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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 2737 | . . 3 β’ (leβπΎ) = (leβπΎ) | |
2 | eqid 2737 | . . 3 β’ (AtomsβπΎ) = (AtomsβπΎ) | |
3 | trlid0.h | . . 3 β’ π» = (LHypβπΎ) | |
4 | 1, 2, 3 | lhpexnle 38472 | . 2 β’ ((πΎ β HL β§ π β π») β βπ β (AtomsβπΎ) Β¬ π(leβπΎ)π) |
5 | simpl 484 | . . 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 486 | . . 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 2737 | . . . . 5 β’ ((LTrnβπΎ)βπ) = ((LTrnβπΎ)βπ) | |
9 | 7, 3, 8 | idltrn 38616 | . . . 4 β’ ((πΎ β HL β§ π β π») β ( I βΎ π΅) β ((LTrnβπΎ)βπ)) |
10 | 9 | adantr 482 | . . 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 2737 | . . . 4 β’ ( I βΎ π΅) = ( I βΎ π΅) | |
12 | 7, 1, 2, 3, 8 | ltrnideq 38641 | . . . . 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 1372 | . . . 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 38636 | . . 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 1375 | . 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 3159 | 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 397 = wceq 1542 β wcel 2107 class class class wbr 5106 I cid 5531 βΎ cres 5636 βcfv 6497 Basecbs 17084 lecple 17141 0.cp0 18313 Atomscatm 37728 HLchlt 37815 LHypclh 38450 LTrncltrn 38567 trLctrl 38624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-map 8768 df-proset 18185 df-poset 18203 df-plt 18220 df-lub 18236 df-glb 18237 df-join 18238 df-meet 18239 df-p0 18315 df-p1 18316 df-lat 18322 df-clat 18389 df-oposet 37641 df-ol 37643 df-oml 37644 df-covers 37731 df-ats 37732 df-atl 37763 df-cvlat 37787 df-hlat 37816 df-lhyp 38454 df-laut 38455 df-ldil 38570 df-ltrn 38571 df-trl 38625 |
This theorem is referenced by: tendoid 39239 tendo0tp 39255 cdlemkid2 39390 cdlemk39s-id 39406 dian0 39505 dihmeetlem4preN 39772 |
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