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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemkfid2N | Structured version Visualization version GIF version | ||
| Description: Lemma for cdlemkfid3N 39796. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| cdlemk5.b | β’ π΅ = (BaseβπΎ) |
| cdlemk5.l | β’ β€ = (leβπΎ) |
| cdlemk5.j | β’ β¨ = (joinβπΎ) |
| cdlemk5.m | β’ β§ = (meetβπΎ) |
| cdlemk5.a | β’ π΄ = (AtomsβπΎ) |
| cdlemk5.h | β’ π» = (LHypβπΎ) |
| cdlemk5.t | β’ π = ((LTrnβπΎ)βπ) |
| cdlemk5.r | β’ π = ((trLβπΎ)βπ) |
| cdlemk5.z | β’ π = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))) |
| Ref | Expression |
|---|---|
| cdlemkfid2N | β’ ((((πΎ β HL β§ π β π») β§ πΉ = π) β§ (πΉ β π β§ πΉ β ( I βΎ π΅) β§ π β π) β§ ((π βπ) β (π βπΉ) β§ (π β π΄ β§ Β¬ π β€ π))) β π = (πβπ)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cdlemk5.z | . 2 β’ π = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))) | |
| 2 | simp1r 1199 | . . . . . 6 β’ ((((πΎ β HL β§ π β π») β§ πΉ = π) β§ (πΉ β π β§ πΉ β ( I βΎ π΅) β§ π β π) β§ ((π βπ) β (π βπΉ) β§ (π β π΄ β§ Β¬ π β€ π))) β πΉ = π) | |
| 3 | 2 | fveq1d 6894 | . . . . 5 β’ ((((πΎ β HL β§ π β π») β§ πΉ = π) β§ (πΉ β π β§ πΉ β ( I βΎ π΅) β§ π β π) β§ ((π βπ) β (π βπΉ) β§ (π β π΄ β§ Β¬ π β€ π))) β (πΉβπ) = (πβπ)) |
| 4 | 3 | oveq1d 7424 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ πΉ = π) β§ (πΉ β π β§ πΉ β ( I βΎ π΅) β§ π β π) β§ ((π βπ) β (π βπΉ) β§ (π β π΄ β§ Β¬ π β€ π))) β ((πΉβπ) β¨ (π β(π β β‘πΉ))) = ((πβπ) β¨ (π β(π β β‘πΉ)))) |
| 5 | 4 | oveq2d 7425 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ πΉ = π) β§ (πΉ β π β§ πΉ β ( I βΎ π΅) β§ π β π) β§ ((π βπ) β (π βπΉ) β§ (π β π΄ β§ Β¬ π β€ π))) β ((π β¨ (π βπ)) β§ ((πΉβπ) β¨ (π β(π β β‘πΉ)))) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ))))) |
| 6 | cdlemk5.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
| 7 | cdlemk5.l | . . . . 5 β’ β€ = (leβπΎ) | |
| 8 | cdlemk5.j | . . . . 5 β’ β¨ = (joinβπΎ) | |
| 9 | cdlemk5.m | . . . . 5 β’ β§ = (meetβπΎ) | |
| 10 | cdlemk5.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
| 11 | cdlemk5.h | . . . . 5 β’ π» = (LHypβπΎ) | |
| 12 | cdlemk5.t | . . . . 5 β’ π = ((LTrnβπΎ)βπ) | |
| 13 | cdlemk5.r | . . . . 5 β’ π = ((trLβπΎ)βπ) | |
| 14 | 6, 7, 8, 9, 10, 11, 12, 13 | cdlemkfid1N 39792 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΉ β ( I βΎ π΅) β§ π β π) β§ ((π βπ) β (π βπΉ) β§ (π β π΄ β§ Β¬ π β€ π))) β ((π β¨ (π βπ)) β§ ((πΉβπ) β¨ (π β(π β β‘πΉ)))) = (πβπ)) |
| 15 | 14 | 3adant1r 1178 | . . 3 β’ ((((πΎ β HL β§ π β π») β§ πΉ = π) β§ (πΉ β π β§ πΉ β ( I βΎ π΅) β§ π β π) β§ ((π βπ) β (π βπΉ) β§ (π β π΄ β§ Β¬ π β€ π))) β ((π β¨ (π βπ)) β§ ((πΉβπ) β¨ (π β(π β β‘πΉ)))) = (πβπ)) |
| 16 | 5, 15 | eqtr3d 2775 | . 2 β’ ((((πΎ β HL β§ π β π») β§ πΉ = π) β§ (πΉ β π β§ πΉ β ( I βΎ π΅) β§ π β π) β§ ((π βπ) β (π βπΉ) β§ (π β π΄ β§ Β¬ π β€ π))) β ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))) = (πβπ)) |
| 17 | 1, 16 | eqtrid 2785 | 1 β’ ((((πΎ β HL β§ π β π») β§ πΉ = π) β§ (πΉ β π β§ πΉ β ( I βΎ π΅) β§ π β π) β§ ((π βπ) β (π βπΉ) β§ (π β π΄ β§ Β¬ π β€ π))) β π = (πβπ)) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2941 class class class wbr 5149 I cid 5574 β‘ccnv 5676 βΎ cres 5679 β ccom 5681 βcfv 6544 (class class class)co 7409 Basecbs 17144 lecple 17204 joincjn 18264 meetcmee 18265 Atomscatm 38133 HLchlt 38220 LHypclh 38855 LTrncltrn 38972 trLctrl 39029 |
| 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 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-riotaBAD 37823 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-undef 8258 df-map 8822 df-proset 18248 df-poset 18266 df-plt 18283 df-lub 18299 df-glb 18300 df-join 18301 df-meet 18302 df-p0 18378 df-p1 18379 df-lat 18385 df-clat 18452 df-oposet 38046 df-ol 38048 df-oml 38049 df-covers 38136 df-ats 38137 df-atl 38168 df-cvlat 38192 df-hlat 38221 df-llines 38369 df-lplanes 38370 df-lvols 38371 df-lines 38372 df-psubsp 38374 df-pmap 38375 df-padd 38667 df-lhyp 38859 df-laut 38860 df-ldil 38975 df-ltrn 38976 df-trl 39030 |
| This theorem is referenced by: cdlemkfid3N 39796 |
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