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Theorem trlid0 39558
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 2726 . . 3 (leβ€˜πΎ) = (leβ€˜πΎ)
2 eqid 2726 . . 3 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
3 trlid0.h . . 3 𝐻 = (LHypβ€˜πΎ)
41, 2, 3lhpexnle 39388 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ βˆƒπ‘ ∈ (Atomsβ€˜πΎ) Β¬ 𝑝(leβ€˜πΎ)π‘Š)
5 simpl 482 . . 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 484 . . 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 2726 . . . . 5 ((LTrnβ€˜πΎ)β€˜π‘Š) = ((LTrnβ€˜πΎ)β€˜π‘Š)
97, 3, 8idltrn 39532 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ( I β†Ύ 𝐡) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š))
109adantr 480 . . 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 2726 . . . 4 ( I β†Ύ 𝐡) = ( I β†Ύ 𝐡)
127, 1, 2, 3, 8ltrnideq 39557 . . . . 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 1368 . . . 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 39552 . . 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 1371 . 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 3155 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 395   = wceq 1533   ∈ wcel 2098   class class class wbr 5141   I cid 5566   β†Ύ cres 5671  β€˜cfv 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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