Theorem cmbr 31655

Description: Binary relation expressing 𝐴 commutes with 𝐵. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
cmbr	⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵)))))

Proof of Theorem cmbr

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

Step	Hyp	Ref	Expression
1		eleq1 2824	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ Cℋ ↔ 𝐴 ∈ Cℋ ))
2	1	anbi1d 632	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝑦 ∈ Cℋ )))
3		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4		ineq1 4153	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝑦))
5		ineq1 4153	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ (⊥‘𝑦)) = (𝐴 ∩ (⊥‘𝑦)))
6	4, 5	oveq12d 7385	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦))) = ((𝐴 ∩ 𝑦) ∨ℋ (𝐴 ∩ (⊥‘𝑦))))
7	3, 6	eqeq12d 2752	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦))) ↔ 𝐴 = ((𝐴 ∩ 𝑦) ∨ℋ (𝐴 ∩ (⊥‘𝑦)))))
8	2, 7	anbi12d 633	. . 3 ⊢ (𝑥 = 𝐴 → (((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ 𝑥 = ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦)))) ↔ ((𝐴 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ 𝐴 = ((𝐴 ∩ 𝑦) ∨ℋ (𝐴 ∩ (⊥‘𝑦))))))
9		eleq1 2824	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ Cℋ ↔ 𝐵 ∈ Cℋ ))
10	9	anbi2d 631	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ )))
11		ineq2 4154	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 ∩ 𝑦) = (𝐴 ∩ 𝐵))
12		fveq2 6840	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵))
13	12	ineq2d 4160	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 ∩ (⊥‘𝑦)) = (𝐴 ∩ (⊥‘𝐵)))
14	11, 13	oveq12d 7385	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴 ∩ 𝑦) ∨ℋ (𝐴 ∩ (⊥‘𝑦))) = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))))
15	14	eqeq2d 2747	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 = ((𝐴 ∩ 𝑦) ∨ℋ (𝐴 ∩ (⊥‘𝑦))) ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵)))))
16	10, 15	anbi12d 633	. . 3 ⊢ (𝑦 = 𝐵 → (((𝐴 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ 𝐴 = ((𝐴 ∩ 𝑦) ∨ℋ (𝐴 ∩ (⊥‘𝑦)))) ↔ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))))))
17		df-cm 31654	. . 3 ⊢ 𝐶ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ 𝑥 = ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦))))}
18	8, 16, 17	brabg 5494	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝐶ℋ 𝐵 ↔ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))))))
19	18	bianabs 541	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∩ cin 3888   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367   C_ℋ cch 31000  ⊥cort 31001   ∨_ℋ chj 31004   𝐶_ℋ ccm 31007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-iota 6454  df-fv 6506  df-ov 7370  df-cm 31654

This theorem is referenced by:  cmbri  31661  cm2j  31691