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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 5613 | . 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 21890 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑌) → (𝑋 × {𝑃}) ∈ (𝐽 Cn 𝐾)) | |
6 | 2, 3, 4, 5 | syl3anc 1367 | . 2 ⊢ (𝜑 → (𝑋 × {𝑃}) ∈ (𝐽 Cn 𝐾)) |
7 | 1, 6 | eqeltrrid 2918 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑃) ∈ (𝐽 Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 {csn 4566 ↦ cmpt 5145 × cxp 5552 ‘cfv 6354 (class class class)co 7155 TopOnctopon 21517 Cn ccn 21831 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-map 8407 df-topgen 16716 df-top 21501 df-topon 21518 df-cn 21834 df-cnp 21835 |
This theorem is referenced by: cnmpt2c 22277 xkoinjcn 22294 txconn 22296 imasnopn 22297 imasncld 22298 imasncls 22299 istgp2 22698 tmdmulg 22699 tmdgsum 22702 tmdlactcn 22709 clsnsg 22717 tgpt0 22726 tlmtgp 22803 nmcn 23451 fsumcn 23477 expcn 23479 divccn 23480 cncfmptc 23518 cdivcncf 23524 iirevcn 23533 iihalf1cn 23535 iihalf2cn 23537 icchmeo 23544 evth 23562 evth2 23563 pcocn 23620 pcopt 23625 pcopt2 23626 pcoass 23627 csscld 23851 clsocv 23852 dvcnvlem 24572 plycn 24850 psercn2 25010 resqrtcn 25329 sqrtcn 25330 atansopn 25509 efrlim 25546 ipasslem7 28612 occllem 29079 rmulccn 31171 cxpcncf1 31866 txsconnlem 32487 cvxpconn 32489 cvmlift2lem2 32551 cvmlift2lem3 32552 cvmliftphtlem 32564 sinccvglem 32915 knoppcnlem10 33841 areacirclem2 34982 fprodcn 41879 |
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