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Mirrors > Home > MPE Home > Th. List > hmeoco | Structured version Visualization version Unicode version |
Description: The composite of two homeomorphisms is a homeomorphism. (Contributed by FL, 9-Mar-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
hmeoco |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmeocn 21563 | . . 3 | |
2 | hmeocn 21563 | . . 3 | |
3 | cnco 21070 | . . 3 | |
4 | 1, 2, 3 | syl2an 494 | . 2 |
5 | cnvco 5308 | . . 3 | |
6 | hmeocnvcn 21564 | . . . 4 | |
7 | hmeocnvcn 21564 | . . . 4 | |
8 | cnco 21070 | . . . 4 | |
9 | 6, 7, 8 | syl2anr 495 | . . 3 |
10 | 5, 9 | syl5eqel 2705 | . 2 |
11 | ishmeo 21562 | . 2 | |
12 | 4, 10, 11 | sylanbrc 698 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wcel 1990 ccnv 5113 ccom 5118 (class class class)co 6650 ccn 21028 chmeo 21556 |
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-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-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-map 7859 df-top 20699 df-topon 20716 df-cn 21031 df-hmeo 21558 |
This theorem is referenced by: hmphtr 21586 xpstopnlem1 21612 tgpconncomp 21916 tsmsxplem1 21956 |
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