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Mirrors > Home > MPE Home > Th. List > cnmpt12f | Structured version Visualization version Unicode version |
Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
cnmptid.j | TopOn |
cnmpt11.a | |
cnmpt1t.b | |
cnmpt12f.f |
Ref | Expression |
---|---|
cnmpt12f |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 6653 | . . 3 | |
2 | 1 | mpteq2i 4741 | . 2 |
3 | cnmptid.j | . . 3 TopOn | |
4 | cnmpt11.a | . . . 4 | |
5 | cnmpt1t.b | . . . 4 | |
6 | 3, 4, 5 | cnmpt1t 21468 | . . 3 |
7 | cnmpt12f.f | . . 3 | |
8 | 3, 6, 7 | cnmpt11f 21467 | . 2 |
9 | 2, 8 | syl5eqel 2705 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wcel 1990 cop 4183 cmpt 4729 cfv 5888 (class class class)co 6650 TopOnctopon 20715 ccn 21028 ctx 21363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 df-topgen 16104 df-top 20699 df-topon 20716 df-bases 20750 df-cn 21031 df-tx 21365 |
This theorem is referenced by: cnmpt12 21470 cnmpt1plusg 21891 istgp2 21895 clsnsg 21913 tgpt0 21922 cnmpt1vsca 21997 cnmpt1ds 22645 fsumcn 22673 expcn 22675 divccn 22676 cncfmpt2f 22717 cdivcncf 22720 iirevcn 22729 iihalf1cn 22731 iihalf2cn 22733 icchmeo 22740 evth 22758 evth2 22759 pcoass 22824 cnmpt1ip 23046 dvcnvlem 23739 plycn 24017 psercn2 24177 atansopn 24659 efrlim 24696 ipasslem7 27691 occllem 28162 hmopidmchi 29010 cvxpconn 31224 cvmlift2lem2 31286 cvmlift2lem3 31287 cvmliftphtlem 31299 sinccvglem 31566 knoppcnlem10 32492 broucube 33443 areacirclem2 33501 fprodcnlem 39831 |
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