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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 | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
cnmptc.k | ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) |
cnmptc.p | ⊢ (𝜑 → 𝑃 ∈ 𝑌) |
Ref | Expression |
---|---|
cnmptc | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑃) ∈ (𝐽 Cn 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fconstmpt 5578 | . 2 ⊢ (𝑋 × {𝑃}) = (𝑥 ∈ 𝑋 ↦ 𝑃) | |
2 | cnmptid.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
3 | cnmptc.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) | |
4 | cnmptc.p | . . 3 ⊢ (𝜑 → 𝑃 ∈ 𝑌) | |
5 | cnconst2 21888 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑌) → (𝑋 × {𝑃}) ∈ (𝐽 Cn 𝐾)) | |
6 | 2, 3, 4, 5 | syl3anc 1368 | . 2 ⊢ (𝜑 → (𝑋 × {𝑃}) ∈ (𝐽 Cn 𝐾)) |
7 | 1, 6 | eqeltrrid 2895 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑃) ∈ (𝐽 Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 {csn 4525 ↦ cmpt 5110 × cxp 5517 ‘cfv 6324 (class class class)co 7135 TopOnctopon 21515 Cn ccn 21829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-map 8391 df-topgen 16709 df-top 21499 df-topon 21516 df-cn 21832 df-cnp 21833 |
This theorem is referenced by: cnmpt2c 22275 xkoinjcn 22292 txconn 22294 imasnopn 22295 imasncld 22296 imasncls 22297 istgp2 22696 tmdmulg 22697 tmdgsum 22700 tmdlactcn 22707 clsnsg 22715 tgpt0 22724 tlmtgp 22801 nmcn 23449 fsumcn 23475 expcn 23477 divccn 23478 cncfmptc 23517 cdivcncf 23526 iirevcn 23535 iihalf1cn 23537 iihalf2cn 23539 icchmeo 23546 evth 23564 evth2 23565 pcocn 23622 pcopt 23627 pcopt2 23628 pcoass 23629 csscld 23853 clsocv 23854 dvcnvlem 24579 plycn 24858 psercn2 25018 resqrtcn 25338 sqrtcn 25339 atansopn 25518 efrlim 25555 ipasslem7 28619 occllem 29086 rmulccn 31281 cxpcncf1 31976 txsconnlem 32600 cvxpconn 32602 cvmlift2lem2 32664 cvmlift2lem3 32665 cvmliftphtlem 32677 sinccvglem 33028 knoppcnlem10 33954 areacirclem2 35146 fprodcn 42242 |
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