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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 5750 | . 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 23306 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑌) → (𝑋 × {𝑃}) ∈ (𝐽 Cn 𝐾)) | |
6 | 2, 3, 4, 5 | syl3anc 1370 | . 2 ⊢ (𝜑 → (𝑋 × {𝑃}) ∈ (𝐽 Cn 𝐾)) |
7 | 1, 6 | eqeltrrid 2843 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑃) ∈ (𝐽 Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 {csn 4630 ↦ cmpt 5230 × cxp 5686 ‘cfv 6562 (class class class)co 7430 TopOnctopon 22931 Cn ccn 23247 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-map 8866 df-topgen 17489 df-top 22915 df-topon 22932 df-cn 23250 df-cnp 23251 |
This theorem is referenced by: cnmpt2c 23693 xkoinjcn 23710 txconn 23712 imasnopn 23713 imasncld 23714 imasncls 23715 istgp2 24114 tmdmulg 24115 tmdgsum 24118 tmdlactcn 24125 clsnsg 24133 tgpt0 24142 tlmtgp 24219 nmcn 24879 fsumcn 24907 expcn 24909 divccn 24910 expcnOLD 24911 divccnOLD 24912 cncfmptc 24951 cdivcncf 24960 iirevcn 24970 iihalf1cn 24972 iihalf1cnOLD 24973 iihalf2cn 24975 iihalf2cnOLD 24976 icchmeo 24984 icchmeoOLD 24985 evth 25004 evth2 25005 pcocn 25063 pcopt 25068 pcopt2 25069 pcoass 25070 csscld 25296 clsocv 25297 dvcnvlem 26028 plycn 26314 plycnOLD 26315 psercn2 26480 psercn2OLD 26481 resqrtcn 26806 sqrtcn 26807 atansopn 26989 efrlim 27026 efrlimOLD 27027 ipasslem7 30864 occllem 31331 rmulccn 33888 cxpcncf1 34588 txsconnlem 35224 cvxpconn 35226 cvmlift2lem2 35288 cvmlift2lem3 35289 cvmliftphtlem 35301 sinccvglem 35656 knoppcnlem10 36484 areacirclem2 37695 fprodcn 45555 |
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