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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleml7 | Structured version Visualization version GIF version |
Description: Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.) |
Ref | Expression |
---|---|
cdleml6.b | β’ π΅ = (BaseβπΎ) |
cdleml6.j | β’ β¨ = (joinβπΎ) |
cdleml6.m | β’ β§ = (meetβπΎ) |
cdleml6.h | β’ π» = (LHypβπΎ) |
cdleml6.t | β’ π = ((LTrnβπΎ)βπ) |
cdleml6.r | β’ π = ((trLβπΎ)βπ) |
cdleml6.p | β’ π = ((ocβπΎ)βπ) |
cdleml6.z | β’ π = ((π β¨ (π βπ)) β§ ((ββπ) β¨ (π β(π β β‘(π ββ))))) |
cdleml6.y | β’ π = ((π β¨ (π βπ)) β§ (π β¨ (π β(π β β‘π)))) |
cdleml6.x | β’ π = (β©π§ β π βπ β π ((π β ( I βΎ π΅) β§ (π βπ) β (π β(π ββ)) β§ (π βπ) β (π βπ)) β (π§βπ) = π)) |
cdleml6.u | β’ π = (π β π β¦ if((π ββ) = β, π, π)) |
cdleml6.e | β’ πΈ = ((TEndoβπΎ)βπ) |
cdleml6.o | β’ 0 = (π β π β¦ ( I βΎ π΅)) |
Ref | Expression |
---|---|
cdleml7 | β’ (((πΎ β HL β§ π β π») β§ β β π β§ (π β πΈ β§ π β 0 )) β ((π β π )ββ) = (( I βΎ π)ββ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleml6.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
2 | cdleml6.j | . . . 4 β’ β¨ = (joinβπΎ) | |
3 | cdleml6.m | . . . 4 β’ β§ = (meetβπΎ) | |
4 | cdleml6.h | . . . 4 β’ π» = (LHypβπΎ) | |
5 | cdleml6.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
6 | cdleml6.r | . . . 4 β’ π = ((trLβπΎ)βπ) | |
7 | cdleml6.p | . . . 4 β’ π = ((ocβπΎ)βπ) | |
8 | cdleml6.z | . . . 4 β’ π = ((π β¨ (π βπ)) β§ ((ββπ) β¨ (π β(π β β‘(π ββ))))) | |
9 | cdleml6.y | . . . 4 β’ π = ((π β¨ (π βπ)) β§ (π β¨ (π β(π β β‘π)))) | |
10 | cdleml6.x | . . . 4 β’ π = (β©π§ β π βπ β π ((π β ( I βΎ π΅) β§ (π βπ) β (π β(π ββ)) β§ (π βπ) β (π βπ)) β (π§βπ) = π)) | |
11 | cdleml6.u | . . . 4 β’ π = (π β π β¦ if((π ββ) = β, π, π)) | |
12 | cdleml6.e | . . . 4 β’ πΈ = ((TEndoβπΎ)βπ) | |
13 | cdleml6.o | . . . 4 β’ 0 = (π β π β¦ ( I βΎ π΅)) | |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 | cdleml6 40363 | . . 3 β’ (((πΎ β HL β§ π β π») β§ β β π β§ (π β πΈ β§ π β 0 )) β (π β πΈ β§ (πβ(π ββ)) = β)) |
15 | 14 | simprd 495 | . 2 β’ (((πΎ β HL β§ π β π») β§ β β π β§ (π β πΈ β§ π β 0 )) β (πβ(π ββ)) = β) |
16 | simp1 1133 | . . 3 β’ (((πΎ β HL β§ π β π») β§ β β π β§ (π β πΈ β§ π β 0 )) β (πΎ β HL β§ π β π»)) | |
17 | 14 | simpld 494 | . . 3 β’ (((πΎ β HL β§ π β π») β§ β β π β§ (π β πΈ β§ π β 0 )) β π β πΈ) |
18 | simp3l 1198 | . . 3 β’ (((πΎ β HL β§ π β π») β§ β β π β§ (π β πΈ β§ π β 0 )) β π β πΈ) | |
19 | simp2 1134 | . . 3 β’ (((πΎ β HL β§ π β π») β§ β β π β§ (π β πΈ β§ π β 0 )) β β β π) | |
20 | 4, 5, 12 | tendocoval 40148 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β πΈ β§ π β πΈ) β§ β β π) β ((π β π )ββ) = (πβ(π ββ))) |
21 | 16, 17, 18, 19, 20 | syl121anc 1372 | . 2 β’ (((πΎ β HL β§ π β π») β§ β β π β§ (π β πΈ β§ π β 0 )) β ((π β π )ββ) = (πβ(π ββ))) |
22 | fvresi 7166 | . . 3 β’ (β β π β (( I βΎ π)ββ) = β) | |
23 | 22 | 3ad2ant2 1131 | . 2 β’ (((πΎ β HL β§ π β π») β§ β β π β§ (π β πΈ β§ π β 0 )) β (( I βΎ π)ββ) = β) |
24 | 15, 21, 23 | 3eqtr4d 2776 | 1 β’ (((πΎ β HL β§ π β π») β§ β β π β§ (π β πΈ β§ π β 0 )) β ((π β π )ββ) = (( I βΎ π)ββ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 βwral 3055 ifcif 4523 β¦ cmpt 5224 I cid 5566 β‘ccnv 5668 βΎ cres 5671 β ccom 5673 βcfv 6536 β©crio 7359 (class class class)co 7404 Basecbs 17151 occoc 17212 joincjn 18274 meetcmee 18275 HLchlt 38731 LHypclh 39366 LTrncltrn 39483 trLctrl 39540 TEndoctendo 40134 |
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 7721 ax-riotaBAD 38334 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 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 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-undef 8256 df-map 8821 df-proset 18258 df-poset 18276 df-plt 18293 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-p0 18388 df-p1 18389 df-lat 18395 df-clat 18462 df-oposet 38557 df-ol 38559 df-oml 38560 df-covers 38647 df-ats 38648 df-atl 38679 df-cvlat 38703 df-hlat 38732 df-llines 38880 df-lplanes 38881 df-lvols 38882 df-lines 38883 df-psubsp 38885 df-pmap 38886 df-padd 39178 df-lhyp 39370 df-laut 39371 df-ldil 39486 df-ltrn 39487 df-trl 39541 df-tendo 40137 |
This theorem is referenced by: cdleml8 40365 |
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