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Mirrors > Home > MPE Home > Th. List > cncfmpt2f | Structured version Visualization version GIF version |
Description: Composition of continuous functions. –cn→ analogue of cnmpt12f 22273. (Contributed by Mario Carneiro, 3-Sep-2014.) |
Ref | Expression |
---|---|
cncfmpt2f.1 | ⊢ 𝐽 = (TopOpen‘ℂfld) |
cncfmpt2f.2 | ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
cncfmpt2f.3 | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) |
cncfmpt2f.4 | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ)) |
Ref | Expression |
---|---|
cncfmpt2f | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncfmpt2f.1 | . . . . 5 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
2 | 1 | cnfldtopon 23390 | . . . 4 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
3 | cncfmpt2f.3 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) | |
4 | cncfrss 23498 | . . . . 5 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ) → 𝑋 ⊆ ℂ) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 ⊆ ℂ) |
6 | resttopon 21768 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑋 ⊆ ℂ) → (𝐽 ↾t 𝑋) ∈ (TopOn‘𝑋)) | |
7 | 2, 5, 6 | sylancr 589 | . . 3 ⊢ (𝜑 → (𝐽 ↾t 𝑋) ∈ (TopOn‘𝑋)) |
8 | ssid 3988 | . . . . 5 ⊢ ℂ ⊆ ℂ | |
9 | eqid 2821 | . . . . . 6 ⊢ (𝐽 ↾t 𝑋) = (𝐽 ↾t 𝑋) | |
10 | 2 | toponrestid 21528 | . . . . . 6 ⊢ 𝐽 = (𝐽 ↾t ℂ) |
11 | 1, 9, 10 | cncfcn 23516 | . . . . 5 ⊢ ((𝑋 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑋–cn→ℂ) = ((𝐽 ↾t 𝑋) Cn 𝐽)) |
12 | 5, 8, 11 | sylancl 588 | . . . 4 ⊢ (𝜑 → (𝑋–cn→ℂ) = ((𝐽 ↾t 𝑋) Cn 𝐽)) |
13 | 3, 12 | eleqtrd 2915 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 ↾t 𝑋) Cn 𝐽)) |
14 | cncfmpt2f.4 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ)) | |
15 | 14, 12 | eleqtrd 2915 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ ((𝐽 ↾t 𝑋) Cn 𝐽)) |
16 | cncfmpt2f.2 | . . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) | |
17 | 7, 13, 15, 16 | cnmpt12f 22273 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ↾t 𝑋) Cn 𝐽)) |
18 | 17, 12 | eleqtrrd 2916 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ↦ cmpt 5145 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 ↾t crest 16693 TopOpenctopn 16694 ℂfldccnfld 20544 TopOnctopon 21517 Cn ccn 21831 ×t ctx 22167 –cn→ccncf 23483 |
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-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 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-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fi 8874 df-sup 8905 df-inf 8906 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-fz 12892 df-seq 13369 df-exp 13429 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-plusg 16577 df-mulr 16578 df-starv 16579 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-rest 16695 df-topn 16696 df-topgen 16716 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-cnfld 20545 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-cn 21834 df-cnp 21835 df-tx 22169 df-xms 22929 df-ms 22930 df-cncf 23485 |
This theorem is referenced by: cncfmpt2ss 23522 addccncf 23523 negcncf 23525 mulcncf 24046 dvcnp2 24516 dvlipcn 24590 dvfsumabs 24619 ftc2 24640 itgparts 24643 taylthlem2 24961 sincn 25031 coscn 25032 logcn 25229 loglesqrt 25338 lgamgulmlem2 25606 pntlem3 26184 logdivsqrle 31921 ftc1cnnclem 34964 ftc2nc 34975 areacirclem4 34984 sub1cncf 35038 sub2cncf 35039 areaquad 39821 subcncf 42150 addcncf 42154 |
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