Theorem cmbrt 9484

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

Assertion

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

Proof of Theorem cmbrt

Step	Hyp	Ref	Expression
1		eleq1 1532	. . . . 5 |- (x = A -> (x e. CH <-> A e. CH))
2	1	anbi1d 616	. . . 4 |- (x = A -> ((x e. CH /\ y e. CH) <-> (A e. CH /\ y e. CH)))
3		id 59	. . . . 5 |- (x = A -> x = A)
4		ineq1 2207	. . . . . 6 |- (x = A -> (x i^i y) = (A i^i y))
5		ineq1 2207	. . . . . 6 |- (x = A -> (x i^i (_|_` y)) = (A i^i (_|_` y)))
6	4, 5	opreq12d 3973	. . . . 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 1487	. . . 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 627	. . 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 1532	. . . . 5 |- (y = B -> (y e. CH <-> B e. CH))
10	9	anbi2d 615	. . . 4 |- (y = B -> ((A e. CH /\ y e. CH) <-> (A e. CH /\ B e. CH)))
11		ineq2 2208	. . . . . 6 |- (y = B -> (A i^i y) = (A i^i B))
12		fveq2 3719	. . . . . . 7 |- (y = B -> (_|_` y) = (_|_` B))
13	12	ineq2d 2214	. . . . . 6 |- (y = B -> (A i^i (_|_` y)) = (A i^i (_|_` B)))
14	11, 13	opreq12d 3973	. . . . 5 |- (y = B -> ((A i^i y) vH (A i^i (_|_` y))) = ((A i^i B) vH (A i^i (_|_` B))))
15	14	eqeq2d 1484	. . . 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 627	. . 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 9483	. . 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 2814	. 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 652	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 146   /\ wa 223   = wceq 955   e. wcel 957   i^i cin 2043   class class class wbr 2615  ` cfv 3178  (class class class)co 3958  CHcch 8753  _|_cort 8754   vH chj 8757   C_H ccm 8760

This theorem is referenced by:  cmbr 9490  cm2jt 9520

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-xp 3180  df-cnv 3182  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fv 3194  df-opr 3960  df-cm 9483

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