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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 39650 | . . . 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 7441 | . 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 39644 | . . . 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 39673 | . . 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 2771 | 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5152 β ccom 5686 βcfv 6553 (class class class)co 7426 lecple 17247 joincjn 18310 Atomscatm 38767 HLchlt 38854 LHypclh 39489 LTrncltrn 39606 trLctrl 39663 |
| 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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-map 8853 df-proset 18294 df-poset 18312 df-plt 18329 df-lub 18345 df-glb 18346 df-join 18347 df-meet 18348 df-p0 18424 df-p1 18425 df-lat 18431 df-clat 18498 df-oposet 38680 df-ol 38682 df-oml 38683 df-covers 38770 df-ats 38771 df-atl 38802 df-cvlat 38826 df-hlat 38855 df-psubsp 39008 df-pmap 39009 df-padd 39301 df-lhyp 39493 df-laut 39494 df-ldil 39609 df-ltrn 39610 df-trl 39664 |
| This theorem is referenced by: cdlemk48 40455 |
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