Theorem isltrn2N 40779

Description: The predicate "is a lattice translation". Version of isltrn 40778 that considers only different 𝑝 and 𝑞. TODO: Can this eliminate some separate proofs for the 𝑝 = 𝑞 case? (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
isltrn2N	⊢ ((𝐾 ∈ 𝐵 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝑇 ↔ (𝐹 ∈ 𝐷 ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑝 ≠ 𝑞) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))))

Distinct variable groups:   𝑞,𝑝,𝐴   𝐾,𝑝,𝑞   𝑊,𝑝,𝑞   𝐹,𝑝,𝑞

Allowed substitution hints:   𝐵(𝑞,𝑝)   𝐷(𝑞,𝑝)   𝑇(𝑞,𝑝)   𝐻(𝑞,𝑝)   ∨ (𝑞,𝑝)   ≤ (𝑞,𝑝)   ∧ (𝑞,𝑝)

Proof of Theorem isltrn2N

Step	Hyp	Ref	Expression
1		ltrnset.l	. . 3 ⊢ ≤ = (le‘𝐾)
2		ltrnset.j	. . 3 ⊢ ∨ = (join‘𝐾)
3		ltrnset.m	. . 3 ⊢ ∧ = (meet‘𝐾)
4		ltrnset.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
5		ltrnset.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
6		ltrnset.d	. . 3 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊)
7		ltrnset.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8	1, 2, 3, 4, 5, 6, 7	isltrn 40778	. 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝑇 ↔ (𝐹 ∈ 𝐷 ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))))
9		3simpa 1164	. . . . . 6 ⊢ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑝 ≠ 𝑞) → (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊))
10	9	imim1i 64	. . . . 5 ⊢ (((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)) → ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑝 ≠ 𝑞) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))
11		3anass 1109	. . . . . . . . 9 ⊢ ((𝑝 ≠ 𝑞 ∧ ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ↔ (𝑝 ≠ 𝑞 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)))
12		3anrot 1115	. . . . . . . . 9 ⊢ ((𝑝 ≠ 𝑞 ∧ ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ↔ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑝 ≠ 𝑞))
13		df-ne 2965	. . . . . . . . . 10 ⊢ (𝑝 ≠ 𝑞 ↔ ¬ 𝑝 = 𝑞)
14	13	anbi1i 635	. . . . . . . . 9 ⊢ ((𝑝 ≠ 𝑞 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)) ↔ (¬ 𝑝 = 𝑞 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)))
15	11, 12, 14	3bitr3i 304	. . . . . . . 8 ⊢ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑝 ≠ 𝑞) ↔ (¬ 𝑝 = 𝑞 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)))
16	15	imbi1i 352	. . . . . . 7 ⊢ (((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑝 ≠ 𝑞) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)) ↔ ((¬ 𝑝 = 𝑞 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))
17		impexp 455	. . . . . . 7 ⊢ (((¬ 𝑝 = 𝑞 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)) ↔ (¬ 𝑝 = 𝑞 → ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊))))
18	16, 17	bitri 278	. . . . . 6 ⊢ (((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑝 ≠ 𝑞) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)) ↔ (¬ 𝑝 = 𝑞 → ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊))))
19		id 23	. . . . . . . . . 10 ⊢ (𝑝 = 𝑞 → 𝑝 = 𝑞)
20		fveq2 6879	. . . . . . . . . 10 ⊢ (𝑝 = 𝑞 → (𝐹‘𝑝) = (𝐹‘𝑞))
21	19, 20	oveq12d 7426	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → (𝑝 ∨ (𝐹‘𝑝)) = (𝑞 ∨ (𝐹‘𝑞)))
22	21	oveq1d 7423	. . . . . . . 8 ⊢ (𝑝 = 𝑞 → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊))
23	22	a1d 26	. . . . . . 7 ⊢ (𝑝 = 𝑞 → ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))
24		pm2.61 194	. . . . . . 7 ⊢ ((𝑝 = 𝑞 → ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊))) → ((¬ 𝑝 = 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))))
25	23, 24	ax-mp 5	. . . . . 6 ⊢ ((¬ 𝑝 = 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)))
26	18, 25	sylbi 220	. . . . 5 ⊢ (((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑝 ≠ 𝑞) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)) → ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))
27	10, 26	impbii 212	. . . 4 ⊢ (((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)) ↔ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑝 ≠ 𝑞) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))
28	27	2ralbii 3146	. . 3 ⊢ (∀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 ≠ 𝑞) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))
29	28	anbi2i 634	. 2 ⊢ ((𝐹 ∈ 𝐷 ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊))) ↔ (𝐹 ∈ 𝐷 ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑝 ≠ 𝑞) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊))))
30	8, 29	bitrdi 290	1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝑇 ↔ (𝐹 ∈ 𝐷 ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑝 ≠ 𝑞) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085   class class class wbr 5110  ‘cfv 6533  (class class class)co 7408  lecple 17313  joincjn 18363  meetcmee 18364  Atomscatm 39922  LHypclh 40643  LDilcldil 40759  LTrncltrn 40760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  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 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7411  df-ltrn 40764

This theorem is referenced by: (None)