Theorem cmbrt 5494

Description: Binary relation expressing A commutes with B. Definition of commutes in [Kalmbach] p. 20.

Assertion

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

Proof of Theorem cmbrt

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . . 5 ⊢ (x = A → (x ∈ Cℋ ↔ A ∈ Cℋ ))
2	1	anbi1d 469	. . . 4 ⊢ (x = A → ((x ∈ Cℋ ∧ y ∈ Cℋ ) ↔ (A ∈ Cℋ ∧ y ∈ Cℋ )))
3		id 9	. . . . 5 ⊢ (x = A → x = A)
4		ineq1 1638	. . . . . 6 ⊢ (x = A → (x ∩ y) = (A ∩ y))
5		ineq1 1638	. . . . . 6 ⊢ (x = A → (x ∩ (⊥ ‘y)) = (A ∩ (⊥ ‘y)))
6	4, 5	opreq12d 3014	. . . . 5 ⊢ (x = A → ((x ∩ y) ∨ℋ (x ∩ (⊥ ‘y))) = ((A ∩ y) ∨ℋ (A ∩ (⊥ ‘y))))
7	3, 6	cleq12d 1115	. . . 4 ⊢ (x = A → (x = ((x ∩ y) ∨ℋ (x ∩ (⊥ ‘y))) ↔ A = ((A ∩ y) ∨ℋ (A ∩ (⊥ ‘y)))))
8	2, 7	anbi12d 476	. . 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 1149	. . . . 5 ⊢ (y = B → (y ∈ Cℋ ↔ B ∈ Cℋ ))
10	9	anbi2d 468	. . . 4 ⊢ (y = B → ((A ∈ Cℋ ∧ y ∈ Cℋ ) ↔ (A ∈ Cℋ ∧ B ∈ Cℋ )))
11		ineq2 1639	. . . . . 6 ⊢ (y = B → (A ∩ y) = (A ∩ B))
12		fveq2 2832	. . . . . . 7 ⊢ (y = B → (⊥ ‘y) = (⊥ ‘B))
13	12	ineq2d 1645	. . . . . 6 ⊢ (y = B → (A ∩ (⊥ ‘y)) = (A ∩ (⊥ ‘B)))
14	11, 13	opreq12d 3014	. . . . 5 ⊢ (y = B → ((A ∩ y) ∨ℋ (A ∩ (⊥ ‘y))) = ((A ∩ B) ∨ℋ (A ∩ (⊥ ‘B))))
15	14	cleq2d 1112	. . . 4 ⊢ (y = B → (A = ((A ∩ y) ∨ℋ (A ∩ (⊥ ‘y))) ↔ A = ((A ∩ B) ∨ℋ (A ∩ (⊥ ‘B)))))
16	10, 15	anbi12d 476	. . 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 5493	. . 3 ⊢ Com = {⟨x, y⟩∣((x ∈ Cℋ ∧ y ∈ Cℋ ) ∧ x = ((x ∩ y) ∨ℋ (x ∩ (⊥ ‘y))))}
18	8, 16, 17	brabg 2116	. 2 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ) → (A Com B ↔ ((A ∈ Cℋ ∧ B ∈ Cℋ ) ∧ A = ((A ∩ B) ∨ℋ (A ∩ (⊥ ‘B))))))
19		ibar 487	. 2 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ) → (A = ((A ∩ B) ∨ℋ (A ∩ (⊥ ‘B))) ↔ ((A ∈ Cℋ ∧ B ∈ Cℋ ) ∧ A = ((A ∩ B) ∨ℋ (A ∩ (⊥ ‘B))))))
20	18, 19	bitr4d 409	1 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ) → (A Com B ↔ A = ((A ∩ B) ∨ℋ (A ∩ (⊥ ‘B)))))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ∩ cin 1486   class class class wbr 2054   ‘cfv 2422  (class class class)co 3001   C_ℋ cch 4968  ⊥cort 4969   ∨_ℋ chj 4972   Com ccm 4975

This theorem is referenced by:  cmbr 5499

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003  df-cm 5493

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