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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 7266 | . 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 22199 | . . . 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 22270 | . 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 1537 ∈ wcel 2114 〈cop 4573 ↦ cmpt 5146 × cxp 5553 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 TopOnctopon 21518 Cn ccn 21832 ×t ctx 22168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-map 8408 df-topgen 16717 df-top 21502 df-topon 21519 df-bases 21554 df-cn 21835 df-cnp 21836 df-tx 22170 |
This theorem is referenced by: cnrehmeo 23557 pcopt 23626 pcopt2 23627 vmcn 28476 dipcn 28497 cvxsconn 32490 |
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