![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dian0 | Structured version Visualization version GIF version |
Description: The value of the partial isomorphism A is not empty. (Contributed by NM, 17-Jan-2014.) |
Ref | Expression |
---|---|
dian0.b | β’ π΅ = (BaseβπΎ) |
dian0.l | β’ β€ = (leβπΎ) |
dian0.h | β’ π» = (LHypβπΎ) |
dian0.i | β’ πΌ = ((DIsoAβπΎ)βπ) |
Ref | Expression |
---|---|
dian0 | β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β (πΌβπ) β β ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dian0.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
2 | dian0.h | . . . . 5 β’ π» = (LHypβπΎ) | |
3 | eqid 2728 | . . . . 5 β’ ((LTrnβπΎ)βπ) = ((LTrnβπΎ)βπ) | |
4 | 1, 2, 3 | idltrn 39623 | . . . 4 β’ ((πΎ β HL β§ π β π») β ( I βΎ π΅) β ((LTrnβπΎ)βπ)) |
5 | 4 | adantr 480 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β ( I βΎ π΅) β ((LTrnβπΎ)βπ)) |
6 | eqid 2728 | . . . . . 6 β’ (0.βπΎ) = (0.βπΎ) | |
7 | eqid 2728 | . . . . . 6 β’ ((trLβπΎ)βπ) = ((trLβπΎ)βπ) | |
8 | 1, 6, 2, 7 | trlid0 39649 | . . . . 5 β’ ((πΎ β HL β§ π β π») β (((trLβπΎ)βπ)β( I βΎ π΅)) = (0.βπΎ)) |
9 | 8 | adantr 480 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β (((trLβπΎ)βπ)β( I βΎ π΅)) = (0.βπΎ)) |
10 | hlatl 38832 | . . . . . 6 β’ (πΎ β HL β πΎ β AtLat) | |
11 | 10 | adantr 480 | . . . . 5 β’ ((πΎ β HL β§ π β π») β πΎ β AtLat) |
12 | simpl 482 | . . . . 5 β’ ((π β π΅ β§ π β€ π) β π β π΅) | |
13 | dian0.l | . . . . . 6 β’ β€ = (leβπΎ) | |
14 | 1, 13, 6 | atl0le 38776 | . . . . 5 β’ ((πΎ β AtLat β§ π β π΅) β (0.βπΎ) β€ π) |
15 | 11, 12, 14 | syl2an 595 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β (0.βπΎ) β€ π) |
16 | 9, 15 | eqbrtrd 5170 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β (((trLβπΎ)βπ)β( I βΎ π΅)) β€ π) |
17 | dian0.i | . . . 4 β’ πΌ = ((DIsoAβπΎ)βπ) | |
18 | 1, 13, 2, 3, 7, 17 | diaelval 40506 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β (( I βΎ π΅) β (πΌβπ) β (( I βΎ π΅) β ((LTrnβπΎ)βπ) β§ (((trLβπΎ)βπ)β( I βΎ π΅)) β€ π))) |
19 | 5, 16, 18 | mpbir2and 712 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β ( I βΎ π΅) β (πΌβπ)) |
20 | 19 | ne0d 4336 | 1 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β€ π)) β (πΌβπ) β β ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β wne 2937 β c0 4323 class class class wbr 5148 I cid 5575 βΎ cres 5680 βcfv 6548 Basecbs 17179 lecple 17239 0.cp0 18414 AtLatcal 38736 HLchlt 38822 LHypclh 39457 LTrncltrn 39574 trLctrl 39631 DIsoAcdia 40501 |
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-rmo 3373 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-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-map 8846 df-proset 18286 df-poset 18304 df-plt 18321 df-lub 18337 df-glb 18338 df-join 18339 df-meet 18340 df-p0 18416 df-p1 18417 df-lat 18423 df-clat 18490 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 df-lhyp 39461 df-laut 39462 df-ldil 39577 df-ltrn 39578 df-trl 39632 df-disoa 40502 |
This theorem is referenced by: dialss 40519 dibn0 40626 |
Copyright terms: Public domain | W3C validator |