Theorem trl0 40758

Description: If an atom not under the fiducial co-atom 𝑊 equals its lattice translation, the trace of the translation is zero. (Contributed by NM, 24-May-2012.)

Hypotheses

Ref	Expression
trl0.l	⊢ ≤ = (le‘𝐾)
trl0.z	⊢ 0 = (0.‘𝐾)
trl0.a	⊢ 𝐴 = (Atoms‘𝐾)
trl0.h	⊢ 𝐻 = (LHyp‘𝐾)
trl0.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
trl0.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

Assertion

Ref	Expression
trl0	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝑅‘𝐹) = 0 )

Proof of Theorem trl0

Step	Hyp	Ref	Expression
1		simp1 1148	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp3l 1214	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → 𝐹 ∈ 𝑇)
3		simp2 1149	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
4		trl0.l	. . . 4 ⊢ ≤ = (le‘𝐾)
5		eqid 2761	. . . 4 ⊢ (join‘𝐾) = (join‘𝐾)
6		eqid 2761	. . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾)
7		trl0.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
8		trl0.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
9		trl0.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
10		trl0.r	. . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
11	4, 5, 6, 7, 8, 9, 10	trlval2 40751	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃(join‘𝐾)(𝐹‘𝑃))(meet‘𝐾)𝑊))
12	1, 2, 3, 11	syl3anc 1389	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝑅‘𝐹) = ((𝑃(join‘𝐾)(𝐹‘𝑃))(meet‘𝐾)𝑊))
13		simp3r 1215	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝐹‘𝑃) = 𝑃)
14	13	oveq2d 7408	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝑃(join‘𝐾)(𝐹‘𝑃)) = (𝑃(join‘𝐾)𝑃))
15		simp1l 1210	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → 𝐾 ∈ HL)
16		simp2l 1212	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → 𝑃 ∈ 𝐴)
17	5, 7	hlatjidm 39957	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → (𝑃(join‘𝐾)𝑃) = 𝑃)
18	15, 16, 17	syl2anc 593	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝑃(join‘𝐾)𝑃) = 𝑃)
19	14, 18	eqtrd 2796	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝑃(join‘𝐾)(𝐹‘𝑃)) = 𝑃)
20	19	oveq1d 7407	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → ((𝑃(join‘𝐾)(𝐹‘𝑃))(meet‘𝐾)𝑊) = (𝑃(meet‘𝐾)𝑊))
21		trl0.z	. . . 4 ⊢ 0 = (0.‘𝐾)
22	4, 6, 21, 7, 8	lhpmat 40618	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃(meet‘𝐾)𝑊) = 0 )
23	1, 3, 22	syl2anc 593	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝑃(meet‘𝐾)𝑊) = 0 )
24	12, 20, 23	3eqtrd 2800	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝑅‘𝐹) = 0 )

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   class class class wbr 5099  ‘cfv 6517  (class class class)co 7392  lecple 17276  joincjn 18326  meetcmee 18327  0.cp0 18436  Atomscatm 39851  HLchlt 39938  LHypclh 40572  LTrncltrn 40689  trLctrl 40746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-map 8805  df-proset 18309  df-poset 18328  df-plt 18343  df-lub 18359  df-glb 18360  df-join 18361  df-meet 18362  df-p0 18438  df-lat 18447  df-covers 39854  df-ats 39855  df-atl 39886  df-cvlat 39910  df-hlat 39939  df-lhyp 40576  df-laut 40577  df-ldil 40692  df-ltrn 40693  df-trl 40747

This theorem is referenced by:  trlator0  40759  ltrnnidn  40762  trlid0  40764  trlnidatb  40765  trlnle  40774  trlval3  40775  trlval4  40776  cdlemc6  40784  cdlemg31d  41288