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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleml9 | 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 |
---|---|
cdleml9 | β’ (((πΎ β HL β§ π β π») β§ (β β π β§ β β ( I βΎ π΅)) β§ (π β πΈ β§ π β 0 )) β π β 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleml6.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
2 | cdleml6.h | . . . 4 β’ π» = (LHypβπΎ) | |
3 | cdleml6.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
4 | cdleml6.e | . . . 4 β’ πΈ = ((TEndoβπΎ)βπ) | |
5 | cdleml6.o | . . . 4 β’ 0 = (π β π β¦ ( I βΎ π΅)) | |
6 | 1, 2, 3, 4, 5 | tendo1ne0 40295 | . . 3 β’ ((πΎ β HL β§ π β π») β ( I βΎ π) β 0 ) |
7 | 6 | 3ad2ant1 1131 | . 2 β’ (((πΎ β HL β§ π β π») β§ (β β π β§ β β ( I βΎ π΅)) β§ (π β πΈ β§ π β 0 )) β ( I βΎ π) β 0 ) |
8 | cdleml6.j | . . . . . . 7 β’ β¨ = (joinβπΎ) | |
9 | cdleml6.m | . . . . . . 7 β’ β§ = (meetβπΎ) | |
10 | cdleml6.r | . . . . . . 7 β’ π = ((trLβπΎ)βπ) | |
11 | cdleml6.p | . . . . . . 7 β’ π = ((ocβπΎ)βπ) | |
12 | cdleml6.z | . . . . . . 7 β’ π = ((π β¨ (π βπ)) β§ ((ββπ) β¨ (π β(π β β‘(π ββ))))) | |
13 | cdleml6.y | . . . . . . 7 β’ π = ((π β¨ (π βπ)) β§ (π β¨ (π β(π β β‘π)))) | |
14 | cdleml6.x | . . . . . . 7 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)) | |
15 | cdleml6.u | . . . . . . 7 β’ π = (π β π β¦ if((π ββ) = β, π, π)) | |
16 | 1, 8, 9, 2, 3, 10, 11, 12, 13, 14, 15, 4, 5 | cdleml8 40450 | . . . . . 6 β’ (((πΎ β HL β§ π β π») β§ (β β π β§ β β ( I βΎ π΅)) β§ (π β πΈ β§ π β 0 )) β (π β π ) = ( I βΎ π)) |
17 | 16 | adantr 480 | . . . . 5 β’ ((((πΎ β HL β§ π β π») β§ (β β π β§ β β ( I βΎ π΅)) β§ (π β πΈ β§ π β 0 )) β§ π = 0 ) β (π β π ) = ( I βΎ π)) |
18 | coeq1 5854 | . . . . . 6 β’ (π = 0 β (π β π ) = ( 0 β π )) | |
19 | simp1 1134 | . . . . . . 7 β’ (((πΎ β HL β§ π β π») β§ (β β π β§ β β ( I βΎ π΅)) β§ (π β πΈ β§ π β 0 )) β (πΎ β HL β§ π β π»)) | |
20 | simp3l 1199 | . . . . . . 7 β’ (((πΎ β HL β§ π β π») β§ (β β π β§ β β ( I βΎ π΅)) β§ (π β πΈ β§ π β 0 )) β π β πΈ) | |
21 | 1, 2, 3, 4, 5 | tendo0mul 40293 | . . . . . . 7 β’ (((πΎ β HL β§ π β π») β§ π β πΈ) β ( 0 β π ) = 0 ) |
22 | 19, 20, 21 | syl2anc 583 | . . . . . 6 β’ (((πΎ β HL β§ π β π») β§ (β β π β§ β β ( I βΎ π΅)) β§ (π β πΈ β§ π β 0 )) β ( 0 β π ) = 0 ) |
23 | 18, 22 | sylan9eqr 2790 | . . . . 5 β’ ((((πΎ β HL β§ π β π») β§ (β β π β§ β β ( I βΎ π΅)) β§ (π β πΈ β§ π β 0 )) β§ π = 0 ) β (π β π ) = 0 ) |
24 | 17, 23 | eqtr3d 2770 | . . . 4 β’ ((((πΎ β HL β§ π β π») β§ (β β π β§ β β ( I βΎ π΅)) β§ (π β πΈ β§ π β 0 )) β§ π = 0 ) β ( I βΎ π) = 0 ) |
25 | 24 | ex 412 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (β β π β§ β β ( I βΎ π΅)) β§ (π β πΈ β§ π β 0 )) β (π = 0 β ( I βΎ π) = 0 )) |
26 | 25 | necon3d 2957 | . 2 β’ (((πΎ β HL β§ π β π») β§ (β β π β§ β β ( I βΎ π΅)) β§ (π β πΈ β§ π β 0 )) β (( I βΎ π) β 0 β π β 0 )) |
27 | 7, 26 | mpd 15 | 1 β’ (((πΎ β HL β§ π β π») β§ (β β π β§ β β ( I βΎ π΅)) β§ (π β πΈ β§ π β 0 )) β π β 0 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 β wne 2936 βwral 3057 ifcif 4524 β¦ cmpt 5225 I cid 5569 β‘ccnv 5671 βΎ cres 5674 β ccom 5676 βcfv 6542 β©crio 7369 (class class class)co 7414 Basecbs 17173 occoc 17234 joincjn 18296 meetcmee 18297 HLchlt 38816 LHypclh 39451 LTrncltrn 39568 trLctrl 39625 TEndoctendo 40219 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-riotaBAD 38419 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-undef 8272 df-map 8840 df-proset 18280 df-poset 18298 df-plt 18315 df-lub 18331 df-glb 18332 df-join 18333 df-meet 18334 df-p0 18410 df-p1 18411 df-lat 18417 df-clat 18484 df-oposet 38642 df-ol 38644 df-oml 38645 df-covers 38732 df-ats 38733 df-atl 38764 df-cvlat 38788 df-hlat 38817 df-llines 38965 df-lplanes 38966 df-lvols 38967 df-lines 38968 df-psubsp 38970 df-pmap 38971 df-padd 39263 df-lhyp 39455 df-laut 39456 df-ldil 39571 df-ltrn 39572 df-trl 39626 df-tendo 40222 |
This theorem is referenced by: (None) |
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