Theorem cmbrt 9444

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 1526	. . . . 5 |- (x = A -> (x e. CH <-> A e. CH))
2	1	anbi1d 615	. . . 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 2200	. . . . . 6 |- (x = A -> (x i^i y) = (A i^i y))
5		ineq1 2200	. . . . . 6 |- (x = A -> (x i^i (_|_` y)) = (A i^i (_|_` y)))
6	4, 5	opreq12d 3963	. . . . 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 1481	. . . 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 626	. . 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 1526	. . . . 5 |- (y = B -> (y e. CH <-> B e. CH))
10	9	anbi2d 614	. . . 4 |- (y = B -> ((A e. CH /\ y e. CH) <-> (A e. CH /\ B e. CH)))
11		ineq2 2201	. . . . . 6 |- (y = B -> (A i^i y) = (A i^i B))
12		fveq2 3709	. . . . . . 7 |- (y = B -> (_|_` y) = (_|_` B))
13	12	ineq2d 2207	. . . . . 6 |- (y = B -> (A i^i (_|_` y)) = (A i^i (_|_` B)))
14	11, 13	opreq12d 3963	. . . . 5 |- (y = B -> ((A i^i y) vH (A i^i (_|_` y))) = ((A i^i B) vH (A i^i (_|_` B))))
15	14	eqeq2d 1478	. . . 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 626	. . 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 9443	. . 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 2807	. 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 651	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 953   e. wcel 955   i^i cin 2036   class class class wbr 2609  ` cfv 3172  (class class class)co 3948  CHcch 8737  _|_cort 8738   vH chj 8741   C_H ccm 8744

This theorem is referenced by:  cmbr 9450  cm2jt 9480

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-xp 3174  df-cnv 3176  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fv 3188  df-opr 3950  df-cm 9443

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