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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 6552 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
2 | 1 | mpteq2i 4669 | . 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 21278 | . . 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 21277 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘〈𝐴, 𝐵〉)) ∈ (𝐽 Cn 𝑀)) |
9 | 2, 8 | syl5eqel 2692 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝐽 Cn 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 〈cop 4131 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 TopOnctopon 20518 Cn ccn 20838 ×t ctx 21173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 df-topgen 15927 df-top 20521 df-bases 20522 df-topon 20523 df-cn 20841 df-tx 21175 |
This theorem is referenced by: cnmpt12 21280 cnmpt1plusg 21701 istgp2 21705 clsnsg 21723 tgpt0 21732 cnmpt1vsca 21807 cnmpt1ds 22453 fsumcn 22481 expcn 22483 divccn 22484 cncfmpt2f 22525 cdivcncf 22528 iirevcn 22537 iihalf1cn 22539 iihalf2cn 22541 icchmeo 22548 evth 22566 evth2 22567 pcoass 22632 cnmpt1ip 22854 dvcnvlem 23543 plycn 23821 psercn2 23981 atansopn 24459 efrlim 24496 ipasslem7 27075 occllem 27546 hmopidmchi 28394 cvxpcon 30478 cvmlift2lem2 30540 cvmlift2lem3 30541 cvmliftphtlem 30553 sinccvglem 30820 knoppcnlem10 31662 broucube 32613 areacirclem2 32671 fprodcnlem 38666 |
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