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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 5747 | . 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 23291 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑌) → (𝑋 × {𝑃}) ∈ (𝐽 Cn 𝐾)) | |
| 6 | 2, 3, 4, 5 | syl3anc 1373 | . 2 ⊢ (𝜑 → (𝑋 × {𝑃}) ∈ (𝐽 Cn 𝐾)) |
| 7 | 1, 6 | eqeltrrid 2846 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑃) ∈ (𝐽 Cn 𝐾)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 {csn 4626 ↦ cmpt 5225 × cxp 5683 ‘cfv 6561 (class class class)co 7431 TopOnctopon 22916 Cn ccn 23232 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-topgen 17488 df-top 22900 df-topon 22917 df-cn 23235 df-cnp 23236 |
| This theorem is referenced by: cnmpt2c 23678 xkoinjcn 23695 txconn 23697 imasnopn 23698 imasncld 23699 imasncls 23700 istgp2 24099 tmdmulg 24100 tmdgsum 24103 tmdlactcn 24110 clsnsg 24118 tgpt0 24127 tlmtgp 24204 nmcn 24866 fsumcn 24894 expcn 24896 divccn 24897 expcnOLD 24898 divccnOLD 24899 cncfmptc 24938 cdivcncf 24947 iirevcn 24957 iihalf1cn 24959 iihalf1cnOLD 24960 iihalf2cn 24962 iihalf2cnOLD 24963 icchmeo 24971 icchmeoOLD 24972 evth 24991 evth2 24992 pcocn 25050 pcopt 25055 pcopt2 25056 pcoass 25057 csscld 25283 clsocv 25284 dvcnvlem 26014 plycn 26300 plycnOLD 26301 psercn2 26466 psercn2OLD 26467 resqrtcn 26792 sqrtcn 26793 atansopn 26975 efrlim 27012 efrlimOLD 27013 ipasslem7 30855 occllem 31322 rmulccn 33927 cxpcncf1 34610 txsconnlem 35245 cvxpconn 35247 cvmlift2lem2 35309 cvmlift2lem3 35310 cvmliftphtlem 35322 sinccvglem 35677 knoppcnlem10 36503 areacirclem2 37716 fprodcn 45615 |
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