Theorem ltrnu 41146

Description: Uniqueness property of a lattice translation value for atoms not under the fiducial co-atom 𝑊. Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 20-May-2012.)

Hypotheses

Ref	Expression
ltrnu.l	⊢ ≤ = (le‘𝐾)
ltrnu.j	⊢ ∨ = (join‘𝐾)
ltrnu.m	⊢ ∧ = (meet‘𝐾)
ltrnu.a	⊢ 𝐴 = (Atoms‘𝐾)
ltrnu.h	⊢ 𝐻 = (LHyp‘𝐾)
ltrnu.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)

Assertion

Ref	Expression
ltrnu	⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘𝑄)) ∧ 𝑊))

Proof of Theorem ltrnu

Dummy variables 𝑞 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		an4 669	. . 3 ⊢ (((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ↔ ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊)))
2		simpr 490	. . . . 5 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴))
3		simplr 781	. . . . . 6 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐹 ∈ 𝑇)
4		ltrnu.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
5		ltrnu.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
6		ltrnu.m	. . . . . . . . 9 ⊢ ∧ = (meet‘𝐾)
7		ltrnu.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
8		ltrnu.h	. . . . . . . . 9 ⊢ 𝐻 = (LHyp‘𝐾)
9		eqid 2761	. . . . . . . . 9 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
10		ltrnu.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
11	4, 5, 6, 7, 8, 9, 10	isltrn 41144	. . . . . . . 8 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝑇 ↔ (𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))))
12	11	ad2antrr 739	. . . . . . 7 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝐹 ∈ 𝑇 ↔ (𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))))
13		simpr 490	. . . . . . 7 ⊢ ((𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊))) → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))
14	12, 13	biimtrdi 256	. . . . . 6 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝐹 ∈ 𝑇 → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊))))
15	3, 14	mpd 16	. . . . 5 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))
16		breq1 5106	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝 ≤ 𝑊 ↔ 𝑃 ≤ 𝑊))
17	16	notbid 321	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (¬ 𝑝 ≤ 𝑊 ↔ ¬ 𝑃 ≤ 𝑊))
18	17	anbi1d 643	. . . . . . 7 ⊢ (𝑝 = 𝑃 → ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ↔ (¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)))
19		id 23	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → 𝑝 = 𝑃)
20		fveq2 6877	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → (𝐹‘𝑝) = (𝐹‘𝑃))
21	19, 20	oveq12d 7430	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝 ∨ (𝐹‘𝑝)) = (𝑃 ∨ (𝐹‘𝑃)))
22	21	oveq1d 7427	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
23	22	eqeq1d 2763	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊) ↔ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))
24	18, 23	imbi12d 347	. . . . . 6 ⊢ (𝑝 = 𝑃 → (((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)) ↔ ((¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊))))
25		breq1 5106	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → (𝑞 ≤ 𝑊 ↔ 𝑄 ≤ 𝑊))
26	25	notbid 321	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → (¬ 𝑞 ≤ 𝑊 ↔ ¬ 𝑄 ≤ 𝑊))
27	26	anbi2d 642	. . . . . . 7 ⊢ (𝑞 = 𝑄 → ((¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ↔ (¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊)))
28		id 23	. . . . . . . . . 10 ⊢ (𝑞 = 𝑄 → 𝑞 = 𝑄)
29		fveq2 6877	. . . . . . . . . 10 ⊢ (𝑞 = 𝑄 → (𝐹‘𝑞) = (𝐹‘𝑄))
30	28, 29	oveq12d 7430	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → (𝑞 ∨ (𝐹‘𝑞)) = (𝑄 ∨ (𝐹‘𝑄)))
31	30	oveq1d 7427	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘𝑄)) ∧ 𝑊))
32	31	eqeq2d 2772	. . . . . . 7 ⊢ (𝑞 = 𝑄 → (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊) ↔ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘𝑄)) ∧ 𝑊)))
33	27, 32	imbi12d 347	. . . . . 6 ⊢ (𝑞 = 𝑄 → (((¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)) ↔ ((¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘𝑄)) ∧ 𝑊))))
34	24, 33	rspc2v 3587	. . . . 5 ⊢ ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)) → ((¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘𝑄)) ∧ 𝑊))))
35	2, 15, 34	sylc 66	. . . 4 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘𝑄)) ∧ 𝑊)))
36	35	impr 460	. . 3 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊))) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘𝑄)) ∧ 𝑊))
37	1, 36	sylan2b 606	. 2 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘𝑄)) ∧ 𝑊))
38	37	3impb 1132	1 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘𝑄)) ∧ 𝑊))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077   class class class wbr 5103  ‘cfv 6531  (class class class)co 7412  lecple 17415  joincjn 18465  meetcmee 18466  Atomscatm 40288  LHypclh 41009  LDilcldil 41125  LTrncltrn 41126

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-ltrn 41130

This theorem is used by:  ltrncnv  41171  trlval2  41188  cdlemg14f  41678  cdlemg14g  41679