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Mirrors > Home > MPE Home > Th. List > latasymd | Structured version Visualization version GIF version |
Description: Deduce equality from lattice ordering. (eqssd 3998 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 1369 | . 2 β’ (π β ((π β€ π β§ π β€ π) β π = π)) |
10 | 1, 2, 9 | mpbi2and 708 | 1 β’ (π β π = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1539 β wcel 2104 class class class wbr 5147 βcfv 6542 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 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-nul 5305 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-xp 5681 df-dm 5685 df-iota 6494 df-fv 6550 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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