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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemk56w | Structured version Visualization version GIF version | ||
| Description: Use a fixed element to eliminate π in cdlemk56 40500. (Contributed by NM, 1-Aug-2013.) |
| Ref | Expression |
|---|---|
| cdlemk6.b | β’ π΅ = (BaseβπΎ) |
| cdlemk6.j | β’ β¨ = (joinβπΎ) |
| cdlemk6.m | β’ β§ = (meetβπΎ) |
| cdlemk6.o | β’ β₯ = (ocβπΎ) |
| cdlemk6.a | β’ π΄ = (AtomsβπΎ) |
| cdlemk6.h | β’ π» = (LHypβπΎ) |
| cdlemk6.t | β’ π = ((LTrnβπΎ)βπ) |
| cdlemk6.r | β’ π = ((trLβπΎ)βπ) |
| cdlemk6.p | β’ π = ( β₯ βπ) |
| cdlemk6.z | β’ π = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))) |
| cdlemk6.y | β’ π = ((π β¨ (π βπ)) β§ (π β¨ (π β(π β β‘π)))) |
| cdlemk6.x | β’ π = (β©π§ β π βπ β π ((π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπ)) β (π§βπ) = π)) |
| cdlemk6.u | β’ π = (π β π β¦ if(πΉ = π, π, π)) |
| cdlemk6.e | β’ πΈ = ((TEndoβπΎ)βπ) |
| Ref | Expression |
|---|---|
| cdlemk56w | β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ π β π) β§ (π βπΉ) = (π βπ)) β (π β πΈ β§ (πβπΉ) = π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp1 1133 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ π β π) β§ (π βπΉ) = (π βπ)) β (πΎ β HL β§ π β π»)) | |
| 2 | simp2l 1196 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ π β π) β§ (π βπΉ) = (π βπ)) β πΉ β π) | |
| 3 | simp2r 1197 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ π β π) β§ (π βπΉ) = (π βπ)) β π β π) | |
| 4 | simp3 1135 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ π β π) β§ (π βπΉ) = (π βπ)) β (π βπΉ) = (π βπ)) | |
| 5 | eqid 2725 | . . . . 5 β’ (leβπΎ) = (leβπΎ) | |
| 6 | cdlemk6.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
| 7 | cdlemk6.h | . . . . 5 β’ π» = (LHypβπΎ) | |
| 8 | cdlemk6.p | . . . . . 6 β’ π = ( β₯ βπ) | |
| 9 | cdlemk6.o | . . . . . . 7 β’ β₯ = (ocβπΎ) | |
| 10 | 9 | fveq1i 6893 | . . . . . 6 β’ ( β₯ βπ) = ((ocβπΎ)βπ) |
| 11 | 8, 10 | eqtri 2753 | . . . . 5 β’ π = ((ocβπΎ)βπ) |
| 12 | 5, 6, 7, 11 | lhpocnel2 39548 | . . . 4 β’ ((πΎ β HL β§ π β π») β (π β π΄ β§ Β¬ π(leβπΎ)π)) |
| 13 | 12 | 3ad2ant1 1130 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ π β π) β§ (π βπΉ) = (π βπ)) β (π β π΄ β§ Β¬ π(leβπΎ)π)) |
| 14 | cdlemk6.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
| 15 | cdlemk6.j | . . . 4 β’ β¨ = (joinβπΎ) | |
| 16 | cdlemk6.m | . . . 4 β’ β§ = (meetβπΎ) | |
| 17 | cdlemk6.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
| 18 | cdlemk6.r | . . . 4 β’ π = ((trLβπΎ)βπ) | |
| 19 | cdlemk6.z | . . . 4 β’ π = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))) | |
| 20 | cdlemk6.y | . . . 4 β’ π = ((π β¨ (π βπ)) β§ (π β¨ (π β(π β β‘π)))) | |
| 21 | cdlemk6.x | . . . 4 β’ π = (β©π§ β π βπ β π ((π β ( I βΎ π΅) β§ (π βπ) β (π βπΉ) β§ (π βπ) β (π βπ)) β (π§βπ) = π)) | |
| 22 | cdlemk6.u | . . . 4 β’ π = (π β π β¦ if(πΉ = π, π, π)) | |
| 23 | cdlemk6.e | . . . 4 β’ πΈ = ((TEndoβπΎ)βπ) | |
| 24 | 14, 5, 15, 16, 6, 7, 17, 18, 19, 20, 21, 22, 23 | cdlemk56 40500 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ πΉ β π β§ π β π) β§ (π βπΉ) = (π βπ) β§ (π β π΄ β§ Β¬ π(leβπΎ)π)) β π β πΈ) |
| 25 | 1, 2, 3, 4, 13, 24 | syl311anc 1381 | . 2 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ π β π) β§ (π βπΉ) = (π βπ)) β π β πΈ) |
| 26 | 14, 15, 16, 9, 6, 7, 17, 18, 8, 19, 20, 21, 22 | cdlemk19w 40501 | . 2 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ π β π) β§ (π βπΉ) = (π βπ)) β (πβπΉ) = π) |
| 27 | 25, 26 | jca 510 | 1 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ π β π) β§ (π βπΉ) = (π βπ)) β (π β πΈ β§ (πβπΉ) = π)) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2930 βwral 3051 ifcif 4524 class class class wbr 5143 β¦ cmpt 5226 I cid 5569 β‘ccnv 5671 βΎ cres 5674 β ccom 5676 βcfv 6543 β©crio 7371 (class class class)co 7416 Basecbs 17179 lecple 17239 occoc 17240 joincjn 18302 meetcmee 18303 Atomscatm 38791 HLchlt 38878 LHypclh 39513 LTrncltrn 39630 trLctrl 39687 TEndoctendo 40281 |
| 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 7738 ax-riotaBAD 38481 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 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-iin 4994 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 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7991 df-2nd 7992 df-undef 8277 df-map 8845 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 38704 df-ol 38706 df-oml 38707 df-covers 38794 df-ats 38795 df-atl 38826 df-cvlat 38850 df-hlat 38879 df-llines 39027 df-lplanes 39028 df-lvols 39029 df-lines 39030 df-psubsp 39032 df-pmap 39033 df-padd 39325 df-lhyp 39517 df-laut 39518 df-ldil 39633 df-ltrn 39634 df-trl 39688 df-tendo 40284 |
| This theorem is referenced by: cdlemk 40503 cdleml6 40510 |
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