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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrncom | Unicode version |
Description: Composition is commutative for translations. Part of proof of Lemma G of [Crawley] p. 116 (Contributed by NM, 5-Jun-2013.) |
Ref | Expression |
---|---|
ltrncom.h |
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ltrncom.t |
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Ref | Expression |
---|---|
ltrncom |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 960 |
. . 3
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2 | simpl2 961 |
. . 3
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3 | simpl3 962 |
. . 3
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4 | simpr 448 |
. . 3
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5 | eqid 2404 |
. . . 4
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6 | ltrncom.h |
. . . 4
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7 | ltrncom.t |
. . . 4
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8 | 5, 6, 7 | cdlemg47a 31216 |
. . 3
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9 | 1, 2, 3, 4, 8 | syl121anc 1189 |
. 2
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10 | simpll1 996 |
. . . 4
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11 | simpll2 997 |
. . . 4
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12 | simpll3 998 |
. . . 4
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13 | simplr 732 |
. . . 4
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14 | simpr 448 |
. . . 4
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15 | eqid 2404 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 5, 6, 7, 15 | cdlemg48 31219 |
. . . 4
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17 | 10, 11, 12, 13, 14, 16 | syl122anc 1193 |
. . 3
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18 | simpll1 996 |
. . . 4
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19 | simpll2 997 |
. . . 4
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20 | simpll3 998 |
. . . 4
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21 | simpr 448 |
. . . 4
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22 | 6, 7, 15 | cdlemg44 31215 |
. . . 4
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23 | 18, 19, 20, 21, 22 | syl121anc 1189 |
. . 3
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24 | 17, 23 | pm2.61dane 2645 |
. 2
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25 | 9, 24 | pm2.61dane 2645 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: ltrnco4 31221 trljco2 31223 tgrpabl 31233 tendoplcom 31264 tendoicl 31278 cdlemk3 31315 cdlemk12 31332 cdlemk12u 31354 cdlemk46 31430 cdlemk49 31433 dvhvaddcomN 31579 cdlemn4 31681 cdlemn8 31687 dihopelvalcpre 31731 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2385 ax-rep 4280 ax-sep 4290 ax-nul 4298 ax-pow 4337 ax-pr 4363 ax-un 4660 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2258 df-mo 2259 df-clab 2391 df-cleq 2397 df-clel 2400 df-nfc 2529 df-ne 2569 df-nel 2570 df-ral 2671 df-rex 2672 df-reu 2673 df-rmo 2674 df-rab 2675 df-v 2918 df-sbc 3122 df-csb 3212 df-dif 3283 df-un 3285 df-in 3287 df-ss 3294 df-nul 3589 df-if 3700 df-pw 3761 df-sn 3780 df-pr 3781 df-op 3783 df-uni 3976 df-iun 4055 df-iin 4056 df-br 4173 df-opab 4227 df-mpt 4228 df-id 4458 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-iota 5377 df-fun 5415 df-fn 5416 df-f 5417 df-f1 5418 df-fo 5419 df-f1o 5420 df-fv 5421 df-ov 6043 df-oprab 6044 df-mpt2 6045 df-1st 6308 df-2nd 6309 df-undef 6502 df-riota 6508 df-map 6979 df-poset 14358 df-plt 14370 df-lub 14386 df-glb 14387 df-join 14388 df-meet 14389 df-p0 14423 df-p1 14424 df-lat 14430 df-clat 14492 df-oposet 29659 df-ol 29661 df-oml 29662 df-covers 29749 df-ats 29750 df-atl 29781 df-cvlat 29805 df-hlat 29834 df-llines 29980 df-lplanes 29981 df-lvols 29982 df-lines 29983 df-psubsp 29985 df-pmap 29986 df-padd 30278 df-lhyp 30470 df-laut 30471 df-ldil 30586 df-ltrn 30587 df-trl 30641 |
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