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Mirrors > Home > MPE Home > Th. List > latmlem1 | Structured version Visualization version Unicode version |
Description: Add meet to both sides of a lattice ordering. (Contributed by NM, 10-Nov-2011.) |
Ref | Expression |
---|---|
latmle.b |
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latmle.l |
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latmle.m |
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Ref | Expression |
---|---|
latmlem1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latmle.b |
. . . . . 6
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2 | latmle.l |
. . . . . 6
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3 | latmle.m |
. . . . . 6
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4 | 1, 2, 3 | latmle1 16334 |
. . . . 5
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5 | 4 | 3adant3r2 1219 |
. . . 4
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6 | simpl 459 |
. . . . 5
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7 | 1, 3 | latmcl 16310 |
. . . . . 6
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8 | 7 | 3adant3r2 1219 |
. . . . 5
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9 | simpr1 1015 |
. . . . 5
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10 | simpr2 1016 |
. . . . 5
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11 | 1, 2 | lattr 16314 |
. . . . 5
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12 | 6, 8, 9, 10, 11 | syl13anc 1271 |
. . . 4
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13 | 5, 12 | mpand 682 |
. . 3
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14 | 1, 2, 3 | latmle2 16335 |
. . . 4
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15 | 14 | 3adant3r2 1219 |
. . 3
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16 | 13, 15 | jctird 547 |
. 2
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17 | simpr3 1017 |
. . . 4
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18 | 8, 10, 17 | 3jca 1189 |
. . 3
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19 | 1, 2, 3 | latlem12 16336 |
. . 3
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20 | 18, 19 | syldan 473 |
. 2
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21 | 16, 20 | sylibd 218 |
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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-rep 4518 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-ral 2744 df-rex 2745 df-reu 2746 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-op 3977 df-uni 4202 df-iun 4283 df-br 4406 df-opab 4465 df-mpt 4466 df-id 4752 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 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-riota 6257 df-ov 6298 df-oprab 6299 df-poset 16203 df-lub 16232 df-glb 16233 df-join 16234 df-meet 16235 df-lat 16304 |
This theorem is referenced by: latmlem2 16340 latmlem12 16341 dalem25 33275 dalawlem2 33449 dalawlem11 33458 dalawlem12 33459 cdleme22d 33922 cdleme30a 33957 cdleme32c 34022 cdleme32e 34024 trlcolem 34305 cdlemk5u 34440 cdlemk39 34495 cdlemm10N 34698 cdlemn2 34775 dihord1 34798 |
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