Theorem ltrnu 40100

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 656	. . 3 ⊢ (((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ↔ ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊)))
2		simpr 484	. . . . 5 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴))
3		simplr 768	. . . . . 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 2729	. . . . . . . . 9 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
10		ltrnu.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
11	4, 5, 6, 7, 8, 9, 10	isltrn 40098	. . . . . . . 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 726	. . . . . . 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 484	. . . . . . 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 253	. . . . . 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 15	. . . . 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 5098	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝 ≤ 𝑊 ↔ 𝑃 ≤ 𝑊))
17	16	notbid 318	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (¬ 𝑝 ≤ 𝑊 ↔ ¬ 𝑃 ≤ 𝑊))
18	17	anbi1d 631	. . . . . . 7 ⊢ (𝑝 = 𝑃 → ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ↔ (¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)))
19		id 22	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → 𝑝 = 𝑃)
20		fveq2 6826	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → (𝐹‘𝑝) = (𝐹‘𝑃))
21	19, 20	oveq12d 7371	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝 ∨ (𝐹‘𝑝)) = (𝑃 ∨ (𝐹‘𝑃)))
22	21	oveq1d 7368	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
23	22	eqeq1d 2731	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊) ↔ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))
24	18, 23	imbi12d 344	. . . . . 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 5098	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → (𝑞 ≤ 𝑊 ↔ 𝑄 ≤ 𝑊))
26	25	notbid 318	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → (¬ 𝑞 ≤ 𝑊 ↔ ¬ 𝑄 ≤ 𝑊))
27	26	anbi2d 630	. . . . . . 7 ⊢ (𝑞 = 𝑄 → ((¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ↔ (¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊)))
28		id 22	. . . . . . . . . 10 ⊢ (𝑞 = 𝑄 → 𝑞 = 𝑄)
29		fveq2 6826	. . . . . . . . . 10 ⊢ (𝑞 = 𝑄 → (𝐹‘𝑞) = (𝐹‘𝑄))
30	28, 29	oveq12d 7371	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → (𝑞 ∨ (𝐹‘𝑞)) = (𝑄 ∨ (𝐹‘𝑄)))
31	30	oveq1d 7368	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘𝑄)) ∧ 𝑊))
32	31	eqeq2d 2740	. . . . . . 7 ⊢ (𝑞 = 𝑄 → (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊) ↔ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘𝑄)) ∧ 𝑊)))
33	27, 32	imbi12d 344	. . . . . 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 65	. . . 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 454	. . 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 594	. 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 1114	1 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘𝑄)) ∧ 𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044   class class class wbr 5095  ‘cfv 6486  (class class class)co 7353  lecple 17186  joincjn 18235  meetcmee 18236  Atomscatm 39241  LHypclh 39963  LDilcldil 40079  LTrncltrn 40080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-ltrn 40084

This theorem is referenced by:  ltrncnv  40125  trlval2  40142  cdlemg14f  40632  cdlemg14g  40633