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Mirrors > Home > MPE Home > Th. List > Mathboxes > latmassOLD | Structured version Visualization version Unicode version |
Description: Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 3823 analog.) (Contributed by NM, 7-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
olmass.b | |
olmass.m |
Ref | Expression |
---|---|
latmassOLD |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 473 | . . . 4 | |
2 | ollat 34500 | . . . . . 6 | |
3 | 2 | adantr 481 | . . . . 5 |
4 | olop 34501 | . . . . . . 7 | |
5 | 4 | adantr 481 | . . . . . 6 |
6 | simpr1 1067 | . . . . . 6 | |
7 | olmass.b | . . . . . . 7 | |
8 | eqid 2622 | . . . . . . 7 | |
9 | 7, 8 | opoccl 34481 | . . . . . 6 |
10 | 5, 6, 9 | syl2anc 693 | . . . . 5 |
11 | simpr2 1068 | . . . . . 6 | |
12 | 7, 8 | opoccl 34481 | . . . . . 6 |
13 | 5, 11, 12 | syl2anc 693 | . . . . 5 |
14 | eqid 2622 | . . . . . 6 | |
15 | 7, 14 | latjcl 17051 | . . . . 5 |
16 | 3, 10, 13, 15 | syl3anc 1326 | . . . 4 |
17 | simpr3 1069 | . . . 4 | |
18 | olmass.m | . . . . 5 | |
19 | 7, 14, 18, 8 | oldmj3 34510 | . . . 4 |
20 | 1, 16, 17, 19 | syl3anc 1326 | . . 3 |
21 | 7, 8 | opoccl 34481 | . . . . . 6 |
22 | 5, 17, 21 | syl2anc 693 | . . . . 5 |
23 | 7, 14 | latjass 17095 | . . . . 5 |
24 | 3, 10, 13, 22, 23 | syl13anc 1328 | . . . 4 |
25 | 24 | fveq2d 6195 | . . 3 |
26 | 7, 14, 18, 8 | oldmj4 34511 | . . . . 5 |
27 | 26 | 3adant3r3 1276 | . . . 4 |
28 | 27 | oveq1d 6665 | . . 3 |
29 | 20, 25, 28 | 3eqtr3rd 2665 | . 2 |
30 | 7, 14 | latjcl 17051 | . . . 4 |
31 | 3, 13, 22, 30 | syl3anc 1326 | . . 3 |
32 | 7, 14, 18, 8 | oldmj2 34509 | . . 3 |
33 | 1, 6, 31, 32 | syl3anc 1326 | . 2 |
34 | 7, 14, 18, 8 | oldmj4 34511 | . . . 4 |
35 | 34 | 3adant3r1 1274 | . . 3 |
36 | 35 | oveq2d 6666 | . 2 |
37 | 29, 33, 36 | 3eqtrd 2660 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 cfv 5888 (class class class)co 6650 cbs 15857 coc 15949 cjn 16944 cmee 16945 clat 17045 cops 34459 col 34461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-preset 16928 df-poset 16946 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-lat 17046 df-oposet 34463 df-ol 34465 |
This theorem is referenced by: latm12 34517 latm32 34518 latmrot 34519 latm4 34520 cmtcomlemN 34535 cmtbr3N 34541 omlfh1N 34545 dalawlem2 35158 dalawlem7 35163 dalawlem11 35167 dalawlem12 35168 lhp2at0 35318 cdleme20d 35600 cdleme23b 35638 cdlemh2 36104 dia2dimlem2 36354 dihmeetbclemN 36593 |
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