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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 2726 | . . . . 5 β’ ((LTrnβπΎ)βπ) = ((LTrnβπΎ)βπ) | |
5 | eqid 2726 | . . . . 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 40528 | . . . 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 2813 | . . 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 8006 | . 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 {csn 4623 class class class wbr 5141 β¦ cmpt 5224 I cid 5566 Γ cxp 5667 βΎ cres 5671 βcfv 6537 1st c1st 7972 Basecbs 17153 lecple 17213 LHypclh 39368 LTrncltrn 39485 DIsoAcdia 40412 DIsoBcdib 40522 |
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 ax-un 7722 |
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-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-1st 7974 df-disoa 40413 df-dib 40523 |
This theorem is referenced by: dibelval1st1 40534 dibelval1st2N 40535 diblss 40554 |
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