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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme50f1o | Structured version Visualization version GIF version |
Description: Part of proof of Lemma D in [Crawley] p. 113. TODO: fix comment. (Contributed by NM, 9-Apr-2013.) |
Ref | Expression |
---|---|
cdlemef50.b | β’ π΅ = (BaseβπΎ) |
cdlemef50.l | β’ β€ = (leβπΎ) |
cdlemef50.j | β’ β¨ = (joinβπΎ) |
cdlemef50.m | β’ β§ = (meetβπΎ) |
cdlemef50.a | β’ π΄ = (AtomsβπΎ) |
cdlemef50.h | β’ π» = (LHypβπΎ) |
cdlemef50.u | β’ π = ((π β¨ π) β§ π) |
cdlemef50.d | β’ π· = ((π‘ β¨ π) β§ (π β¨ ((π β¨ π‘) β§ π))) |
cdlemefs50.e | β’ πΈ = ((π β¨ π) β§ (π· β¨ ((π β¨ π‘) β§ π))) |
cdlemef50.f | β’ πΉ = (π₯ β π΅ β¦ if((π β π β§ Β¬ π₯ β€ π), (β©π§ β π΅ βπ β π΄ ((Β¬ π β€ π β§ (π β¨ (π₯ β§ π)) = π₯) β π§ = (if(π β€ (π β¨ π), (β©π¦ β π΅ βπ‘ β π΄ ((Β¬ π‘ β€ π β§ Β¬ π‘ β€ (π β¨ π)) β π¦ = πΈ)), β¦π / π‘β¦π·) β¨ (π₯ β§ π)))), π₯)) |
Ref | Expression |
---|---|
cdleme50f1o | β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β πΉ:π΅β1-1-ontoβπ΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemef50.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | cdlemef50.l | . . 3 β’ β€ = (leβπΎ) | |
3 | cdlemef50.j | . . 3 β’ β¨ = (joinβπΎ) | |
4 | cdlemef50.m | . . 3 β’ β§ = (meetβπΎ) | |
5 | cdlemef50.a | . . 3 β’ π΄ = (AtomsβπΎ) | |
6 | cdlemef50.h | . . 3 β’ π» = (LHypβπΎ) | |
7 | cdlemef50.u | . . 3 β’ π = ((π β¨ π) β§ π) | |
8 | cdlemef50.d | . . 3 β’ π· = ((π‘ β¨ π) β§ (π β¨ ((π β¨ π‘) β§ π))) | |
9 | cdlemefs50.e | . . 3 β’ πΈ = ((π β¨ π) β§ (π· β¨ ((π β¨ π‘) β§ π))) | |
10 | cdlemef50.f | . . 3 β’ πΉ = (π₯ β π΅ β¦ if((π β π β§ Β¬ π₯ β€ π), (β©π§ β π΅ βπ β π΄ ((Β¬ π β€ π β§ (π β¨ (π₯ β§ π)) = π₯) β π§ = (if(π β€ (π β¨ π), (β©π¦ β π΅ βπ‘ β π΄ ((Β¬ π‘ β€ π β§ Β¬ π‘ β€ (π β¨ π)) β π¦ = πΈ)), β¦π / π‘β¦π·) β¨ (π₯ β§ π)))), π₯)) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | cdleme50f1 39205 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β πΉ:π΅β1-1βπ΅) |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | cdleme50rn 39207 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β ran πΉ = π΅) |
13 | dff1o5 6828 | . 2 β’ (πΉ:π΅β1-1-ontoβπ΅ β (πΉ:π΅β1-1βπ΅ β§ ran πΉ = π΅)) | |
14 | 11, 12, 13 | sylanbrc 583 | 1 β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β πΉ:π΅β1-1-ontoβπ΅) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2939 βwral 3060 β¦csb 3888 ifcif 4521 class class class wbr 5140 β¦ cmpt 5223 ran crn 5669 β1-1βwf1 6528 β1-1-ontoβwf1o 6530 βcfv 6531 β©crio 7347 (class class class)co 7392 Basecbs 17125 lecple 17185 joincjn 18245 meetcmee 18246 Atomscatm 37924 HLchlt 38011 LHypclh 38646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5277 ax-sep 5291 ax-nul 5298 ax-pow 5355 ax-pr 5419 ax-un 7707 ax-riotaBAD 37614 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3474 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4991 df-iin 4992 df-br 5141 df-opab 5203 df-mpt 5224 df-id 5566 df-xp 5674 df-rel 5675 df-cnv 5676 df-co 5677 df-dm 5678 df-rn 5679 df-res 5680 df-ima 5681 df-iota 6483 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7348 df-ov 7395 df-oprab 7396 df-mpo 7397 df-1st 7956 df-2nd 7957 df-undef 8239 df-proset 18229 df-poset 18247 df-plt 18264 df-lub 18280 df-glb 18281 df-join 18282 df-meet 18283 df-p0 18359 df-p1 18360 df-lat 18366 df-clat 18433 df-oposet 37837 df-ol 37839 df-oml 37840 df-covers 37927 df-ats 37928 df-atl 37959 df-cvlat 37983 df-hlat 38012 df-llines 38160 df-lplanes 38161 df-lvols 38162 df-lines 38163 df-psubsp 38165 df-pmap 38166 df-padd 38458 df-lhyp 38650 |
This theorem is referenced by: cdleme50laut 39209 cdleme51finvfvN 39217 cdleme51finvN 39218 |
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