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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dibelval1st1 | 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 |
---|---|
dibelval1st1.b | β’ π΅ = (BaseβπΎ) |
dibelval1st1.l | β’ β€ = (leβπΎ) |
dibelval1st1.h | β’ π» = (LHypβπΎ) |
dibelval1st1.t | β’ π = ((LTrnβπΎ)βπ) |
dibelval1st1.i | β’ πΌ = ((DIsoBβπΎ)βπ) |
Ref | Expression |
---|---|
dibelval1st1 | β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π) β§ π β (πΌβπ)) β (1st βπ) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dibelval1st1.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | dibelval1st1.l | . . 3 β’ β€ = (leβπΎ) | |
3 | dibelval1st1.h | . . 3 β’ π» = (LHypβπΎ) | |
4 | eqid 2727 | . . 3 β’ ((DIsoAβπΎ)βπ) = ((DIsoAβπΎ)βπ) | |
5 | dibelval1st1.i | . . 3 β’ πΌ = ((DIsoBβπΎ)βπ) | |
6 | 1, 2, 3, 4, 5 | dibelval1st 40546 | . 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 | dibelval1st1.t | . . 3 β’ π = ((LTrnβπΎ)βπ) | |
8 | 1, 2, 3, 7, 4 | diael 40440 | . 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 (1st βπ) β π) |
9 | 6, 8 | syld3an3 1407 | 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 π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 class class class wbr 5142 βcfv 6542 1st c1st 7983 Basecbs 17165 lecple 17225 LHypclh 39381 LTrncltrn 39498 DIsoAcdia 40425 DIsoBcdib 40535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 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 5143 df-opab 5205 df-mpt 5226 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 7985 df-disoa 40426 df-dib 40536 |
This theorem is referenced by: diblss 40567 |
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