Theorem mdbrt 10159

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

Assertion

Ref	Expression
mdbrt	|- ((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 groups:   x,A   x,B

Proof of Theorem mdbrt

Step	Hyp	Ref	Expression
1		eleq1 1531	. . . . 5 |- (y = A -> (y e. CH <-> A e. CH))
2	1	anbi1d 616	. . . 4 |- (y = A -> ((y e. CH /\ z e. CH) <-> (A e. CH /\ z e. CH)))
3		opreq2 3960	. . . . . . . 8 |- (y = A -> (x vH y) = (x vH A))
4	3	ineq1d 2212	. . . . . . 7 |- (y = A -> ((x vH y) i^i z) = ((x vH A) i^i z))
5		ineq1 2206	. . . . . . . 8 |- (y = A -> (y i^i z) = (A i^i z))
6	5	opreq2d 3967	. . . . . . 7 |- (y = A -> (x vH (y i^i z)) = (x vH (A i^i z)))
7	4, 6	eqeq12d 1486	. . . . . 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 611	. . . . 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	ralbidv 1660	. . . 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 627	. . 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 1531	. . . . 5 |- (z = B -> (z e. CH <-> B e. CH))
12	11	anbi2d 615	. . . 4 |- (z = B -> ((A e. CH /\ z e. CH) <-> (A e. CH /\ B e. CH)))
13		sseq2 2079	. . . . . 6 |- (z = B -> (x (_ z <-> x (_ B))
14		ineq2 2207	. . . . . . 7 |- (z = B -> ((x vH A) i^i z) = ((x vH A) i^i B))
15		ineq2 2207	. . . . . . . 8 |- (z = B -> (A i^i z) = (A i^i B))
16	15	opreq2d 3967	. . . . . . 7 |- (z = B -> (x vH (A i^i z)) = (x vH (A i^i B)))
17	14, 16	eqeq12d 1486	. . . . . 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 625	. . . . 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	ralbidv 1660	. . . 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 627	. . 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 10145	. . 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 2813	. 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	22	bianabs 652	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 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  A.wral 1642   i^i cin 2042   (_ wss 2043   class class class wbr 2614  (class class class)co 3954  CHcch 8737   vH chj 8741   MH cmd 8774

This theorem is referenced by:  mdit 10160  mdbr2 10161  mdbr3 10162  dmdmdt 10165  mddmd 10173  mdsl1 10185

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-xp 3179  df-cnv 3181  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fv 3193  df-opr 3956  df-md 10145

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