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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dibelval1st | 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.) |
Ref | Expression |
---|---|
dibelval1.b | β’ π΅ = (BaseβπΎ) |
dibelval1.l | β’ β€ = (leβπΎ) |
dibelval1.h | β’ π» = (LHypβπΎ) |
dibelval1.j | β’ π½ = ((DIsoAβπΎ)βπ) |
dibelval1.i | β’ πΌ = ((DIsoBβπΎ)βπ) |
Ref | Expression |
---|---|
dibelval1st | β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π) β§ π β (πΌβπ)) β (1st βπ) β (π½βπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dibelval1.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
2 | dibelval1.l | . . . . 5 β’ β€ = (leβπΎ) | |
3 | dibelval1.h | . . . . 5 β’ π» = (LHypβπΎ) | |
4 | eqid 2725 | . . . . 5 β’ ((LTrnβπΎ)βπ) = ((LTrnβπΎ)βπ) | |
5 | eqid 2725 | . . . . 5 β’ (π β ((LTrnβπΎ)βπ) β¦ ( I βΎ π΅)) = (π β ((LTrnβπΎ)βπ) β¦ ( I βΎ π΅)) | |
6 | dibelval1.j | . . . . 5 β’ π½ = ((DIsoAβπΎ)βπ) | |
7 | dibelval1.i | . . . . 5 β’ πΌ = ((DIsoBβπΎ)βπ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | dibval2 40672 | . . . 4 β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π)) β (πΌβπ) = ((π½βπ) Γ {(π β ((LTrnβπΎ)βπ) β¦ ( I βΎ π΅))})) |
9 | 8 | eleq2d 2811 | . . 3 β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π)) β (π β (πΌβπ) β π β ((π½βπ) Γ {(π β ((LTrnβπΎ)βπ) β¦ ( I βΎ π΅))}))) |
10 | 9 | biimp3a 1465 | . 2 β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π) β§ π β (πΌβπ)) β π β ((π½βπ) Γ {(π β ((LTrnβπΎ)βπ) β¦ ( I βΎ π΅))})) |
11 | xp1st 8021 | . 2 β’ (π β ((π½βπ) Γ {(π β ((LTrnβπΎ)βπ) β¦ ( I βΎ π΅))}) β (1st βπ) β (π½βπ)) | |
12 | 10, 11 | syl 17 | 1 β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π) β§ π β (πΌβπ)) β (1st βπ) β (π½βπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 {csn 4624 class class class wbr 5143 β¦ cmpt 5226 I cid 5569 Γ cxp 5670 βΎ cres 5674 βcfv 6542 1st c1st 7987 Basecbs 17177 lecple 17237 LHypclh 39512 LTrncltrn 39629 DIsoAcdia 40556 DIsoBcdib 40666 |
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 ax-un 7737 |
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 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 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 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-1st 7989 df-disoa 40557 df-dib 40667 |
This theorem is referenced by: dibelval1st1 40678 dibelval1st2N 40679 diblss 40698 |
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