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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mhmhmeotmd | Structured version Visualization version Unicode version |
Description: Deduce a Topological Monoid using mapping that is both a homeomorphism and a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.) |
Ref | Expression |
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mhmhmeotmd.m |
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mhmhmeotmd.h |
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mhmhmeotmd.t |
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mhmhmeotmd.s |
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Ref | Expression |
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mhmhmeotmd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mhmhmeotmd.m |
. . 3
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2 | mhmrcl2 16664 |
. . 3
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3 | 1, 2 | ax-mp 5 |
. 2
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4 | mhmhmeotmd.s |
. 2
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5 | mhmhmeotmd.h |
. . 3
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6 | mhmrcl1 16663 |
. . . . 5
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7 | 1, 6 | ax-mp 5 |
. . . 4
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8 | eqid 2471 |
. . . . 5
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9 | eqid 2471 |
. . . . 5
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10 | 8, 9 | mndplusf 16633 |
. . . 4
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11 | 7, 10 | ax-mp 5 |
. . 3
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12 | eqid 2471 |
. . . . 5
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13 | eqid 2471 |
. . . . 5
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14 | 12, 13 | mndplusf 16633 |
. . . 4
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15 | 3, 14 | ax-mp 5 |
. . 3
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16 | mhmhmeotmd.t |
. . . 4
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17 | eqid 2471 |
. . . . 5
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18 | 17, 8 | tmdtopon 21174 |
. . . 4
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19 | 16, 18 | ax-mp 5 |
. . 3
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20 | eqid 2471 |
. . . . 5
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21 | 12, 20 | istps 20028 |
. . . 4
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22 | 4, 21 | mpbi 213 |
. . 3
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23 | eqid 2471 |
. . . . . 6
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24 | eqid 2471 |
. . . . . 6
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25 | 8, 23, 24 | mhmlin 16667 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 1, 25 | mp3an1 1377 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 8, 23, 9 | plusfval 16572 |
. . . . 5
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28 | 27 | fveq2d 5883 |
. . . 4
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29 | 8, 12 | mhmf 16665 |
. . . . . . 7
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30 | 1, 29 | ax-mp 5 |
. . . . . 6
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31 | 30 | ffvelrni 6036 |
. . . . 5
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32 | 30 | ffvelrni 6036 |
. . . . 5
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33 | 12, 24, 13 | plusfval 16572 |
. . . . 5
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34 | 31, 32, 33 | syl2an 485 |
. . . 4
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35 | 26, 28, 34 | 3eqtr4d 2515 |
. . 3
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36 | 17, 9 | tmdcn 21176 |
. . . 4
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37 | 16, 36 | ax-mp 5 |
. . 3
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38 | 5, 11, 15, 19, 22, 35, 37 | mndpluscn 28806 |
. 2
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39 | 13, 20 | istmd 21167 |
. 2
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40 | 3, 4, 38, 39 | mpbir3an 1212 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 ax-un 6602 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-ral 2761 df-rex 2762 df-rab 2765 df-v 3033 df-sbc 3256 df-csb 3350 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-pw 3944 df-sn 3960 df-pr 3962 df-op 3966 df-uni 4191 df-iun 4271 df-br 4396 df-opab 4455 df-mpt 4456 df-id 4754 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 df-iota 5553 df-fun 5591 df-fn 5592 df-f 5593 df-f1 5594 df-fo 5595 df-f1o 5596 df-fv 5597 df-ov 6311 df-oprab 6312 df-mpt2 6313 df-1st 6812 df-2nd 6813 df-map 7492 df-topgen 15420 df-plusf 16565 df-mgm 16566 df-sgrp 16605 df-mnd 16615 df-mhm 16660 df-top 19998 df-bases 19999 df-topon 20000 df-topsp 20001 df-cn 20320 df-tx 20654 df-hmeo 20847 df-tmd 21165 |
This theorem is referenced by: xrge0tmd 28826 |
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