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Mirrors > Home > MPE Home > Th. List > latasymd | Structured version Visualization version GIF version |
Description: Deduce equality from lattice ordering. (eqssd 3999 analog.) (Contributed by NM, 18-Nov-2011.) |
Ref | Expression |
---|---|
latasymd.b | β’ π΅ = (BaseβπΎ) |
latasymd.l | β’ β€ = (leβπΎ) |
latasymd.3 | β’ (π β πΎ β Lat) |
latasymd.4 | β’ (π β π β π΅) |
latasymd.5 | β’ (π β π β π΅) |
latasymd.6 | β’ (π β π β€ π) |
latasymd.7 | β’ (π β π β€ π) |
Ref | Expression |
---|---|
latasymd | β’ (π β π = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latasymd.6 | . 2 β’ (π β π β€ π) | |
2 | latasymd.7 | . 2 β’ (π β π β€ π) | |
3 | latasymd.3 | . . 3 β’ (π β πΎ β Lat) | |
4 | latasymd.4 | . . 3 β’ (π β π β π΅) | |
5 | latasymd.5 | . . 3 β’ (π β π β π΅) | |
6 | latasymd.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
7 | latasymd.l | . . . 4 β’ β€ = (leβπΎ) | |
8 | 6, 7 | latasymb 18399 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β ((π β€ π β§ π β€ π) β π = π)) |
9 | 3, 4, 5, 8 | syl3anc 1371 | . 2 β’ (π β ((π β€ π β§ π β€ π) β π = π)) |
10 | 1, 2, 9 | mpbi2and 710 | 1 β’ (π β π = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 class class class wbr 5148 βcfv 6543 Basecbs 17148 lecple 17208 Latclat 18388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-xp 5682 df-dm 5686 df-iota 6495 df-fv 6551 df-proset 18252 df-poset 18270 df-lat 18389 |
This theorem is referenced by: latjidm 18419 latmidm 18431 latjass 18440 oldmm1 38390 olj01 38398 olm01 38409 cvlcvr1 38512 llnmlplnN 38713 2llnjaN 38740 2lplnja 38793 cdlema1N 38965 hlmod1i 39030 lautj 39267 lautm 39268 cdleme19a 39477 cdleme28b 39545 trljco 39914 dochvalr 40531 |
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