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Mirrors > Home > ILE Home > Th. List > cncfmpt1f | GIF version |
Description: Composition of continuous functions. βcnβ analogue of cnmpt11f 13924. (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 14204 | . . . . 5 β’ ((π₯ β π β¦ π΄) β (πβcnββ) β (π₯ β π β¦ π΄):πβΆβ) | |
3 | 1, 2 | syl 14 | . . . 4 β’ (π β (π₯ β π β¦ π΄):πβΆβ) |
4 | eqid 2177 | . . . . 5 β’ (π₯ β π β¦ π΄) = (π₯ β π β¦ π΄) | |
5 | 4 | fmpt 5669 | . . . 4 β’ (βπ₯ β π π΄ β β β (π₯ β π β¦ π΄):πβΆβ) |
6 | 3, 5 | sylibr 134 | . . 3 β’ (π β βπ₯ β π π΄ β β) |
7 | eqidd 2178 | . . 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 14204 | . . . . 5 β’ (πΉ β (ββcnββ) β πΉ:ββΆβ) | |
10 | 8, 9 | syl 14 | . . . 4 β’ (π β πΉ:ββΆβ) |
11 | 10 | feqmptd 5572 | . . 3 β’ (π β πΉ = (π¦ β β β¦ (πΉβπ¦))) |
12 | fveq2 5517 | . . 3 β’ (π¦ = π΄ β (πΉβπ¦) = (πΉβπ΄)) | |
13 | 6, 7, 11, 12 | fmptcof 5686 | . 2 β’ (π β (πΉ β (π₯ β π β¦ π΄)) = (π₯ β π β¦ (πΉβπ΄))) |
14 | 1, 8 | cncfco 14218 | . 2 β’ (π β (πΉ β (π₯ β π β¦ π΄)) β (πβcnββ)) |
15 | 13, 14 | eqeltrrd 2255 | 1 β’ (π β (π₯ β π β¦ (πΉβπ΄)) β (πβcnββ)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β wcel 2148 βwral 2455 β¦ cmpt 4066 β ccom 4632 βΆwf 5214 βcfv 5218 (class class class)co 5878 βcc 7812 βcnβccncf 14197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-mulrcl 7913 ax-addcom 7914 ax-mulcom 7915 ax-addass 7916 ax-mulass 7917 ax-distr 7918 ax-i2m1 7919 ax-0lt1 7920 ax-1rid 7921 ax-0id 7922 ax-rnegex 7923 ax-precex 7924 ax-cnre 7925 ax-pre-ltirr 7926 ax-pre-ltwlin 7927 ax-pre-lttrn 7928 ax-pre-apti 7929 ax-pre-ltadd 7930 ax-pre-mulgt0 7931 ax-pre-mulext 7932 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-po 4298 df-iso 4299 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-map 6653 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 df-sub 8133 df-neg 8134 df-reap 8535 df-ap 8542 df-div 8633 df-2 8981 df-cj 10854 df-re 10855 df-im 10856 df-rsqrt 11010 df-abs 11011 df-cncf 14198 |
This theorem is referenced by: sincn 14330 coscn 14331 |
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