![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 39610 | . . 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 2824 | . 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 5670 | . . 3 β’ (β¨πΉ, πβ© β ((π½βπ) Γ { 0 }) β (πΉ β (π½βπ) β§ π β { 0 })) | |
11 | 4 | fvexi 6857 | . . . . . . 7 β’ π β V |
12 | 11 | mptex 7174 | . . . . . 6 β’ (π β π β¦ ( I βΎ π΅)) β V |
13 | 5, 12 | eqeltri 2834 | . . . . 5 β’ 0 β V |
14 | 13 | elsn2 4626 | . . . 4 β’ (π β { 0 } β π = 0 ) |
15 | 14 | anbi2i 624 | . . 3 β’ ((πΉ β (π½βπ) β§ π β { 0 }) β (πΉ β (π½βπ) β§ π = 0 )) |
16 | 10, 15 | bitri 275 | . 2 β’ (β¨πΉ, πβ© β ((π½βπ) Γ { 0 }) β (πΉ β (π½βπ) β§ π = 0 )) |
17 | 9, 16 | bitrdi 287 | 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 397 = wceq 1542 β wcel 2107 Vcvv 3446 {csn 4587 β¨cop 4593 class class class wbr 5106 β¦ cmpt 5189 I cid 5531 Γ cxp 5632 βΎ cres 5636 βcfv 6497 Basecbs 17084 lecple 17141 LHypclh 38450 LTrncltrn 38567 DIsoAcdia 39494 DIsoBcdib 39604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-disoa 39495 df-dib 39605 |
This theorem is referenced by: dibopelval3 39614 dibglbN 39632 diblsmopel 39637 dib2dim 39709 dih2dimbALTN 39711 dihord6apre 39722 |
Copyright terms: Public domain | W3C validator |