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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 2735 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 𝑃 = 𝑃) | |
2 | 1 | mpompt 7546 | . 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 23614 | . . . 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 23685 | . 2 ⊢ (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) |
10 | 2, 9 | eqeltrrid 2843 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 〈cop 4636 ↦ cmpt 5230 × cxp 5686 ‘cfv 6562 (class class class)co 7430 ∈ cmpo 7432 TopOnctopon 22931 Cn ccn 23247 ×t ctx 23583 |
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-bases 22968 df-cn 23250 df-cnp 23251 df-tx 23585 |
This theorem is referenced by: cnrehmeo 24997 cnrehmeoOLD 24998 pcopt 25068 pcopt2 25069 vmcn 30727 dipcn 30748 cvxsconn 35227 |
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