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Mirrors > Home > MPE Home > Th. List > cnmpt12f | Structured version Visualization version GIF version |
Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
cnmptid.j | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
cnmpt11.a | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) |
cnmpt1t.b | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿)) |
cnmpt12f.f | ⊢ (𝜑 → 𝐹 ∈ ((𝐾 ×t 𝐿) Cn 𝑀)) |
Ref | Expression |
---|---|
cnmpt12f | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝐽 Cn 𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 6693 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | 1 | mpteq2i 4774 | . 2 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐹‘〈𝐴, 𝐵〉)) |
3 | cnmptid.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
4 | cnmpt11.a | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) | |
5 | cnmpt1t.b | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿)) | |
6 | 3, 4, 5 | cnmpt1t 21516 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) ∈ (𝐽 Cn (𝐾 ×t 𝐿))) |
7 | cnmpt12f.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐾 ×t 𝐿) Cn 𝑀)) | |
8 | 3, 6, 7 | cnmpt11f 21515 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘〈𝐴, 𝐵〉)) ∈ (𝐽 Cn 𝑀)) |
9 | 2, 8 | syl5eqel 2734 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝐽 Cn 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2030 〈cop 4216 ↦ cmpt 4762 ‘cfv 5926 (class class class)co 6690 TopOnctopon 20763 Cn ccn 21076 ×t ctx 21411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-1st 7210 df-2nd 7211 df-map 7901 df-topgen 16151 df-top 20747 df-topon 20764 df-bases 20798 df-cn 21079 df-tx 21413 |
This theorem is referenced by: cnmpt12 21518 cnmpt1plusg 21938 istgp2 21942 clsnsg 21960 tgpt0 21969 cnmpt1vsca 22044 cnmpt1ds 22692 fsumcn 22720 expcn 22722 divccn 22723 cncfmpt2f 22764 cdivcncf 22767 iirevcn 22776 iihalf1cn 22778 iihalf2cn 22780 icchmeo 22787 evth 22805 evth2 22806 pcoass 22870 cnmpt1ip 23092 dvcnvlem 23784 plycn 24062 psercn2 24222 atansopn 24704 efrlim 24741 ipasslem7 27819 occllem 28290 hmopidmchi 29138 cvxpconn 31350 cvmlift2lem2 31412 cvmlift2lem3 31413 cvmliftphtlem 31425 sinccvglem 31692 knoppcnlem10 32617 broucube 33573 areacirclem2 33631 fprodcnlem 40149 |
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