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| Mirrors > Home > MPE Home > Th. List > Mathboxes > trlcoabs | Structured version Visualization version GIF version | ||
| Description: Absorption into a composition by joining with trace. (Contributed by NM, 22-Jul-2013.) |
| Ref | Expression |
|---|---|
| trlcoabs.l | β’ β€ = (leβπΎ) |
| trlcoabs.j | β’ β¨ = (joinβπΎ) |
| trlcoabs.a | β’ π΄ = (AtomsβπΎ) |
| trlcoabs.h | β’ π» = (LHypβπΎ) |
| trlcoabs.t | β’ π = ((LTrnβπΎ)βπ) |
| trlcoabs.r | β’ π = ((trLβπΎ)βπ) |
| Ref | Expression |
|---|---|
| trlcoabs | β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π β π΄ β§ Β¬ π β€ π)) β (((πΉ β πΊ)βπ) β¨ (π βπΉ)) = ((πΊβπ) β¨ (π βπΉ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | trlcoabs.l | . . . . 5 β’ β€ = (leβπΎ) | |
| 2 | trlcoabs.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
| 3 | trlcoabs.h | . . . . 5 β’ π» = (LHypβπΎ) | |
| 4 | trlcoabs.t | . . . . 5 β’ π = ((LTrnβπΎ)βπ) | |
| 5 | 1, 2, 3, 4 | ltrncoval 39529 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ π β π΄) β ((πΉ β πΊ)βπ) = (πΉβ(πΊβπ))) |
| 6 | 5 | 3adant3r 1178 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π β π΄ β§ Β¬ π β€ π)) β ((πΉ β πΊ)βπ) = (πΉβ(πΊβπ))) |
| 7 | 6 | oveq1d 7420 | . 2 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π β π΄ β§ Β¬ π β€ π)) β (((πΉ β πΊ)βπ) β¨ (π βπΉ)) = ((πΉβ(πΊβπ)) β¨ (π βπΉ))) |
| 8 | simp1 1133 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π β π΄ β§ Β¬ π β€ π)) β (πΎ β HL β§ π β π»)) | |
| 9 | simp2l 1196 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π β π΄ β§ Β¬ π β€ π)) β πΉ β π) | |
| 10 | 1, 2, 3, 4 | ltrnel 39523 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ πΊ β π β§ (π β π΄ β§ Β¬ π β€ π)) β ((πΊβπ) β π΄ β§ Β¬ (πΊβπ) β€ π)) |
| 11 | 10 | 3adant2l 1175 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π β π΄ β§ Β¬ π β€ π)) β ((πΊβπ) β π΄ β§ Β¬ (πΊβπ) β€ π)) |
| 12 | trlcoabs.j | . . . 4 β’ β¨ = (joinβπΎ) | |
| 13 | trlcoabs.r | . . . 4 β’ π = ((trLβπΎ)βπ) | |
| 14 | 1, 12, 2, 3, 4, 13 | trljat3 39552 | . . 3 β’ (((πΎ β HL β§ π β π») β§ πΉ β π β§ ((πΊβπ) β π΄ β§ Β¬ (πΊβπ) β€ π)) β ((πΊβπ) β¨ (π βπΉ)) = ((πΉβ(πΊβπ)) β¨ (π βπΉ))) |
| 15 | 8, 9, 11, 14 | syl3anc 1368 | . 2 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π β π΄ β§ Β¬ π β€ π)) β ((πΊβπ) β¨ (π βπΉ)) = ((πΉβ(πΊβπ)) β¨ (π βπΉ))) |
| 16 | 7, 15 | eqtr4d 2769 | 1 β’ (((πΎ β HL β§ π β π») β§ (πΉ β π β§ πΊ β π) β§ (π β π΄ β§ Β¬ π β€ π)) β (((πΉ β πΊ)βπ) β¨ (π βπΉ)) = ((πΊβπ) β¨ (π βπΉ))) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5141 β ccom 5673 βcfv 6537 (class class class)co 7405 lecple 17213 joincjn 18276 Atomscatm 38646 HLchlt 38733 LHypclh 39368 LTrncltrn 39485 trLctrl 39542 |
| 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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-map 8824 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-p1 18391 df-lat 18397 df-clat 18464 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 df-psubsp 38887 df-pmap 38888 df-padd 39180 df-lhyp 39372 df-laut 39373 df-ldil 39488 df-ltrn 39489 df-trl 39543 |
| This theorem is referenced by: cdlemk48 40334 |
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