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Theorem trlid0 38642
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 2737 . . 3 (leβ€˜πΎ) = (leβ€˜πΎ)
2 eqid 2737 . . 3 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
3 trlid0.h . . 3 𝐻 = (LHypβ€˜πΎ)
41, 2, 3lhpexnle 38472 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ βˆƒπ‘ ∈ (Atomsβ€˜πΎ) Β¬ 𝑝(leβ€˜πΎ)π‘Š)
5 simpl 484 . . 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 486 . . 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 2737 . . . . 5 ((LTrnβ€˜πΎ)β€˜π‘Š) = ((LTrnβ€˜πΎ)β€˜π‘Š)
97, 3, 8idltrn 38616 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ( I β†Ύ 𝐡) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š))
109adantr 482 . . 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 2737 . . . 4 ( I β†Ύ 𝐡) = ( I β†Ύ 𝐡)
127, 1, 2, 3, 8ltrnideq 38641 . . . . 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 1372 . . . 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 38636 . . 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 1375 . 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 3159 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 397   = wceq 1542   ∈ wcel 2107   class class class wbr 5106   I cid 5531   β†Ύ cres 5636  β€˜cfv 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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