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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 38745 | . . . 4 β’ (πΎ β HL β πΎ β OP) | |
2 | 1 | adantr 480 | . . 3 β’ ((πΎ β HL β§ π β π») β πΎ β OP) |
3 | eqid 2726 | . . . 4 β’ (BaseβπΎ) = (BaseβπΎ) | |
4 | dia0eldm.z | . . . 4 β’ 0 = (0.βπΎ) | |
5 | 3, 4 | op0cl 38567 | . . 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 39382 | . . 3 β’ (π β π» β π β (BaseβπΎ)) |
9 | eqid 2726 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
10 | 3, 9, 4 | op0le 38569 | . . 3 β’ ((πΎ β OP β§ π β (BaseβπΎ)) β 0 (leβπΎ)π) |
11 | 1, 8, 10 | syl2an 595 | . 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 40420 | . 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 710 | 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 395 = wceq 1533 β wcel 2098 class class class wbr 5141 dom cdm 5669 βcfv 6537 Basecbs 17153 lecple 17213 0.cp0 18388 OPcops 38555 HLchlt 38733 LHypclh 39368 DIsoAcdia 40412 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-glb 18312 df-p0 18390 df-oposet 38559 df-ol 38561 df-oml 38562 df-hlat 38734 df-lhyp 39372 df-disoa 40413 |
This theorem is referenced by: (None) |
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