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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dibval3N | Structured version Visualization version GIF version |
Description: Value of the partial isomorphism B for a lattice πΎ. (Contributed by NM, 24-Feb-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dibval3.b | β’ π΅ = (BaseβπΎ) |
dibval3.l | β’ β€ = (leβπΎ) |
dibval3.h | β’ π» = (LHypβπΎ) |
dibval3.t | β’ π = ((LTrnβπΎ)βπ) |
dibval3.r | β’ π = ((trLβπΎ)βπ) |
dibval3.o | β’ 0 = (π β π β¦ ( I βΎ π΅)) |
dibval3.i | β’ πΌ = ((DIsoBβπΎ)βπ) |
Ref | Expression |
---|---|
dibval3N | β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π)) β (πΌβπ) = ({π β π β£ (π βπ) β€ π} Γ { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dibval3.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | dibval3.l | . . 3 β’ β€ = (leβπΎ) | |
3 | dibval3.h | . . 3 β’ π» = (LHypβπΎ) | |
4 | dibval3.t | . . 3 β’ π = ((LTrnβπΎ)βπ) | |
5 | dibval3.o | . . 3 β’ 0 = (π β π β¦ ( I βΎ π΅)) | |
6 | eqid 2728 | . . 3 β’ ((DIsoAβπΎ)βπ) = ((DIsoAβπΎ)βπ) | |
7 | dibval3.i | . . 3 β’ πΌ = ((DIsoBβπΎ)βπ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | dibval2 40649 | . 2 β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π)) β (πΌβπ) = ((((DIsoAβπΎ)βπ)βπ) Γ { 0 })) |
9 | dibval3.r | . . . 4 β’ π = ((trLβπΎ)βπ) | |
10 | 1, 2, 3, 4, 9, 6 | diaval 40537 | . . 3 β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π)) β (((DIsoAβπΎ)βπ)βπ) = {π β π β£ (π βπ) β€ π}) |
11 | 10 | xpeq1d 5711 | . 2 β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π)) β ((((DIsoAβπΎ)βπ)βπ) Γ { 0 }) = ({π β π β£ (π βπ) β€ π} Γ { 0 })) |
12 | 8, 11 | eqtrd 2768 | 1 β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π)) β (πΌβπ) = ({π β π β£ (π βπ) β€ π} Γ { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 {crab 3430 {csn 4632 class class class wbr 5152 β¦ cmpt 5235 I cid 5579 Γ cxp 5680 βΎ cres 5684 βcfv 6553 Basecbs 17187 lecple 17247 LHypclh 39489 LTrncltrn 39606 trLctrl 39663 DIsoAcdia 40533 DIsoBcdib 40643 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-disoa 40534 df-dib 40644 |
This theorem is referenced by: (None) |
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