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Theorem trlid0 39649
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 2728 . . 3 (leβ€˜πΎ) = (leβ€˜πΎ)
2 eqid 2728 . . 3 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
3 trlid0.h . . 3 𝐻 = (LHypβ€˜πΎ)
41, 2, 3lhpexnle 39479 . 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 2728 . . . . 5 ((LTrnβ€˜πΎ)β€˜π‘Š) = ((LTrnβ€˜πΎ)β€˜π‘Š)
97, 3, 8idltrn 39623 . . . 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 2728 . . . 4 ( I β†Ύ 𝐡) = ( I β†Ύ 𝐡)
127, 1, 2, 3, 8ltrnideq 39648 . . . . 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 1369 . . . 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 39643 . . 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 1372 . 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 3158 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 1534   ∈ wcel 2099   class class class wbr 5148   I cid 5575   β†Ύ cres 5680  β€˜cfv 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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