Theorem ltrnu 41271

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 2760	. . . . . . . . 9 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
10		ltrnu.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
11	4, 5, 6, 7, 8, 9, 10	isltrn 41269	. . . . . . . 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 5105	. . . . . . . . 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 6885	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → (𝐹‘𝑝) = (𝐹‘𝑃))
21	19, 20	oveq12d 7438	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝 ∨ (𝐹‘𝑝)) = (𝑃 ∨ (𝐹‘𝑃)))
22	21	oveq1d 7435	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
23	22	eqeq1d 2762	. . . . . . 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 5105	. . . . . . . . 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 6885	. . . . . . . . . 10 ⊢ (𝑞 = 𝑄 → (𝐹‘𝑞) = (𝐹‘𝑄))
30	28, 29	oveq12d 7438	. . . . . . . . 9 ⊢ (𝑞 = 𝑄 → (𝑞 ∨ (𝐹‘𝑞)) = (𝑄 ∨ (𝐹‘𝑄)))
31	30	oveq1d 7435	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘𝑄)) ∧ 𝑊))
32	31	eqeq2d 2771	. . . . . . 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 3586	. . . . 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 1569   ∈ wcel 2144  ∀wral 3076   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  lecple 17440  joincjn 18490  meetcmee 18491  Atomscatm 40413  LHypclh 41134  LDilcldil 41250  LTrncltrn 41251

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-ltrn 41255

This theorem is used by:  ltrncnv  41296  trlval2  41313  cdlemg14f  41803  cdlemg14g  41804