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Mirrors > Home > MPE Home > Th. List > cncfmpt1f | Structured version Visualization version GIF version |
Description: Composition of continuous functions. βcnβ analogue of cnmpt11f 23168. (Contributed by Mario Carneiro, 3-Sep-2014.) |
Ref | Expression |
---|---|
cncfmpt1f.1 | β’ (π β πΉ β (ββcnββ)) |
cncfmpt1f.2 | β’ (π β (π₯ β π β¦ π΄) β (πβcnββ)) |
Ref | Expression |
---|---|
cncfmpt1f | β’ (π β (π₯ β π β¦ (πΉβπ΄)) β (πβcnββ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncfmpt1f.2 | . . . . 5 β’ (π β (π₯ β π β¦ π΄) β (πβcnββ)) | |
2 | cncff 24409 | . . . . 5 β’ ((π₯ β π β¦ π΄) β (πβcnββ) β (π₯ β π β¦ π΄):πβΆβ) | |
3 | 1, 2 | syl 17 | . . . 4 β’ (π β (π₯ β π β¦ π΄):πβΆβ) |
4 | eqid 2733 | . . . . 5 β’ (π₯ β π β¦ π΄) = (π₯ β π β¦ π΄) | |
5 | 4 | fmpt 7110 | . . . 4 β’ (βπ₯ β π π΄ β β β (π₯ β π β¦ π΄):πβΆβ) |
6 | 3, 5 | sylibr 233 | . . 3 β’ (π β βπ₯ β π π΄ β β) |
7 | eqidd 2734 | . . 3 β’ (π β (π₯ β π β¦ π΄) = (π₯ β π β¦ π΄)) | |
8 | cncfmpt1f.1 | . . . . 5 β’ (π β πΉ β (ββcnββ)) | |
9 | cncff 24409 | . . . . 5 β’ (πΉ β (ββcnββ) β πΉ:ββΆβ) | |
10 | 8, 9 | syl 17 | . . . 4 β’ (π β πΉ:ββΆβ) |
11 | 10 | feqmptd 6961 | . . 3 β’ (π β πΉ = (π¦ β β β¦ (πΉβπ¦))) |
12 | fveq2 6892 | . . 3 β’ (π¦ = π΄ β (πΉβπ¦) = (πΉβπ΄)) | |
13 | 6, 7, 11, 12 | fmptcof 7128 | . 2 β’ (π β (πΉ β (π₯ β π β¦ π΄)) = (π₯ β π β¦ (πΉβπ΄))) |
14 | 1, 8 | cncfco 24423 | . 2 β’ (π β (πΉ β (π₯ β π β¦ π΄)) β (πβcnββ)) |
15 | 13, 14 | eqeltrrd 2835 | 1 β’ (π β (π₯ β π β¦ (πΉβπ΄)) β (πβcnββ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2107 βwral 3062 β¦ cmpt 5232 β ccom 5681 βΆwf 6540 βcfv 6544 (class class class)co 7409 βcc 11108 βcnβccncf 24392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-2 12275 df-cj 15046 df-re 15047 df-im 15048 df-abs 15183 df-cncf 24394 |
This theorem is referenced by: taylthlem2 25886 sincn 25956 coscn 25957 pige3ALT 26029 efmul2picn 33608 itgexpif 33618 ftc1cnnclem 36559 ftc2nc 36570 itgcoscmulx 44685 itgsincmulx 44690 dirkeritg 44818 dirkercncflem2 44820 dirkercncflem4 44822 fourierdlem16 44839 fourierdlem21 44844 fourierdlem22 44845 fourierdlem39 44862 fourierdlem58 44880 fourierdlem62 44884 fourierdlem68 44890 fourierdlem73 44895 fourierdlem76 44898 fourierdlem78 44900 fourierdlem83 44905 sqwvfoura 44944 sqwvfourb 44945 etransclem18 44968 etransclem46 44996 |
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