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Mirrors > Home > MPE Home > Th. List > luble | Structured version Visualization version GIF version |
Description: The greatest lower bound is the least element. (Contributed by NM, 22-Oct-2011.) (Revised by NM, 7-Sep-2018.) |
Ref | Expression |
---|---|
lubprop.b | β’ π΅ = (BaseβπΎ) |
lubprop.l | β’ β€ = (leβπΎ) |
lubprop.u | β’ π = (lubβπΎ) |
lubprop.k | β’ (π β πΎ β π) |
lubprop.s | β’ (π β π β dom π) |
luble.x | β’ (π β π β π) |
Ref | Expression |
---|---|
luble | β’ (π β π β€ (πβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5151 | . 2 β’ (π¦ = π β (π¦ β€ (πβπ) β π β€ (πβπ))) | |
2 | lubprop.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
3 | lubprop.l | . . . 4 β’ β€ = (leβπΎ) | |
4 | lubprop.u | . . . 4 β’ π = (lubβπΎ) | |
5 | lubprop.k | . . . 4 β’ (π β πΎ β π) | |
6 | lubprop.s | . . . 4 β’ (π β π β dom π) | |
7 | 2, 3, 4, 5, 6 | lubprop 18349 | . . 3 β’ (π β (βπ¦ β π π¦ β€ (πβπ) β§ βπ§ β π΅ (βπ¦ β π π¦ β€ π§ β (πβπ) β€ π§))) |
8 | 7 | simpld 494 | . 2 β’ (π β βπ¦ β π π¦ β€ (πβπ)) |
9 | luble.x | . 2 β’ (π β π β π) | |
10 | 1, 8, 9 | rspcdva 3610 | 1 β’ (π β π β€ (πβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 βwral 3058 class class class wbr 5148 dom cdm 5678 βcfv 6548 Basecbs 17179 lecple 17239 lubclub 18300 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-lub 18337 |
This theorem is referenced by: ple1 18421 lubsscl 47979 |
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