Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trlid0 Structured version   Visualization version   GIF version

Theorem trlid0 39035
Description: The trace of the identity translation is zero. (Contributed by NM, 11-Jun-2013.)
Hypotheses
Ref Expression
trlid0.b 𝐡 = (Baseβ€˜πΎ)
trlid0.z 0 = (0.β€˜πΎ)
trlid0.h 𝐻 = (LHypβ€˜πΎ)
trlid0.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
trlid0 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (π‘…β€˜( I β†Ύ 𝐡)) = 0 )

Proof of Theorem trlid0
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . 3 (leβ€˜πΎ) = (leβ€˜πΎ)
2 eqid 2732 . . 3 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
3 trlid0.h . . 3 𝐻 = (LHypβ€˜πΎ)
41, 2, 3lhpexnle 38865 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ βˆƒπ‘ ∈ (Atomsβ€˜πΎ) Β¬ 𝑝(leβ€˜πΎ)π‘Š)
5 simpl 483 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
6 simpr 485 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š))
7 trlid0.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
8 eqid 2732 . . . . 5 ((LTrnβ€˜πΎ)β€˜π‘Š) = ((LTrnβ€˜πΎ)β€˜π‘Š)
97, 3, 8idltrn 39009 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ( I β†Ύ 𝐡) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š))
109adantr 481 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ ( I β†Ύ 𝐡) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š))
11 eqid 2732 . . . 4 ( I β†Ύ 𝐡) = ( I β†Ύ 𝐡)
127, 1, 2, 3, 8ltrnideq 39034 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ( I β†Ύ 𝐡) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ (( I β†Ύ 𝐡) = ( I β†Ύ 𝐡) ↔ (( I β†Ύ 𝐡)β€˜π‘) = 𝑝))
135, 10, 6, 12syl3anc 1371 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ (( I β†Ύ 𝐡) = ( I β†Ύ 𝐡) ↔ (( I β†Ύ 𝐡)β€˜π‘) = 𝑝))
1411, 13mpbii 232 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ (( I β†Ύ 𝐡)β€˜π‘) = 𝑝)
15 trlid0.z . . . 4 0 = (0.β€˜πΎ)
16 trlid0.r . . . 4 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
171, 15, 2, 3, 8, 16trl0 39029 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š) ∧ (( I β†Ύ 𝐡) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∧ (( I β†Ύ 𝐡)β€˜π‘) = 𝑝)) β†’ (π‘…β€˜( I β†Ύ 𝐡)) = 0 )
185, 6, 10, 14, 17syl112anc 1374 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ (π‘…β€˜( I β†Ύ 𝐡)) = 0 )
194, 18rexlimddv 3161 1 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (π‘…β€˜( I β†Ύ 𝐡)) = 0 )
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   class class class wbr 5147   I cid 5572   β†Ύ cres 5677  β€˜cfv 6540  Basecbs 17140  lecple 17200  0.cp0 18372  Atomscatm 38121  HLchlt 38208  LHypclh 38843  LTrncltrn 38960  trLctrl 39017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8818  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-p1 18375  df-lat 18381  df-clat 18448  df-oposet 38034  df-ol 38036  df-oml 38037  df-covers 38124  df-ats 38125  df-atl 38156  df-cvlat 38180  df-hlat 38209  df-lhyp 38847  df-laut 38848  df-ldil 38963  df-ltrn 38964  df-trl 39018
This theorem is referenced by:  tendoid  39632  tendo0tp  39648  cdlemkid2  39783  cdlemk39s-id  39799  dian0  39898  dihmeetlem4preN  40165
  Copyright terms: Public domain W3C validator