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| Mirrors > Home > MPE Home > Th. List > cnmpt2c | 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 |
|---|---|
| cnmpt21.j | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
| cnmpt21.k | ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) |
| cnmpt2c.l | ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) |
| cnmpt2c.p | ⊢ (𝜑 → 𝑃 ∈ 𝑍) |
| Ref | Expression |
|---|---|
| cnmpt2c | ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqidd 2738 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 𝑃 = 𝑃) | |
| 2 | 1 | mpompt 7547 | . 2 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑃) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑃) |
| 3 | cnmpt21.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
| 4 | cnmpt21.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) | |
| 5 | txtopon 23599 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) | |
| 6 | 3, 4, 5 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) |
| 7 | cnmpt2c.l | . . 3 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) | |
| 8 | cnmpt2c.p | . . 3 ⊢ (𝜑 → 𝑃 ∈ 𝑍) | |
| 9 | 6, 7, 8 | cnmptc 23670 | . 2 ⊢ (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) |
| 10 | 2, 9 | eqeltrrid 2846 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 〈cop 4632 ↦ cmpt 5225 × cxp 5683 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 TopOnctopon 22916 Cn ccn 23232 ×t ctx 23568 |
| 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-bases 22953 df-cn 23235 df-cnp 23236 df-tx 23570 |
| This theorem is referenced by: cnrehmeo 24984 cnrehmeoOLD 24985 pcopt 25055 pcopt2 25056 vmcn 30718 dipcn 30739 cvxsconn 35248 |
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