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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 39837. (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 1136 | . . 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 1199 | . . 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 1200 | . . 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 1138 | . . 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 2732 | . . . . 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 6892 | . . . . . 6 β’ ( β₯ βπ) = ((ocβπΎ)βπ) |
| 11 | 8, 10 | eqtri 2760 | . . . . 5 β’ π = ((ocβπΎ)βπ) |
| 12 | 5, 6, 7, 11 | lhpocnel2 38885 | . . . 4 β’ ((πΎ β HL β§ π β π») β (π β π΄ β§ Β¬ π(leβπΎ)π)) |
| 13 | 12 | 3ad2ant1 1133 | . . 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 39837 | . . 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 1384 | . 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 39838 | . 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 512 | 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 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β wne 2940 βwral 3061 ifcif 4528 class class class wbr 5148 β¦ cmpt 5231 I cid 5573 β‘ccnv 5675 βΎ cres 5678 β ccom 5680 βcfv 6543 β©crio 7363 (class class class)co 7408 Basecbs 17143 lecple 17203 occoc 17204 joincjn 18263 meetcmee 18264 Atomscatm 38128 HLchlt 38215 LHypclh 38850 LTrncltrn 38967 trLctrl 39024 TEndoctendo 39618 |
| 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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-riotaBAD 37818 |
| 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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-undef 8257 df-map 8821 df-proset 18247 df-poset 18265 df-plt 18282 df-lub 18298 df-glb 18299 df-join 18300 df-meet 18301 df-p0 18377 df-p1 18378 df-lat 18384 df-clat 18451 df-oposet 38041 df-ol 38043 df-oml 38044 df-covers 38131 df-ats 38132 df-atl 38163 df-cvlat 38187 df-hlat 38216 df-llines 38364 df-lplanes 38365 df-lvols 38366 df-lines 38367 df-psubsp 38369 df-pmap 38370 df-padd 38662 df-lhyp 38854 df-laut 38855 df-ldil 38970 df-ltrn 38971 df-trl 39025 df-tendo 39621 |
| This theorem is referenced by: cdlemk 39840 cdleml6 39847 |
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