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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dibopelval2 | Structured version Visualization version GIF version |
Description: Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.) (Revised by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
dibval2.b | β’ π΅ = (BaseβπΎ) |
dibval2.l | β’ β€ = (leβπΎ) |
dibval2.h | β’ π» = (LHypβπΎ) |
dibval2.t | β’ π = ((LTrnβπΎ)βπ) |
dibval2.o | β’ 0 = (π β π β¦ ( I βΎ π΅)) |
dibval2.j | β’ π½ = ((DIsoAβπΎ)βπ) |
dibval2.i | β’ πΌ = ((DIsoBβπΎ)βπ) |
Ref | Expression |
---|---|
dibopelval2 | β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π)) β (β¨πΉ, πβ© β (πΌβπ) β (πΉ β (π½βπ) β§ π = 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dibval2.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
2 | dibval2.l | . . . 4 β’ β€ = (leβπΎ) | |
3 | dibval2.h | . . . 4 β’ π» = (LHypβπΎ) | |
4 | dibval2.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
5 | dibval2.o | . . . 4 β’ 0 = (π β π β¦ ( I βΎ π΅)) | |
6 | dibval2.j | . . . 4 β’ π½ = ((DIsoAβπΎ)βπ) | |
7 | dibval2.i | . . . 4 β’ πΌ = ((DIsoBβπΎ)βπ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | dibval2 40672 | . . 3 β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π)) β (πΌβπ) = ((π½βπ) Γ { 0 })) |
9 | 8 | eleq2d 2811 | . 2 β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π)) β (β¨πΉ, πβ© β (πΌβπ) β β¨πΉ, πβ© β ((π½βπ) Γ { 0 }))) |
10 | opelxp 5708 | . . 3 β’ (β¨πΉ, πβ© β ((π½βπ) Γ { 0 }) β (πΉ β (π½βπ) β§ π β { 0 })) | |
11 | 4 | fvexi 6905 | . . . . . . 7 β’ π β V |
12 | 11 | mptex 7230 | . . . . . 6 β’ (π β π β¦ ( I βΎ π΅)) β V |
13 | 5, 12 | eqeltri 2821 | . . . . 5 β’ 0 β V |
14 | 13 | elsn2 4663 | . . . 4 β’ (π β { 0 } β π = 0 ) |
15 | 14 | anbi2i 621 | . . 3 β’ ((πΉ β (π½βπ) β§ π β { 0 }) β (πΉ β (π½βπ) β§ π = 0 )) |
16 | 10, 15 | bitri 274 | . 2 β’ (β¨πΉ, πβ© β ((π½βπ) Γ { 0 }) β (πΉ β (π½βπ) β§ π = 0 )) |
17 | 9, 16 | bitrdi 286 | 1 β’ (((πΎ β π β§ π β π») β§ (π β π΅ β§ π β€ π)) β (β¨πΉ, πβ© β (πΌβπ) β (πΉ β (π½βπ) β§ π = 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 Vcvv 3463 {csn 4624 β¨cop 4630 class class class wbr 5143 β¦ cmpt 5226 I cid 5569 Γ cxp 5670 βΎ cres 5674 βcfv 6542 Basecbs 17177 lecple 17237 LHypclh 39512 LTrncltrn 39629 DIsoAcdia 40556 DIsoBcdib 40666 |
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-disoa 40557 df-dib 40667 |
This theorem is referenced by: dibopelval3 40676 dibglbN 40694 diblsmopel 40699 dib2dim 40771 dih2dimbALTN 40773 dihord6apre 40784 |
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