Theorem ltrnu 40581

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 657	. . 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 769	. . . . . 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 2737	. . . . . . . . 9 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
10		ltrnu.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
11	4, 5, 6, 7, 8, 9, 10	isltrn 40579	. . . . . . . 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 727	. . . . . . 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 5089	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝 ≤ 𝑊 ↔ 𝑃 ≤ 𝑊))
17	16	notbid 318	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (¬ 𝑝 ≤ 𝑊 ↔ ¬ 𝑃 ≤ 𝑊))
18	17	anbi1d 632	. . . . . . 7 ⊢ (𝑝 = 𝑃 → ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ↔ (¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)))
19		id 22	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → 𝑝 = 𝑃)
20		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → (𝐹‘𝑝) = (𝐹‘𝑃))
21	19, 20	oveq12d 7378	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝 ∨ (𝐹‘𝑝)) = (𝑃 ∨ (𝐹‘𝑃)))
22	21	oveq1d 7375	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
23	22	eqeq1d 2739	. . . . . . 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 5089	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → (𝑞 ≤ 𝑊 ↔ 𝑄 ≤ 𝑊))
26	25	notbid 318	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → (¬ 𝑞 ≤ 𝑊 ↔ ¬ 𝑄 ≤ 𝑊))
27	26	anbi2d 631	. . . . . . 7 ⊢ (𝑞 = 𝑄 → ((¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ↔ (¬ 𝑃 ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊)))
28		id 22	. . . . . . . . . 10 ⊢ (𝑞 = 𝑄 → 𝑞 = 𝑄)
29		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑞 = 𝑄 → (𝐹‘𝑞) = (𝐹‘𝑄))
30	28, 29	oveq12d 7378	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → (𝑞 ∨ (𝐹‘𝑞)) = (𝑄 ∨ (𝐹‘𝑄)))
31	30	oveq1d 7375	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘𝑄)) ∧ 𝑊))
32	31	eqeq2d 2748	. . . . . . 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 3576	. . . . 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 595	. 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 1115	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 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360  lecple 17218  joincjn 18268  meetcmee 18269  Atomscatm 39723  LHypclh 40444  LDilcldil 40560  LTrncltrn 40561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-ltrn 40565

This theorem is referenced by:  ltrncnv  40606  trlval2  40623  cdlemg14f  41113  cdlemg14g  41114