmlift2lem3 33963 cvmliftphtlem 33975 sinccvglem 34324 knoppcnlem10 35018 areacirclem2 36217 fprodcn 43931 |
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