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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > exidresid | Structured version Visualization version Unicode version |
Description: The restriction of a binary operation with identity to a subset containing the identity has the same identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.) |
Ref | Expression |
---|---|
exidres.1 |
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exidres.2 |
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exidres.3 |
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Ref | Expression |
---|---|
exidresid |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exidres.3 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | resexg 5146 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | 1, 2 | syl5eqel 2532 |
. . . . 5
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4 | eqid 2450 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 4 | gidval 25934 |
. . . . 5
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6 | 3, 5 | syl 17 |
. . . 4
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7 | 6 | 3ad2ant1 1028 |
. . 3
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8 | 7 | adantr 467 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | exidres.1 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() | |
10 | exidres.2 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 9, 10, 1 | exidreslem 32168 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 11 | simprd 465 |
. . . . 5
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13 | 12 | adantr 467 |
. . . 4
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14 | 9, 10, 1 | exidres 32169 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | elin 3616 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | rngopidOLD 26044 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 15, 16 | sylbir 217 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 17 | ancoms 455 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 14, 18 | sylan 474 |
. . . . 5
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20 | 19 | raleqdv 2992 |
. . . 4
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21 | 13, 20 | mpbird 236 |
. . 3
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22 | 11 | simpld 461 |
. . . . . 6
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23 | 22 | adantr 467 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 23, 19 | eleqtrrd 2531 |
. . . 4
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25 | 4 | exidu1 26047 |
. . . . . . 7
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26 | 15, 25 | sylbir 217 |
. . . . . 6
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27 | 26 | ancoms 455 |
. . . . 5
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28 | 14, 27 | sylan 474 |
. . . 4
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29 | oveq1 6295 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
30 | 29 | eqeq1d 2452 |
. . . . . . 7
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31 | oveq2 6296 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
32 | 31 | eqeq1d 2452 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 30, 32 | anbi12d 716 |
. . . . . 6
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34 | 33 | ralbidv 2826 |
. . . . 5
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35 | 34 | riota2 6272 |
. . . 4
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36 | 24, 28, 35 | syl2anc 666 |
. . 3
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37 | 21, 36 | mpbid 214 |
. 2
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38 | 8, 37 | eqtrd 2484 |
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 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-sep 4524 ax-nul 4533 ax-pr 4638 ax-un 6580 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-ral 2741 df-rex 2742 df-reu 2743 df-rmo 2744 df-rab 2745 df-v 3046 df-sbc 3267 df-csb 3363 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-nul 3731 df-if 3881 df-sn 3968 df-pr 3970 df-op 3974 df-uni 4198 df-iun 4279 df-br 4402 df-opab 4461 df-mpt 4462 df-id 4748 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-fo 5587 df-fv 5589 df-riota 6250 df-ov 6291 df-gid 25913 df-exid 26036 df-mgmOLD 26040 |
This theorem is referenced by: isdrngo2 32190 |
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