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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemkfid2N | Structured version Visualization version GIF version | ||
| Description: Lemma for cdlemkfid3N 40099. (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 1196 | . . . . . 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 6892 | . . . . 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 7426 | . . . 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 7427 | . . 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 40095 | . . . 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 1175 | . . 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 2772 | . 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 2782 | 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 394 β§ w3a 1085 = wceq 1539 β wcel 2104 β wne 2938 class class class wbr 5147 I cid 5572 β‘ccnv 5674 βΎ cres 5677 β ccom 5679 βcfv 6542 (class class class)co 7411 Basecbs 17148 lecple 17208 joincjn 18268 meetcmee 18269 Atomscatm 38436 HLchlt 38523 LHypclh 39158 LTrncltrn 39275 trLctrl 39332 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-riotaBAD 38126 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-undef 8260 df-map 8824 df-proset 18252 df-poset 18270 df-plt 18287 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-p0 18382 df-p1 18383 df-lat 18389 df-clat 18456 df-oposet 38349 df-ol 38351 df-oml 38352 df-covers 38439 df-ats 38440 df-atl 38471 df-cvlat 38495 df-hlat 38524 df-llines 38672 df-lplanes 38673 df-lvols 38674 df-lines 38675 df-psubsp 38677 df-pmap 38678 df-padd 38970 df-lhyp 39162 df-laut 39163 df-ldil 39278 df-ltrn 39279 df-trl 39333 |
| This theorem is referenced by: cdlemkfid3N 40099 |
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