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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > atmod3i1 | Structured version Visualization version Unicode version |
Description: Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 4-Jun-2012.) (Revised by Mario Carneiro, 10-May-2013.) |
Ref | Expression |
---|---|
atmod.b |
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atmod.l |
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atmod.j |
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atmod.m |
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atmod.a |
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Ref | Expression |
---|---|
atmod3i1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1008 |
. . 3
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2 | simp21 1041 |
. . 3
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3 | simp23 1043 |
. . 3
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4 | simp22 1042 |
. . 3
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5 | simp3 1010 |
. . 3
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6 | atmod.b |
. . . 4
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7 | atmod.l |
. . . 4
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8 | atmod.j |
. . . 4
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9 | atmod.m |
. . . 4
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10 | atmod.a |
. . . 4
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11 | 6, 7, 8, 9, 10 | atmod1i1 33422 |
. . 3
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12 | 1, 2, 3, 4, 5, 11 | syl131anc 1281 |
. 2
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13 | hllat 32929 |
. . . . 5
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14 | 13 | 3ad2ant1 1029 |
. . . 4
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15 | 6, 9 | latmcom 16321 |
. . . 4
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16 | 14, 4, 3, 15 | syl3anc 1268 |
. . 3
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17 | 16 | oveq2d 6306 |
. 2
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18 | 6, 10 | atbase 32855 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 2, 18 | syl 17 |
. . . 4
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20 | 6, 8 | latjcl 16297 |
. . . 4
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21 | 14, 19, 3, 20 | syl3anc 1268 |
. . 3
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22 | 6, 9 | latmcom 16321 |
. . 3
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23 | 14, 4, 21, 22 | syl3anc 1268 |
. 2
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24 | 12, 17, 23 | 3eqtr4d 2495 |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-rep 4515 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-reu 2744 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-op 3975 df-uni 4199 df-iun 4280 df-iin 4281 df-br 4403 df-opab 4462 df-mpt 4463 df-id 4749 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-riota 6252 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-1st 6793 df-2nd 6794 df-preset 16173 df-poset 16191 df-plt 16204 df-lub 16220 df-glb 16221 df-join 16222 df-meet 16223 df-p0 16285 df-lat 16292 df-clat 16354 df-oposet 32742 df-ol 32744 df-oml 32745 df-covers 32832 df-ats 32833 df-atl 32864 df-cvlat 32888 df-hlat 32917 df-psubsp 33068 df-pmap 33069 df-padd 33361 |
This theorem is referenced by: dalawlem2 33437 dalawlem3 33438 dalawlem6 33441 lhpmcvr3 33590 cdleme0cp 33780 cdleme0cq 33781 cdleme1 33793 cdleme4 33804 cdleme5 33806 cdleme8 33816 cdleme9 33819 cdleme10 33820 cdleme15b 33841 cdleme22e 33911 cdleme22eALTN 33912 cdleme23c 33918 cdleme35b 34017 cdleme35e 34020 cdleme42a 34038 trlcoabs2N 34289 cdlemi1 34385 cdlemk4 34401 dia2dimlem1 34632 dia2dimlem2 34633 cdlemn10 34774 dihglbcpreN 34868 |
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