Theorem cmtvalN 38813

Description: Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 31466 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cmtfval.b	⊢ 𝐵 = (Base‘𝐾)
cmtfval.j	⊢ ∨ = (join‘𝐾)
cmtfval.m	⊢ ∧ = (meet‘𝐾)
cmtfval.o	⊢ ⊥ = (oc‘𝐾)
cmtfval.c	⊢ 𝐶 = (cm‘𝐾)

Assertion

Ref	Expression
cmtvalN	⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑋 = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌)))))

Proof of Theorem cmtvalN

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cmtfval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
2		cmtfval.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
3		cmtfval.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
4		cmtfval.o	. . . . . 6 ⊢ ⊥ = (oc‘𝐾)
5		cmtfval.c	. . . . . 6 ⊢ 𝐶 = (cm‘𝐾)
6	1, 2, 3, 4, 5	cmtfvalN 38812	. . . . 5 ⊢ (𝐾 ∈ 𝐴 → 𝐶 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))})
7		df-3an 1086	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦)))) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦)))))
8	7	opabbii 5216	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))}
9	6, 8	eqtrdi 2781	. . . 4 ⊢ (𝐾 ∈ 𝐴 → 𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))})
10	9	breqd 5160	. . 3 ⊢ (𝐾 ∈ 𝐴 → (𝑋𝐶𝑌 ↔ 𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))}𝑌))
11	10	3ad2ant1 1130	. 2 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))}𝑌))
12		df-br 5150	. . . 4 ⊢ (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))})
13		id 22	. . . . . 6 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
14		oveq1 7426	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ∧ 𝑦) = (𝑋 ∧ 𝑦))
15		oveq1 7426	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ∧ ( ⊥ ‘𝑦)) = (𝑋 ∧ ( ⊥ ‘𝑦)))
16	14, 15	oveq12d 7437	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))) = ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ ( ⊥ ‘𝑦))))
17	13, 16	eqeq12d 2741	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))) ↔ 𝑋 = ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ ( ⊥ ‘𝑦)))))
18		oveq2 7427	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋 ∧ 𝑦) = (𝑋 ∧ 𝑌))
19		fveq2 6896	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ( ⊥ ‘𝑦) = ( ⊥ ‘𝑌))
20	19	oveq2d 7435	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋 ∧ ( ⊥ ‘𝑦)) = (𝑋 ∧ ( ⊥ ‘𝑌)))
21	18, 20	oveq12d 7437	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ ( ⊥ ‘𝑦))) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))))
22	21	eqeq2d 2736	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 = ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ ( ⊥ ‘𝑦))) ↔ 𝑋 = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌)))))
23	17, 22	opelopab2 5543	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))} ↔ 𝑋 = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌)))))
24	12, 23	bitrid 282	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))}𝑌 ↔ 𝑋 = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌)))))
25	24	3adant1 1127	. 2 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))}𝑌 ↔ 𝑋 = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌)))))
26	11, 25	bitrd 278	1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑋 = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌)))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ⟨cop 4636   class class class wbr 5149  {copab 5211  ‘cfv 6549  (class class class)co 7419  Basecbs 17183  occoc 17244  joincjn 18306  meetcmee 18307  cmccmtN 38775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-cmtN 38779

This theorem is referenced by:  cmtcomlemN  38850  cmt2N  38852  cmtbr2N  38855  cmtbr3N  38856