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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dia0eldmN | Structured version Visualization version GIF version |
Description: The lattice zero belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dia0eldm.z | β’ 0 = (0.βπΎ) |
dia0eldm.h | β’ π» = (LHypβπΎ) |
dia0eldm.i | β’ πΌ = ((DIsoAβπΎ)βπ) |
Ref | Expression |
---|---|
dia0eldmN | β’ ((πΎ β HL β§ π β π») β 0 β dom πΌ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlop 38890 | . . . 4 β’ (πΎ β HL β πΎ β OP) | |
2 | 1 | adantr 479 | . . 3 β’ ((πΎ β HL β§ π β π») β πΎ β OP) |
3 | eqid 2725 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
4 | dia0eldm.z | . . . 4 β’ 0 = (0.βπΎ) | |
5 | 3, 4 | op0cl 38712 | . . 3 β’ (πΎ β OP β 0 β (BaseβπΎ)) |
6 | 2, 5 | syl 17 | . 2 β’ ((πΎ β HL β§ π β π») β 0 β (BaseβπΎ)) |
7 | dia0eldm.h | . . . 4 β’ π» = (LHypβπΎ) | |
8 | 3, 7 | lhpbase 39527 | . . 3 β’ (π β π» β π β (BaseβπΎ)) |
9 | eqid 2725 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
10 | 3, 9, 4 | op0le 38714 | . . 3 β’ ((πΎ β OP β§ π β (BaseβπΎ)) β 0 (leβπΎ)π) |
11 | 1, 8, 10 | syl2an 594 | . 2 β’ ((πΎ β HL β§ π β π») β 0 (leβπΎ)π) |
12 | dia0eldm.i | . . 3 β’ πΌ = ((DIsoAβπΎ)βπ) | |
13 | 3, 9, 7, 12 | diaeldm 40565 | . 2 β’ ((πΎ β HL β§ π β π») β ( 0 β dom πΌ β ( 0 β (BaseβπΎ) β§ 0 (leβπΎ)π))) |
14 | 6, 11, 13 | mpbir2and 711 | 1 β’ ((πΎ β HL β§ π β π») β 0 β dom πΌ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 class class class wbr 5143 dom cdm 5672 βcfv 6543 Basecbs 17179 lecple 17239 0.cp0 18414 OPcops 38700 HLchlt 38878 LHypclh 39513 DIsoAcdia 40557 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-glb 18338 df-p0 18416 df-oposet 38704 df-ol 38706 df-oml 38707 df-hlat 38879 df-lhyp 39517 df-disoa 40558 |
This theorem is referenced by: (None) |
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