Theorem isltrn2N 40122

Description: The predicate "is a lattice translation". Version of isltrn 40121 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 40121	. 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 1149	. . . . . 6 ⊢ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑝 ≠ 𝑞) → (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊))
10	9	imim1i 63	. . . . 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 1095	. . . . . . . . 9 ⊢ ((𝑝 ≠ 𝑞 ∧ ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ↔ (𝑝 ≠ 𝑞 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)))
12		3anrot 1100	. . . . . . . . 9 ⊢ ((𝑝 ≠ 𝑞 ∧ ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ↔ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑝 ≠ 𝑞))
13		df-ne 2941	. . . . . . . . . 10 ⊢ (𝑝 ≠ 𝑞 ↔ ¬ 𝑝 = 𝑞)
14	13	anbi1i 624	. . . . . . . . 9 ⊢ ((𝑝 ≠ 𝑞 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)) ↔ (¬ 𝑝 = 𝑞 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)))
15	11, 12, 14	3bitr3i 301	. . . . . . . 8 ⊢ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑝 ≠ 𝑞) ↔ (¬ 𝑝 = 𝑞 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)))
16	15	imbi1i 349	. . . . . . 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 450	. . . . . . 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 275	. . . . . 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 22	. . . . . . . . . 10 ⊢ (𝑝 = 𝑞 → 𝑝 = 𝑞)
20		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑝 = 𝑞 → (𝐹‘𝑝) = (𝐹‘𝑞))
21	19, 20	oveq12d 7449	. . . . . . . . 9 ⊢ (𝑝 = 𝑞 → (𝑝 ∨ (𝐹‘𝑝)) = (𝑞 ∨ (𝐹‘𝑞)))
22	21	oveq1d 7446	. . . . . . . 8 ⊢ (𝑝 = 𝑞 → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊))
23	22	a1d 25	. . . . . . 7 ⊢ (𝑝 = 𝑞 → ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) → ((𝑝 ∨ (𝐹‘𝑝)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))
24		pm2.61 192	. . . . . . 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 217	. . . . 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 209	. . . 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 3128	. . 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 623	. 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 287	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431  lecple 17304  joincjn 18357  meetcmee 18358  Atomscatm 39264  LHypclh 39986  LDilcldil 40102  LTrncltrn 40103

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-ltrn 40107

This theorem is referenced by: (None)