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Mirrors > Home > MPE Home > Th. List > cncfmpt2ss | Structured version Visualization version GIF version |
Description: Composition of continuous functions in a subset. (Contributed by Mario Carneiro, 17-May-2016.) |
Ref | Expression |
---|---|
cncfmpt2ss.1 | β’ π½ = (TopOpenββfld) |
cncfmpt2ss.2 | β’ πΉ β ((π½ Γt π½) Cn π½) |
cncfmpt2ss.3 | β’ (π β (π₯ β π β¦ π΄) β (πβcnβπ)) |
cncfmpt2ss.4 | β’ (π β (π₯ β π β¦ π΅) β (πβcnβπ)) |
cncfmpt2ss.5 | β’ π β β |
cncfmpt2ss.6 | β’ ((π΄ β π β§ π΅ β π) β (π΄πΉπ΅) β π) |
Ref | Expression |
---|---|
cncfmpt2ss | β’ (π β (π₯ β π β¦ (π΄πΉπ΅)) β (πβcnβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncfmpt2ss.3 | . . . . . 6 β’ (π β (π₯ β π β¦ π΄) β (πβcnβπ)) | |
2 | cncff 24634 | . . . . . 6 β’ ((π₯ β π β¦ π΄) β (πβcnβπ) β (π₯ β π β¦ π΄):πβΆπ) | |
3 | 1, 2 | syl 17 | . . . . 5 β’ (π β (π₯ β π β¦ π΄):πβΆπ) |
4 | 3 | fvmptelcdm 7114 | . . . 4 β’ ((π β§ π₯ β π) β π΄ β π) |
5 | cncfmpt2ss.4 | . . . . . 6 β’ (π β (π₯ β π β¦ π΅) β (πβcnβπ)) | |
6 | cncff 24634 | . . . . . 6 β’ ((π₯ β π β¦ π΅) β (πβcnβπ) β (π₯ β π β¦ π΅):πβΆπ) | |
7 | 5, 6 | syl 17 | . . . . 5 β’ (π β (π₯ β π β¦ π΅):πβΆπ) |
8 | 7 | fvmptelcdm 7114 | . . . 4 β’ ((π β§ π₯ β π) β π΅ β π) |
9 | cncfmpt2ss.6 | . . . 4 β’ ((π΄ β π β§ π΅ β π) β (π΄πΉπ΅) β π) | |
10 | 4, 8, 9 | syl2anc 583 | . . 3 β’ ((π β§ π₯ β π) β (π΄πΉπ΅) β π) |
11 | 10 | fmpttd 7116 | . 2 β’ (π β (π₯ β π β¦ (π΄πΉπ΅)):πβΆπ) |
12 | cncfmpt2ss.5 | . . 3 β’ π β β | |
13 | cncfmpt2ss.1 | . . . 4 β’ π½ = (TopOpenββfld) | |
14 | cncfmpt2ss.2 | . . . . 5 β’ πΉ β ((π½ Γt π½) Cn π½) | |
15 | 14 | a1i 11 | . . . 4 β’ (π β πΉ β ((π½ Γt π½) Cn π½)) |
16 | ssid 4004 | . . . . . 6 β’ β β β | |
17 | cncfss 24640 | . . . . . 6 β’ ((π β β β§ β β β) β (πβcnβπ) β (πβcnββ)) | |
18 | 12, 16, 17 | mp2an 689 | . . . . 5 β’ (πβcnβπ) β (πβcnββ) |
19 | 18, 1 | sselid 3980 | . . . 4 β’ (π β (π₯ β π β¦ π΄) β (πβcnββ)) |
20 | 18, 5 | sselid 3980 | . . . 4 β’ (π β (π₯ β π β¦ π΅) β (πβcnββ)) |
21 | 13, 15, 19, 20 | cncfmpt2f 24656 | . . 3 β’ (π β (π₯ β π β¦ (π΄πΉπ΅)) β (πβcnββ)) |
22 | cncfcdm 24639 | . . 3 β’ ((π β β β§ (π₯ β π β¦ (π΄πΉπ΅)) β (πβcnββ)) β ((π₯ β π β¦ (π΄πΉπ΅)) β (πβcnβπ) β (π₯ β π β¦ (π΄πΉπ΅)):πβΆπ)) | |
23 | 12, 21, 22 | sylancr 586 | . 2 β’ (π β ((π₯ β π β¦ (π΄πΉπ΅)) β (πβcnβπ) β (π₯ β π β¦ (π΄πΉπ΅)):πβΆπ)) |
24 | 11, 23 | mpbird 257 | 1 β’ (π β (π₯ β π β¦ (π΄πΉπ΅)) β (πβcnβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1540 β wcel 2105 β wss 3948 β¦ cmpt 5231 βΆwf 6539 βcfv 6543 (class class class)co 7412 βcc 11112 TopOpenctopn 17372 βfldccnfld 21145 Cn ccn 22949 Γt ctx 23285 βcnβccncf 24617 |
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 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 |
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 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fi 9410 df-sup 9441 df-inf 9442 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-fz 13490 df-seq 13972 df-exp 14033 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-struct 17085 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-mulr 17216 df-starv 17217 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-rest 17373 df-topn 17374 df-topgen 17394 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cn 22952 df-cnp 22953 df-tx 23287 df-xms 24047 df-ms 24048 df-cncf 24619 |
This theorem is referenced by: cmvth 25744 dvle 25760 dvfsumle 25774 dvfsumge 25775 dvfsumlem2 25780 gg-cmvth 35467 gg-dvfsumle 35469 gg-dvfsumlem2 35470 |
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