Theorem ltrnu 41002

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 2762	. . . . . . . . 9 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
10		ltrnu.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
11	4, 5, 6, 7, 8, 9, 10	isltrn 41000	. . . . . . . 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 5110	. . . . . . . . 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 6882	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → (𝐹‘𝑝) = (𝐹‘𝑃))
21	19, 20	oveq12d 7435	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝 ∨ (𝐹‘𝑝)) = (𝑃 ∨ (𝐹‘𝑃)))
22	21	oveq1d 7432	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
23	22	eqeq1d 2764	. . . . . . 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 5110	. . . . . . . . 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 6882	. . . . . . . . . 10 ⊢ (𝑞 = 𝑄 → (𝐹‘𝑞) = (𝐹‘𝑄))
30	28, 29	oveq12d 7435	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → (𝑞 ∨ (𝐹‘𝑞)) = (𝑄 ∨ (𝐹‘𝑄)))
31	30	oveq1d 7432	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘𝑄)) ∧ 𝑊))
32	31	eqeq2d 2773	. . . . . . 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 3590	. . . . 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 3078   class class class wbr 5107  ‘cfv 6537  (class class class)co 7417  lecple 17355  joincjn 18405  meetcmee 18406  Atomscatm 40144  LHypclh 40865  LDilcldil 40981  LTrncltrn 40982

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-ltrn 40986

This theorem is used by:  ltrncnv  41027  trlval2  41044  cdlemg14f  41534  cdlemg14g  41535