| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 7434 | . . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
| 2 | 1 | mpteq2i 5247 | . 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 23673 | . . 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 23672 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘〈𝐴, 𝐵〉)) ∈ (𝐽 Cn 𝑀)) |
| 9 | 2, 8 | eqeltrid 2845 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝐽 Cn 𝑀)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 〈cop 4632 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 TopOnctopon 22916 Cn ccn 23232 ×t ctx 23568 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-topgen 17488 df-top 22900 df-topon 22917 df-bases 22953 df-cn 23235 df-tx 23570 |
| This theorem is referenced by: cnmpt12 23675 cnmpt1plusg 24095 istgp2 24099 clsnsg 24118 tgpt0 24127 cnmpt1vsca 24202 cnmpt1ds 24864 fsumcn 24894 expcn 24896 expcnOLD 24898 divccnOLD 24899 cncfmpt2f 24941 cdivcncf 24947 iirevcn 24957 iihalf1cnOLD 24960 iihalf2cn 24962 iihalf2cnOLD 24963 icchmeo 24971 icchmeoOLD 24972 evth 24991 evth2 24992 pcoass 25057 cnmpt1ip 25281 dvcnvlem 26014 plycnOLD 26301 psercn2OLD 26467 atansopn 26975 efrlim 27012 efrlimOLD 27013 ipasslem7 30855 occllem 31322 hmopidmchi 32170 cvxpconn 35247 cvmlift2lem2 35309 cvmlift2lem3 35310 cvmliftphtlem 35322 sinccvglem 35677 broucube 37661 areacirclem2 37716 |
| Copyright terms: Public domain | W3C validator |