Theorem cmbr 31674

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 2827	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ Cℋ ↔ 𝐴 ∈ Cℋ ))
2	1	anbi1d 637	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝑦 ∈ Cℋ )))
3		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4		ineq1 4143	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝑦))
5		ineq1 4143	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ (⊥‘𝑦)) = (𝐴 ∩ (⊥‘𝑦)))
6	4, 5	oveq12d 7375	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦))) = ((𝐴 ∩ 𝑦) ∨ℋ (𝐴 ∩ (⊥‘𝑦))))
7	3, 6	eqeq12d 2755	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦))) ↔ 𝐴 = ((𝐴 ∩ 𝑦) ∨ℋ (𝐴 ∩ (⊥‘𝑦)))))
8	2, 7	anbi12d 638	. . 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 2827	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ Cℋ ↔ 𝐵 ∈ Cℋ ))
10	9	anbi2d 636	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ )))
11		ineq2 4144	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 ∩ 𝑦) = (𝐴 ∩ 𝐵))
12		fveq2 6828	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵))
13	12	ineq2d 4150	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 ∩ (⊥‘𝑦)) = (𝐴 ∩ (⊥‘𝐵)))
14	11, 13	oveq12d 7375	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴 ∩ 𝑦) ∨ℋ (𝐴 ∩ (⊥‘𝑦))) = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))))
15	14	eqeq2d 2750	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 = ((𝐴 ∩ 𝑦) ∨ℋ (𝐴 ∩ (⊥‘𝑦))) ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵)))))
16	10, 15	anbi12d 638	. . 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 31673	. . 3 ⊢ 𝐶ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ 𝑥 = ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦))))}
18	8, 16, 17	brabg 5482	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝐶ℋ 𝐵 ↔ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))))))
19	18	bianabs 546	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵)))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ∩ cin 3882   class class class wbr 5073  ‘cfv 6486  (class class class)co 7357   C_ℋ cch 31019  ⊥cort 31020   ∨_ℋ chj 31023   𝐶_ℋ ccm 31026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5219  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-opab 5136  df-iota 6442  df-fv 6494  df-ov 7360  df-cm 31673

This theorem is referenced by:  cmbri  31680  cm2j  31710