Theorem cmtvalN 35231

Description: Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 28967 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 35230	. . . . 5 ⊢ (𝐾 ∈ 𝐴 → 𝐶 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))})
7		df-3an 1110	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦)))) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦)))))
8	7	opabbii 4911	. . . . 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	syl6eq 2850	. . . 4 ⊢ (𝐾 ∈ 𝐴 → 𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))})
10	9	breqd 4855	. . 3 ⊢ (𝐾 ∈ 𝐴 → (𝑋𝐶𝑌 ↔ 𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))}𝑌))
11	10	3ad2ant1 1164	. 2 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))}𝑌))
12		df-br 4845	. . . 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 6886	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ∧ 𝑦) = (𝑋 ∧ 𝑦))
15		oveq1 6886	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ∧ ( ⊥ ‘𝑦)) = (𝑋 ∧ ( ⊥ ‘𝑦)))
16	14, 15	oveq12d 6897	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))) = ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ ( ⊥ ‘𝑦))))
17	13, 16	eqeq12d 2815	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))) ↔ 𝑋 = ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ ( ⊥ ‘𝑦)))))
18		oveq2 6887	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋 ∧ 𝑦) = (𝑋 ∧ 𝑌))
19		fveq2 6412	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ( ⊥ ‘𝑦) = ( ⊥ ‘𝑌))
20	19	oveq2d 6895	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋 ∧ ( ⊥ ‘𝑦)) = (𝑋 ∧ ( ⊥ ‘𝑌)))
21	18, 20	oveq12d 6897	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ ( ⊥ ‘𝑦))) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌))))
22	21	eqeq2d 2810	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 = ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ ( ⊥ ‘𝑦))) ↔ 𝑋 = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌)))))
23	17, 22	opelopab2 5193	. . . 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	syl5bb 275	. . 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 1161	. 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 271	1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑋 = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌)))))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   ∧ w3a 1108   = wceq 1653   ∈ wcel 2157  ⟨cop 4375   class class class wbr 4844  {copab 4906  ‘cfv 6102  (class class class)co 6879  Basecbs 16183  occoc 16274  joincjn 17258  meetcmee 17259  cmccmtN 35193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-iota 6065  df-fun 6104  df-fv 6110  df-ov 6882  df-cmtN 35197

This theorem is referenced by:  cmtcomlemN  35268  cmt2N  35270  cmtbr2N  35273  cmtbr3N  35274