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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 40614 | . . . 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 6899 | . 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 6897 | . . . . 5 β’ (π₯ = π β (π½βπ₯) = (π½βπ)) | |
11 | 10 | xpeq1d 5707 | . . . 4 β’ (π₯ = π β ((π½βπ₯) Γ { 0 }) = ((π½βπ) Γ { 0 })) |
12 | eqid 2728 | . . . 4 β’ (π₯ β dom π½ β¦ ((π½βπ₯) Γ { 0 })) = (π₯ β dom π½ β¦ ((π½βπ₯) Γ { 0 })) | |
13 | fvex 6910 | . . . . 5 β’ (π½βπ) β V | |
14 | snex 5433 | . . . . 5 β’ { 0 } β V | |
15 | 13, 14 | xpex 7755 | . . . 4 β’ ((π½βπ) Γ { 0 }) β V |
16 | 11, 12, 15 | fvmpt 7005 | . . 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 2768 | 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 1534 β wcel 2099 {csn 4629 β¦ cmpt 5231 I cid 5575 Γ cxp 5676 dom cdm 5678 βΎ cres 5680 βcfv 6548 Basecbs 17180 LHypclh 39457 LTrncltrn 39574 DIsoAcdia 40501 DIsoBcdib 40611 |
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 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-dib 40612 |
This theorem is referenced by: dibopelvalN 40616 dibval2 40617 dibvalrel 40636 |
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