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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 5740 | . 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 23231 | . . 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 2830 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑃) ∈ (𝐽 Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 {csn 4630 ↦ cmpt 5232 × cxp 5676 ‘cfv 6549 (class class class)co 7419 TopOnctopon 22856 Cn ccn 23172 |
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 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-map 8847 df-topgen 17428 df-top 22840 df-topon 22857 df-cn 23175 df-cnp 23176 |
This theorem is referenced by: cnmpt2c 23618 xkoinjcn 23635 txconn 23637 imasnopn 23638 imasncld 23639 imasncls 23640 istgp2 24039 tmdmulg 24040 tmdgsum 24043 tmdlactcn 24050 clsnsg 24058 tgpt0 24067 tlmtgp 24144 nmcn 24804 fsumcn 24832 expcn 24834 divccn 24835 expcnOLD 24836 divccnOLD 24837 cncfmptc 24876 cdivcncf 24885 iirevcn 24895 iihalf1cn 24897 iihalf1cnOLD 24898 iihalf2cn 24900 iihalf2cnOLD 24901 icchmeo 24909 icchmeoOLD 24910 evth 24929 evth2 24930 pcocn 24988 pcopt 24993 pcopt2 24994 pcoass 24995 csscld 25221 clsocv 25222 dvcnvlem 25952 plycn 26240 plycnOLD 26241 psercn2 26404 psercn2OLD 26405 resqrtcn 26729 sqrtcn 26730 atansopn 26909 efrlim 26946 efrlimOLD 26947 ipasslem7 30718 occllem 31185 rmulccn 33657 cxpcncf1 34355 txsconnlem 34978 cvxpconn 34980 cvmlift2lem2 35042 cvmlift2lem3 35043 cvmliftphtlem 35055 sinccvglem 35404 knoppcnlem10 36105 areacirclem2 37310 fprodcn 45123 |
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