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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dibval | Structured version Visualization version GIF version |
Description: The partial isomorphism B for a lattice πΎ. (Contributed by NM, 8-Dec-2013.) |
Ref | Expression |
---|---|
dibval.b | β’ π΅ = (BaseβπΎ) |
dibval.h | β’ π» = (LHypβπΎ) |
dibval.t | β’ π = ((LTrnβπΎ)βπ) |
dibval.o | β’ 0 = (π β π β¦ ( I βΎ π΅)) |
dibval.j | β’ π½ = ((DIsoAβπΎ)βπ) |
dibval.i | β’ πΌ = ((DIsoBβπΎ)βπ) |
Ref | Expression |
---|---|
dibval | β’ (((πΎ β π β§ π β π») β§ π β dom π½) β (πΌβπ) = ((π½βπ) Γ { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dibval.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
2 | dibval.h | . . . . 5 β’ π» = (LHypβπΎ) | |
3 | dibval.t | . . . . 5 β’ π = ((LTrnβπΎ)βπ) | |
4 | dibval.o | . . . . 5 β’ 0 = (π β π β¦ ( I βΎ π΅)) | |
5 | dibval.j | . . . . 5 β’ π½ = ((DIsoAβπΎ)βπ) | |
6 | dibval.i | . . . . 5 β’ πΌ = ((DIsoBβπΎ)βπ) | |
7 | 1, 2, 3, 4, 5, 6 | dibfval 40523 | . . . 4 β’ ((πΎ β π β§ π β π») β πΌ = (π₯ β dom π½ β¦ ((π½βπ₯) Γ { 0 }))) |
8 | 7 | adantr 480 | . . 3 β’ (((πΎ β π β§ π β π») β§ π β dom π½) β πΌ = (π₯ β dom π½ β¦ ((π½βπ₯) Γ { 0 }))) |
9 | 8 | fveq1d 6886 | . 2 β’ (((πΎ β π β§ π β π») β§ π β dom π½) β (πΌβπ) = ((π₯ β dom π½ β¦ ((π½βπ₯) Γ { 0 }))βπ)) |
10 | fveq2 6884 | . . . . 5 β’ (π₯ = π β (π½βπ₯) = (π½βπ)) | |
11 | 10 | xpeq1d 5698 | . . . 4 β’ (π₯ = π β ((π½βπ₯) Γ { 0 }) = ((π½βπ) Γ { 0 })) |
12 | eqid 2726 | . . . 4 β’ (π₯ β dom π½ β¦ ((π½βπ₯) Γ { 0 })) = (π₯ β dom π½ β¦ ((π½βπ₯) Γ { 0 })) | |
13 | fvex 6897 | . . . . 5 β’ (π½βπ) β V | |
14 | snex 5424 | . . . . 5 β’ { 0 } β V | |
15 | 13, 14 | xpex 7736 | . . . 4 β’ ((π½βπ) Γ { 0 }) β V |
16 | 11, 12, 15 | fvmpt 6991 | . . 3 β’ (π β dom π½ β ((π₯ β dom π½ β¦ ((π½βπ₯) Γ { 0 }))βπ) = ((π½βπ) Γ { 0 })) |
17 | 16 | adantl 481 | . 2 β’ (((πΎ β π β§ π β π») β§ π β dom π½) β ((π₯ β dom π½ β¦ ((π½βπ₯) Γ { 0 }))βπ) = ((π½βπ) Γ { 0 })) |
18 | 9, 17 | eqtrd 2766 | 1 β’ (((πΎ β π β§ π β π») β§ π β dom π½) β (πΌβπ) = ((π½βπ) Γ { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 {csn 4623 β¦ cmpt 5224 I cid 5566 Γ cxp 5667 dom cdm 5669 βΎ cres 5671 βcfv 6536 Basecbs 17151 LHypclh 39366 LTrncltrn 39483 DIsoAcdia 40410 DIsoBcdib 40520 |
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 7721 |
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 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-dib 40521 |
This theorem is referenced by: dibopelvalN 40525 dibval2 40526 dibvalrel 40545 |
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