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Mirrors > Home > MPE Home > Th. List > mulcncf | Structured version Visualization version GIF version |
Description: The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Ref | Expression |
---|---|
mulcncf.1 | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) |
mulcncf.2 | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ)) |
Ref | Expression |
---|---|
mulcncf | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝑋–cn→ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2795 | . 2 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
2 | 1 | mulcn 23158 | . . 3 ⊢ · ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)) |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → · ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))) |
4 | mulcncf.1 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) | |
5 | mulcncf.2 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ)) | |
6 | 1, 3, 4, 5 | cncfmpt2f 23205 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝑋–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2081 ↦ cmpt 5041 ‘cfv 6225 (class class class)co 7016 ℂcc 10381 · cmul 10388 TopOpenctopn 16524 ℂfldccnfld 20227 Cn ccn 21516 ×t ctx 21852 –cn→ccncf 23167 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 ax-mulf 10463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-iin 4828 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-of 7267 df-om 7437 df-1st 7545 df-2nd 7546 df-supp 7682 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-oadd 7957 df-er 8139 df-map 8258 df-ixp 8311 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-fsupp 8680 df-fi 8721 df-sup 8752 df-inf 8753 df-oi 8820 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-z 11830 df-dec 11948 df-uz 12094 df-q 12198 df-rp 12240 df-xneg 12357 df-xadd 12358 df-xmul 12359 df-icc 12595 df-fz 12743 df-fzo 12884 df-seq 13220 df-exp 13280 df-hash 13541 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-starv 16409 df-sca 16410 df-vsca 16411 df-ip 16412 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-hom 16418 df-cco 16419 df-rest 16525 df-topn 16526 df-0g 16544 df-gsum 16545 df-topgen 16546 df-pt 16547 df-prds 16550 df-xrs 16604 df-qtop 16609 df-imas 16610 df-xps 16612 df-mre 16686 df-mrc 16687 df-acs 16689 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-submnd 17775 df-mulg 17982 df-cntz 18188 df-cmn 18635 df-psmet 20219 df-xmet 20220 df-met 20221 df-bl 20222 df-mopn 20223 df-cnfld 20228 df-top 21186 df-topon 21203 df-topsp 21225 df-bases 21238 df-cn 21519 df-cnp 21520 df-tx 21854 df-hmeo 22047 df-xms 22613 df-ms 22614 df-tms 22615 df-cncf 23169 |
This theorem is referenced by: divcncf 23731 dvlipcn 24274 dvfsumabs 24303 itgparts 24327 itgsubstlem 24328 lgamgulmlem2 25289 pntlem3 25867 efmul2picn 31484 circlemeth 31528 logdivsqrle 31538 ftc1cnnclem 34496 ftc2nc 34507 areacirclem3 34515 areacirclem4 34516 areacirc 34518 itgpowd 39306 areaquad 39308 mulcncff 41692 fprodcncf 41725 itgsinexplem1 41780 itgcoscmulx 41795 itgsincmulx 41800 dirkercncflem2 41931 dirkercncflem4 41933 fourierdlem16 41950 fourierdlem18 41952 fourierdlem21 41955 fourierdlem22 41956 fourierdlem39 41973 fourierdlem40 41974 fourierdlem62 41995 fourierdlem68 42001 fourierdlem73 42006 fourierdlem76 42009 fourierdlem78 42011 fourierdlem83 42016 fourierdlem84 42017 fourierdlem101 42034 fourierdlem111 42044 sqwvfoura 42055 sqwvfourb 42056 etransclem18 42079 etransclem22 42083 etransclem34 42095 etransclem46 42107 |
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