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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > trlid0 | Structured version Visualization version GIF version |
Description: The trace of the identity translation is zero. (Contributed by NM, 11-Jun-2013.) |
Ref | Expression |
---|---|
trlid0.b | β’ π΅ = (BaseβπΎ) |
trlid0.z | β’ 0 = (0.βπΎ) |
trlid0.h | β’ π» = (LHypβπΎ) |
trlid0.r | β’ π = ((trLβπΎ)βπ) |
Ref | Expression |
---|---|
trlid0 | β’ ((πΎ β HL β§ π β π») β (π β( I βΎ π΅)) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . 3 β’ (leβπΎ) = (leβπΎ) | |
2 | eqid 2726 | . . 3 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
3 | trlid0.h | . . 3 β’ π» = (LHypβπΎ) | |
4 | 1, 2, 3 | lhpexnle 39388 | . 2 β’ ((πΎ β HL β§ π β π») β βπ β (AtomsβπΎ) Β¬ π(leβπΎ)π) |
5 | simpl 482 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β (AtomsβπΎ) β§ Β¬ π(leβπΎ)π)) β (πΎ β HL β§ π β π»)) | |
6 | simpr 484 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β (AtomsβπΎ) β§ Β¬ π(leβπΎ)π)) β (π β (AtomsβπΎ) β§ Β¬ π(leβπΎ)π)) | |
7 | trlid0.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
8 | eqid 2726 | . . . . 5 β’ ((LTrnβπΎ)βπ) = ((LTrnβπΎ)βπ) | |
9 | 7, 3, 8 | idltrn 39532 | . . . 4 β’ ((πΎ β HL β§ π β π») β ( I βΎ π΅) β ((LTrnβπΎ)βπ)) |
10 | 9 | adantr 480 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β (AtomsβπΎ) β§ Β¬ π(leβπΎ)π)) β ( I βΎ π΅) β ((LTrnβπΎ)βπ)) |
11 | eqid 2726 | . . . 4 β’ ( I βΎ π΅) = ( I βΎ π΅) | |
12 | 7, 1, 2, 3, 8 | ltrnideq 39557 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ ( I βΎ π΅) β ((LTrnβπΎ)βπ) β§ (π β (AtomsβπΎ) β§ Β¬ π(leβπΎ)π)) β (( I βΎ π΅) = ( I βΎ π΅) β (( I βΎ π΅)βπ) = π)) |
13 | 5, 10, 6, 12 | syl3anc 1368 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β (AtomsβπΎ) β§ Β¬ π(leβπΎ)π)) β (( I βΎ π΅) = ( I βΎ π΅) β (( I βΎ π΅)βπ) = π)) |
14 | 11, 13 | mpbii 232 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β (AtomsβπΎ) β§ Β¬ π(leβπΎ)π)) β (( I βΎ π΅)βπ) = π) |
15 | trlid0.z | . . . 4 β’ 0 = (0.βπΎ) | |
16 | trlid0.r | . . . 4 β’ π = ((trLβπΎ)βπ) | |
17 | 1, 15, 2, 3, 8, 16 | trl0 39552 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β (AtomsβπΎ) β§ Β¬ π(leβπΎ)π) β§ (( I βΎ π΅) β ((LTrnβπΎ)βπ) β§ (( I βΎ π΅)βπ) = π)) β (π β( I βΎ π΅)) = 0 ) |
18 | 5, 6, 10, 14, 17 | syl112anc 1371 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β (AtomsβπΎ) β§ Β¬ π(leβπΎ)π)) β (π β( I βΎ π΅)) = 0 ) |
19 | 4, 18 | rexlimddv 3155 | 1 β’ ((πΎ β HL β§ π β π») β (π β( I βΎ π΅)) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 class class class wbr 5141 I cid 5566 βΎ cres 5671 βcfv 6536 Basecbs 17151 lecple 17211 0.cp0 18386 Atomscatm 38644 HLchlt 38731 LHypclh 39366 LTrncltrn 39483 trLctrl 39540 |
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 |
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-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-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-lhyp 39370 df-laut 39371 df-ldil 39486 df-ltrn 39487 df-trl 39541 |
This theorem is referenced by: tendoid 40155 tendo0tp 40171 cdlemkid2 40306 cdlemk39s-id 40322 dian0 40421 dihmeetlem4preN 40688 |
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