Theorem cmbr 31520

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 2817	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ Cℋ ↔ 𝐴 ∈ Cℋ ))
2	1	anbi1d 631	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝑦 ∈ Cℋ )))
3		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4		ineq1 4179	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝑦))
5		ineq1 4179	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ (⊥‘𝑦)) = (𝐴 ∩ (⊥‘𝑦)))
6	4, 5	oveq12d 7408	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦))) = ((𝐴 ∩ 𝑦) ∨ℋ (𝐴 ∩ (⊥‘𝑦))))
7	3, 6	eqeq12d 2746	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦))) ↔ 𝐴 = ((𝐴 ∩ 𝑦) ∨ℋ (𝐴 ∩ (⊥‘𝑦)))))
8	2, 7	anbi12d 632	. . 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 2817	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ Cℋ ↔ 𝐵 ∈ Cℋ ))
10	9	anbi2d 630	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ )))
11		ineq2 4180	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 ∩ 𝑦) = (𝐴 ∩ 𝐵))
12		fveq2 6861	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵))
13	12	ineq2d 4186	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 ∩ (⊥‘𝑦)) = (𝐴 ∩ (⊥‘𝐵)))
14	11, 13	oveq12d 7408	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴 ∩ 𝑦) ∨ℋ (𝐴 ∩ (⊥‘𝑦))) = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))))
15	14	eqeq2d 2741	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 = ((𝐴 ∩ 𝑦) ∨ℋ (𝐴 ∩ (⊥‘𝑦))) ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵)))))
16	10, 15	anbi12d 632	. . 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 31519	. . 3 ⊢ 𝐶ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ 𝑥 = ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦))))}
18	8, 16, 17	brabg 5502	. 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 1540   ∈ wcel 2109   ∩ cin 3916   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390   C_ℋ cch 30865  ⊥cort 30866   ∨_ℋ chj 30869   𝐶_ℋ ccm 30872

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-iota 6467  df-fv 6522  df-ov 7393  df-cm 31519

This theorem is referenced by:  cmbri  31526  cm2j  31556