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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 5142 | . 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 18319 | . . 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 3605 | 1 β’ (π β π β€ (πβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βwral 3053 class class class wbr 5139 dom cdm 5667 βcfv 6534 Basecbs 17149 lecple 17209 lubclub 18270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-lub 18307 |
This theorem is referenced by: ple1 18391 lubsscl 47840 |
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