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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dibelval1st2N | Structured version Visualization version GIF version |
Description: Membership in value of the partial isomorphism B for a lattice πΎ. (Contributed by NM, 13-Feb-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dibelval1st2.b | β’ π΅ = (BaseβπΎ) |
dibelval1st2.l | β’ β€ = (leβπΎ) |
dibelval1st2.h | β’ π» = (LHypβπΎ) |
dibelval1st2.t | β’ π = ((LTrnβπΎ)βπ) |
dibelval1st2.r | β’ π = ((trLβπΎ)βπ) |
dibelval1st2.i | β’ πΌ = ((DIsoBβπΎ)βπ) |
Ref | Expression |
---|---|
dibelval1st2N | β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π) β§ π β (πΌβπ)) β (π β(1st βπ)) β€ π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dibelval1st2.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | dibelval1st2.l | . . 3 β’ β€ = (leβπΎ) | |
3 | dibelval1st2.h | . . 3 β’ π» = (LHypβπΎ) | |
4 | eqid 2725 | . . 3 β’ ((DIsoAβπΎ)βπ) = ((DIsoAβπΎ)βπ) | |
5 | dibelval1st2.i | . . 3 β’ πΌ = ((DIsoBβπΎ)βπ) | |
6 | 1, 2, 3, 4, 5 | dibelval1st 40674 | . 2 β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π) β§ π β (πΌβπ)) β (1st βπ) β (((DIsoAβπΎ)βπ)βπ)) |
7 | dibelval1st2.t | . . 3 β’ π = ((LTrnβπΎ)βπ) | |
8 | dibelval1st2.r | . . 3 β’ π = ((trLβπΎ)βπ) | |
9 | 1, 2, 3, 7, 8, 4 | diatrl 40569 | . 2 β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π) β§ (1st βπ) β (((DIsoAβπΎ)βπ)βπ)) β (π β(1st βπ)) β€ π) |
10 | 6, 9 | syld3an3 1406 | 1 β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π) β§ π β (πΌβπ)) β (π β(1st βπ)) β€ π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5144 βcfv 6543 1st c1st 7985 Basecbs 17174 lecple 17234 LHypclh 39509 LTrncltrn 39626 trLctrl 39683 DIsoAcdia 40553 DIsoBcdib 40663 |
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 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 |
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-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 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-1st 7987 df-disoa 40554 df-dib 40664 |
This theorem is referenced by: (None) |
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