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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ople1 | Structured version Visualization version GIF version |
Description: Any element is less than the orthoposet unity. (chss 30174 analog.) (Contributed by NM, 23-Oct-2011.) |
Ref | Expression |
---|---|
ople1.b | β’ π΅ = (BaseβπΎ) |
ople1.l | β’ β€ = (leβπΎ) |
ople1.u | β’ 1 = (1.βπΎ) |
Ref | Expression |
---|---|
ople1 | β’ ((πΎ β OP β§ π β π΅) β π β€ 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ople1.b | . 2 β’ π΅ = (BaseβπΎ) | |
2 | eqid 2737 | . 2 β’ (lubβπΎ) = (lubβπΎ) | |
3 | ople1.l | . 2 β’ β€ = (leβπΎ) | |
4 | ople1.u | . 2 β’ 1 = (1.βπΎ) | |
5 | simpl 484 | . 2 β’ ((πΎ β OP β§ π β π΅) β πΎ β OP) | |
6 | simpr 486 | . 2 β’ ((πΎ β OP β§ π β π΅) β π β π΅) | |
7 | eqid 2737 | . . . . 5 β’ (glbβπΎ) = (glbβπΎ) | |
8 | 1, 2, 7 | op01dm 37648 | . . . 4 β’ (πΎ β OP β (π΅ β dom (lubβπΎ) β§ π΅ β dom (glbβπΎ))) |
9 | 8 | simpld 496 | . . 3 β’ (πΎ β OP β π΅ β dom (lubβπΎ)) |
10 | 9 | adantr 482 | . 2 β’ ((πΎ β OP β§ π β π΅) β π΅ β dom (lubβπΎ)) |
11 | 1, 2, 3, 4, 5, 6, 10 | ple1 18320 | 1 β’ ((πΎ β OP β§ π β π΅) β π β€ 1 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 class class class wbr 5106 dom cdm 5634 βcfv 6497 Basecbs 17084 lecple 17141 lubclub 18199 glbcglb 18200 1.cp1 18314 OPcops 37637 |
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 |
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-lub 18236 df-p1 18316 df-oposet 37641 |
This theorem is referenced by: op1le 37657 glb0N 37658 opoc1 37667 ncvr1 37737 1cvrat 37942 pmap1N 38233 pol1N 38376 dih1 39752 dihjatc 39883 |
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