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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 23491. (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 24620 | . . . 4 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
3 | cncfmpt2f.3 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) | |
4 | cncfrss 24732 | . . . . 5 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ) → 𝑋 ⊆ ℂ) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 ⊆ ℂ) |
6 | resttopon 22986 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑋 ⊆ ℂ) → (𝐽 ↾t 𝑋) ∈ (TopOn‘𝑋)) | |
7 | 2, 5, 6 | sylancr 586 | . . 3 ⊢ (𝜑 → (𝐽 ↾t 𝑋) ∈ (TopOn‘𝑋)) |
8 | ssid 4004 | . . . . 5 ⊢ ℂ ⊆ ℂ | |
9 | eqid 2731 | . . . . . 6 ⊢ (𝐽 ↾t 𝑋) = (𝐽 ↾t 𝑋) | |
10 | 2 | toponrestid 22744 | . . . . . 6 ⊢ 𝐽 = (𝐽 ↾t ℂ) |
11 | 1, 9, 10 | cncfcn 24751 | . . . . 5 ⊢ ((𝑋 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑋–cn→ℂ) = ((𝐽 ↾t 𝑋) Cn 𝐽)) |
12 | 5, 8, 11 | sylancl 585 | . . . 4 ⊢ (𝜑 → (𝑋–cn→ℂ) = ((𝐽 ↾t 𝑋) Cn 𝐽)) |
13 | 3, 12 | eleqtrd 2834 | . . 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 2834 | . . 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 23491 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ↾t 𝑋) Cn 𝐽)) |
18 | 17, 12 | eleqtrrd 2835 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ⊆ wss 3948 ↦ cmpt 5231 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 ↾t crest 17373 TopOpenctopn 17374 ℂfldccnfld 21234 TopOnctopon 22733 Cn ccn 23049 ×t ctx 23385 –cn→ccncf 24717 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fi 9412 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-fz 13492 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-struct 17087 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-mulr 17218 df-starv 17219 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-rest 17375 df-topn 17376 df-topgen 17396 df-psmet 21226 df-xmet 21227 df-met 21228 df-bl 21229 df-mopn 21230 df-cnfld 21235 df-top 22717 df-topon 22734 df-topsp 22756 df-bases 22770 df-cn 23052 df-cnp 23053 df-tx 23387 df-xms 24147 df-ms 24148 df-cncf 24719 |
This theorem is referenced by: cncfmpt2ss 24757 addccncf 24758 sub1cncf 24760 sub2cncf 24761 negcncf 24763 negcncfOLD 24764 addcncf 25293 subcncf 25294 mulcncfOLD 25296 dvcnp2OLD 25771 dvlipcn 25848 dvfsumabs 25878 ftc2 25900 itgparts 25903 taylthlem2 26226 sincn 26297 coscn 26298 logcn 26496 loglesqrt 26608 lgamgulmlem2 26877 pntlem3 27457 logdivsqrle 34128 gg-taylthlem2 35634 ftc1cnnclem 37026 ftc2nc 37037 areacirclem4 37046 areaquad 42431 |
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