Theorem mdbr 5726

Description: Binary relation expressing <.A, B>. is a modular pair. Definition 1.1 of [MaedaMaeda] p. 1.

Assertion

Ref	Expression
mdbr	|- ((A e. CH /\ B e. CH) -> (A MH B <-> A.x e. CH (x (_ B -> ((x vH A) i^i B) = (x vH (A i^i B)))))

Distinct variable group(s):   x,A   x,B

Proof of Theorem mdbr

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . . 5 |- (y = A -> (y e. CH <-> A e. CH))
2	1	anbi1d 469	. . . 4 |- (y = A -> ((y e. CH /\ z e. CH) <-> (A e. CH /\ z e. CH)))
3		opreq2 3007	. . . . . . . 8 |- (y = A -> (x vH y) = (x vH A))
4	3	ineq1d 1644	. . . . . . 7 |- (y = A -> ((x vH y) i^i z) = ((x vH A) i^i z))
5		ineq1 1638	. . . . . . . 8 |- (y = A -> (y i^i z) = (A i^i z))
6	5	opreq2d 3013	. . . . . . 7 |- (y = A -> (x vH (y i^i z)) = (x vH (A i^i z)))
7	4, 6	cleq12d 1115	. . . . . 6 |- (y = A -> (((x vH y) i^i z) = (x vH (y i^i z)) <-> ((x vH A) i^i z) = (x vH (A i^i z))))
8	7	imbi2d 464	. . . . 5 |- (y = A -> ((x (_ z -> ((x vH y) i^i z) = (x vH (y i^i z))) <-> (x (_ z -> ((x vH A) i^i z) = (x vH (A i^i z)))))
9	8	biraldv 1219	. . . 4 |- (y = A -> (A.x e. CH (x (_ z -> ((x vH y) i^i z) = (x vH (y i^i z))) <-> A.x e. CH (x (_ z -> ((x vH A) i^i z) = (x vH (A i^i z)))))
10	2, 9	anbi12d 476	. . 3 |- (y = A -> (((y e. CH /\ z e. CH) /\ A.x e. CH (x (_ z -> ((x vH y) i^i z) = (x vH (y i^i z)))) <-> ((A e. CH /\ z e. CH) /\ A.x e. CH (x (_ z -> ((x vH A) i^i z) = (x vH (A i^i z))))))
11		eleq1 1149	. . . . 5 |- (z = B -> (z e. CH <-> B e. CH))
12	11	anbi2d 468	. . . 4 |- (z = B -> ((A e. CH /\ z e. CH) <-> (A e. CH /\ B e. CH)))
13		sseq2 1522	. . . . . 6 |- (z = B -> (x (_ z <-> x (_ B))
14		ineq2 1639	. . . . . . 7 |- (z = B -> ((x vH A) i^i z) = ((x vH A) i^i B))
15		ineq2 1639	. . . . . . . 8 |- (z = B -> (A i^i z) = (A i^i B))
16	15	opreq2d 3013	. . . . . . 7 |- (z = B -> (x vH (A i^i z)) = (x vH (A i^i B)))
17	14, 16	cleq12d 1115	. . . . . 6 |- (z = B -> (((x vH A) i^i z) = (x vH (A i^i z)) <-> ((x vH A) i^i B) = (x vH (A i^i B))))
18	13, 17	imbi12d 474	. . . . 5 |- (z = B -> ((x (_ z -> ((x vH A) i^i z) = (x vH (A i^i z))) <-> (x (_ B -> ((x vH A) i^i B) = (x vH (A i^i B)))))
19	18	biraldv 1219	. . . 4 |- (z = B -> (A.x e. CH (x (_ z -> ((x vH A) i^i z) = (x vH (A i^i z))) <-> A.x e. CH (x (_ B -> ((x vH A) i^i B) = (x vH (A i^i B)))))
20	12, 19	anbi12d 476	. . 3 |- (z = B -> (((A e. CH /\ z e. CH) /\ A.x e. CH (x (_ z -> ((x vH A) i^i z) = (x vH (A i^i z)))) <-> ((A e. CH /\ B e. CH) /\ A.x e. CH (x (_ B -> ((x vH A) i^i B) = (x vH (A i^i B))))))
21		df-md 5713	. . 3 |- MH = {<.y, z>. | ((y e. CH /\ z e. CH) /\ A.x e. CH (x (_ z -> ((x vH y) i^i z) = (x vH (y i^i z))))}
22	10, 20, 21	brabg 2116	. 2 |- ((A e. CH /\ B e. CH) -> (A MH B <-> ((A e. CH /\ B e. CH) /\ A.x e. CH (x (_ B -> ((x vH A) i^i B) = (x vH (A i^i B))))))
23		ibar 487	. 2 |- ((A e. CH /\ B e. CH) -> (A.x e. CH (x (_ B -> ((x vH A) i^i B) = (x vH (A i^i B))) <-> ((A e. CH /\ B e. CH) /\ A.x e. CH (x (_ B -> ((x vH A) i^i B) = (x vH (A i^i B))))))
24	22, 23	bitr4d 409	1 |- ((A e. CH /\ B e. CH) -> (A MH B <-> A.x e. CH (x (_ B -> ((x vH A) i^i B) = (x vH (A i^i B)))))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  A.wral 1201   i^i cin 1486   (_ wss 1487   class class class wbr 2054  (class class class)co 3001  CHcch 4968   vH chj 4972   MH cmd 4982

This theorem is referenced by:  mdi 5727  mdbr2 5728  mdbr3 5729  dmdbr 5731

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-ral 1205  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-md 5713

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