Theorem cmbr 28443

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 2689	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ Cℋ ↔ 𝐴 ∈ Cℋ ))
2	1	anbi1d 741	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝑦 ∈ Cℋ )))
3		id 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4		ineq1 3807	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝑦))
5		ineq1 3807	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ (⊥‘𝑦)) = (𝐴 ∩ (⊥‘𝑦)))
6	4, 5	oveq12d 6668	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦))) = ((𝐴 ∩ 𝑦) ∨ℋ (𝐴 ∩ (⊥‘𝑦))))
7	3, 6	eqeq12d 2637	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦))) ↔ 𝐴 = ((𝐴 ∩ 𝑦) ∨ℋ (𝐴 ∩ (⊥‘𝑦)))))
8	2, 7	anbi12d 747	. . 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 2689	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ Cℋ ↔ 𝐵 ∈ Cℋ ))
10	9	anbi2d 740	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ↔ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ )))
11		ineq2 3808	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 ∩ 𝑦) = (𝐴 ∩ 𝐵))
12		fveq2 6191	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵))
13	12	ineq2d 3814	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 ∩ (⊥‘𝑦)) = (𝐴 ∩ (⊥‘𝐵)))
14	11, 13	oveq12d 6668	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴 ∩ 𝑦) ∨ℋ (𝐴 ∩ (⊥‘𝑦))) = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))))
15	14	eqeq2d 2632	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 = ((𝐴 ∩ 𝑦) ∨ℋ (𝐴 ∩ (⊥‘𝑦))) ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵)))))
16	10, 15	anbi12d 747	. . 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 28442	. . 3 ⊢ 𝐶ℋ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ Cℋ ) ∧ 𝑥 = ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦))))}
18	8, 16, 17	brabg 4994	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝐶ℋ 𝐵 ↔ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) ∧ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))))))
19	18	bianabs 924	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵)))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ∩ cin 3573   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650   C_ℋ cch 27786  ⊥cort 27787   ∨_ℋ chj 27790   𝐶_ℋ ccm 27793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-iota 5851  df-fv 5896  df-ov 6653  df-cm 28442

This theorem is referenced by:  cmbri  28449  cm2j  28479