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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > olm12 | Structured version Visualization version GIF version |
Description: The meet of an ortholattice element with one equals itself. (Contributed by NM, 22-May-2012.) |
Ref | Expression |
---|---|
olm1.b | β’ π΅ = (BaseβπΎ) |
olm1.m | β’ β§ = (meetβπΎ) |
olm1.u | β’ 1 = (1.βπΎ) |
Ref | Expression |
---|---|
olm12 | β’ ((πΎ β OL β§ π β π΅) β ( 1 β§ π) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ollat 37678 | . . . 4 β’ (πΎ β OL β πΎ β Lat) | |
2 | 1 | adantr 482 | . . 3 β’ ((πΎ β OL β§ π β π΅) β πΎ β Lat) |
3 | olop 37679 | . . . . 5 β’ (πΎ β OL β πΎ β OP) | |
4 | 3 | adantr 482 | . . . 4 β’ ((πΎ β OL β§ π β π΅) β πΎ β OP) |
5 | olm1.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
6 | olm1.u | . . . . 5 β’ 1 = (1.βπΎ) | |
7 | 5, 6 | op1cl 37650 | . . . 4 β’ (πΎ β OP β 1 β π΅) |
8 | 4, 7 | syl 17 | . . 3 β’ ((πΎ β OL β§ π β π΅) β 1 β π΅) |
9 | simpr 486 | . . 3 β’ ((πΎ β OL β§ π β π΅) β π β π΅) | |
10 | olm1.m | . . . 4 β’ β§ = (meetβπΎ) | |
11 | 5, 10 | latmcom 18353 | . . 3 β’ ((πΎ β Lat β§ 1 β π΅ β§ π β π΅) β ( 1 β§ π) = (π β§ 1 )) |
12 | 2, 8, 9, 11 | syl3anc 1372 | . 2 β’ ((πΎ β OL β§ π β π΅) β ( 1 β§ π) = (π β§ 1 )) |
13 | 5, 10, 6 | olm11 37692 | . 2 β’ ((πΎ β OL β§ π β π΅) β (π β§ 1 ) = π) |
14 | 12, 13 | eqtrd 2777 | 1 β’ ((πΎ β OL β§ π β π΅) β ( 1 β§ π) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βcfv 6497 (class class class)co 7358 Basecbs 17084 meetcmee 18202 1.cp1 18314 Latclat 18321 OPcops 37637 OLcol 37639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-proset 18185 df-poset 18203 df-lub 18236 df-glb 18237 df-join 18238 df-meet 18239 df-p0 18315 df-p1 18316 df-lat 18322 df-oposet 37641 df-ol 37643 |
This theorem is referenced by: dih1 39752 dihjatc 39883 |
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