Theorem cmbr 11160

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

Assertion

Ref	Expression
cmbr	|- ((A e. CH /\ B e. CH) -> (A C_H B <-> A = ((A i^i B) vH (A i^i (_|_` B)))))

Proof of Theorem cmbr

Step	Hyp	Ref	Expression
1		eleq1 1957	. . . . 5 |- (x = A -> (x e. CH <-> A e. CH))
2	1	anbi1d 679	. . . 4 |- (x = A -> ((x e. CH /\ y e. CH) <-> (A e. CH /\ y e. CH)))
3		id 73	. . . . 5 |- (x = A -> x = A)
4		ineq1 2789	. . . . . 6 |- (x = A -> (x i^i y) = (A i^i y))
5		ineq1 2789	. . . . . 6 |- (x = A -> (x i^i (_|_` y)) = (A i^i (_|_` y)))
6	4, 5	opreq12d 4900	. . . . 5 |- (x = A -> ((x i^i y) vH (x i^i (_|_` y))) = ((A i^i y) vH (A i^i (_|_` y))))
7	3, 6	eqeq12d 1899	. . . 4 |- (x = A -> (x = ((x i^i y) vH (x i^i (_|_` y))) <-> A = ((A i^i y) vH (A i^i (_|_` y)))))
8	2, 7	anbi12d 690	. . 3 |- (x = A -> (((x e. CH /\ y e. CH) /\ x = ((x i^i y) vH (x i^i (_|_` y)))) <-> ((A e. CH /\ y e. CH) /\ A = ((A i^i y) vH (A i^i (_|_` y))))))
9		eleq1 1957	. . . . 5 |- (y = B -> (y e. CH <-> B e. CH))
10	9	anbi2d 678	. . . 4 |- (y = B -> ((A e. CH /\ y e. CH) <-> (A e. CH /\ B e. CH)))
11		ineq2 2790	. . . . . 6 |- (y = B -> (A i^i y) = (A i^i B))
12		fveq2 4681	. . . . . . 7 |- (y = B -> (_|_` y) = (_|_` B))
13	12	ineq2d 2796	. . . . . 6 |- (y = B -> (A i^i (_|_` y)) = (A i^i (_|_` B)))
14	11, 13	opreq12d 4900	. . . . 5 |- (y = B -> ((A i^i y) vH (A i^i (_|_` y))) = ((A i^i B) vH (A i^i (_|_` B))))
15	14	eqeq2d 1895	. . . 4 |- (y = B -> (A = ((A i^i y) vH (A i^i (_|_` y))) <-> A = ((A i^i B) vH (A i^i (_|_` B)))))
16	10, 15	anbi12d 690	. . 3 |- (y = B -> (((A e. CH /\ y e. CH) /\ A = ((A i^i y) vH (A i^i (_|_` y)))) <-> ((A e. CH /\ B e. CH) /\ A = ((A i^i B) vH (A i^i (_|_` B))))))
17		df-cm 11159	. . 3 |- C_H = {<.x, y>. | ((x e. CH /\ y e. CH) /\ x = ((x i^i y) vH (x i^i (_|_` y))))}
18	8, 16, 17	brabg 3568	. 2 |- ((A e. CH /\ B e. CH) -> (A C_H B <-> ((A e. CH /\ B e. CH) /\ A = ((A i^i B) vH (A i^i (_|_` B))))))
19	18	bianabs 715	1 |- ((A e. CH /\ B e. CH) -> (A C_H B <-> A = ((A i^i B) vH (A i^i (_|_` B)))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   i^i cin 2592   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CHcch 10430  _|_cort 10431   vH chj 10434   C_H ccm 10437

This theorem is referenced by:  cmbri 11166  cm2j 11196

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886  df-cm 11159

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