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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 2822 | . . 3 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 𝑃 = 𝑃) | |
2 | 1 | mpompt 7260 | . 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 22193 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌))) | |
6 | 3, 4, 5 | syl2anc 586 | . . 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 22264 | . 2 ⊢ (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) |
10 | 2, 9 | eqeltrrid 2918 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 〈cop 4566 ↦ cmpt 5138 × cxp 5547 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 TopOnctopon 21512 Cn ccn 21826 ×t ctx 22162 |
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 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
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 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-map 8402 df-topgen 16711 df-top 21496 df-topon 21513 df-bases 21548 df-cn 21829 df-cnp 21830 df-tx 22164 |
This theorem is referenced by: cnrehmeo 23551 pcopt 23620 pcopt2 23621 vmcn 28470 dipcn 28491 cvxsconn 32485 |
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