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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 2728 | . . 3 β’ (leβπΎ) = (leβπΎ) | |
2 | eqid 2728 | . . 3 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
3 | trlid0.h | . . 3 β’ π» = (LHypβπΎ) | |
4 | 1, 2, 3 | lhpexnle 39479 | . 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 2728 | . . . . 5 β’ ((LTrnβπΎ)βπ) = ((LTrnβπΎ)βπ) | |
9 | 7, 3, 8 | idltrn 39623 | . . . 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 2728 | . . . 4 β’ ( I βΎ π΅) = ( I βΎ π΅) | |
12 | 7, 1, 2, 3, 8 | ltrnideq 39648 | . . . . 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 1369 | . . . 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 39643 | . . 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 1372 | . 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 3158 | 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 1534 β wcel 2099 class class class wbr 5148 I cid 5575 βΎ cres 5680 βcfv 6548 Basecbs 17180 lecple 17240 0.cp0 18415 Atomscatm 38735 HLchlt 38822 LHypclh 39457 LTrncltrn 39574 trLctrl 39631 |
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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 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 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-map 8847 df-proset 18287 df-poset 18305 df-plt 18322 df-lub 18338 df-glb 18339 df-join 18340 df-meet 18341 df-p0 18417 df-p1 18418 df-lat 18424 df-clat 18491 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 df-lhyp 39461 df-laut 39462 df-ldil 39577 df-ltrn 39578 df-trl 39632 |
This theorem is referenced by: tendoid 40246 tendo0tp 40262 cdlemkid2 40397 cdlemk39s-id 40413 dian0 40512 dihmeetlem4preN 40779 |
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