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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dicopelval2 | Structured version Visualization version GIF version |
Description: Membership in value of the partial isomorphism C for a lattice πΎ. (Contributed by NM, 20-Feb-2014.) |
Ref | Expression |
---|---|
dicval.l | β’ β€ = (leβπΎ) |
dicval.a | β’ π΄ = (AtomsβπΎ) |
dicval.h | β’ π» = (LHypβπΎ) |
dicval.p | β’ π = ((ocβπΎ)βπ) |
dicval.t | β’ π = ((LTrnβπΎ)βπ) |
dicval.e | β’ πΈ = ((TEndoβπΎ)βπ) |
dicval.i | β’ πΌ = ((DIsoCβπΎ)βπ) |
dicval2.g | β’ πΊ = (β©π β π (πβπ) = π) |
dicelval2.f | β’ πΉ β V |
dicelval2.s | β’ π β V |
Ref | Expression |
---|---|
dicopelval2 | β’ (((πΎ β π β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β (β¨πΉ, πβ© β (πΌβπ) β (πΉ = (πβπΊ) β§ π β πΈ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dicval.l | . . 3 β’ β€ = (leβπΎ) | |
2 | dicval.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
3 | dicval.h | . . 3 β’ π» = (LHypβπΎ) | |
4 | dicval.p | . . 3 β’ π = ((ocβπΎ)βπ) | |
5 | dicval.t | . . 3 β’ π = ((LTrnβπΎ)βπ) | |
6 | dicval.e | . . 3 β’ πΈ = ((TEndoβπΎ)βπ) | |
7 | dicval.i | . . 3 β’ πΌ = ((DIsoCβπΎ)βπ) | |
8 | dicelval2.f | . . 3 β’ πΉ β V | |
9 | dicelval2.s | . . 3 β’ π β V | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | dicopelval 40705 | . 2 β’ (((πΎ β π β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β (β¨πΉ, πβ© β (πΌβπ) β (πΉ = (πβ(β©π β π (πβπ) = π)) β§ π β πΈ))) |
11 | dicval2.g | . . . . 5 β’ πΊ = (β©π β π (πβπ) = π) | |
12 | 11 | fveq2i 6894 | . . . 4 β’ (πβπΊ) = (πβ(β©π β π (πβπ) = π)) |
13 | 12 | eqeq2i 2738 | . . 3 β’ (πΉ = (πβπΊ) β πΉ = (πβ(β©π β π (πβπ) = π))) |
14 | 13 | anbi1i 622 | . 2 β’ ((πΉ = (πβπΊ) β§ π β πΈ) β (πΉ = (πβ(β©π β π (πβπ) = π)) β§ π β πΈ)) |
15 | 10, 14 | bitr4di 288 | 1 β’ (((πΎ β π β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β (β¨πΉ, πβ© β (πΌβπ) β (πΉ = (πβπΊ) β§ π β πΈ))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 Vcvv 3463 β¨cop 4630 class class class wbr 5143 βcfv 6542 β©crio 7370 lecple 17237 occoc 17238 Atomscatm 38790 LHypclh 39512 LTrncltrn 39629 TEndoctendo 40280 DIsoCcdic 40700 |
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-riota 7371 df-dic 40701 |
This theorem is referenced by: diclspsn 40722 cdlemn11a 40735 dihopelvalcqat 40774 dihopelvalcpre 40776 dihord6apre 40784 |
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