Theorem mdbrt 10221

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 1534	. . . . 5 |- (y = A -> (y e. CH <-> A e. CH))
2	1	anbi1d 617	. . . 4 |- (y = A -> ((y e. CH /\ z e. CH) <-> (A e. CH /\ z e. CH)))
3		opreq2 3969	. . . . . . . 8 |- (y = A -> (x vH y) = (x vH A))
4	3	ineq1d 2216	. . . . . . 7 |- (y = A -> ((x vH y) i^i z) = ((x vH A) i^i z))
5		ineq1 2210	. . . . . . . 8 |- (y = A -> (y i^i z) = (A i^i z))
6	5	opreq2d 3976	. . . . . . 7 |- (y = A -> (x vH (y i^i z)) = (x vH (A i^i z)))
7	4, 6	eqeq12d 1489	. . . . . 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 612	. . . . 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 1663	. . . 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 628	. . 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 1534	. . . . 5 |- (z = B -> (z e. CH <-> B e. CH))
12	11	anbi2d 616	. . . 4 |- (z = B -> ((A e. CH /\ z e. CH) <-> (A e. CH /\ B e. CH)))
13		sseq2 2083	. . . . . 6 |- (z = B -> (x (_ z <-> x (_ B))
14		ineq2 2211	. . . . . . 7 |- (z = B -> ((x vH A) i^i z) = ((x vH A) i^i B))
15		ineq2 2211	. . . . . . . 8 |- (z = B -> (A i^i z) = (A i^i B))
16	15	opreq2d 3976	. . . . . . 7 |- (z = B -> (x vH (A i^i z)) = (x vH (A i^i B)))
17	14, 16	eqeq12d 1489	. . . . . 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 626	. . . . 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 1663	. . . 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 628	. . 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 10207	. . 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 2818	. 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 653	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 956   e. wcel 958  A.wral 1645   i^i cin 2046   (_ wss 2047   class class class wbr 2619  (class class class)co 3963  CHcch 8798   vH chj 8802   MH cmd 8835

This theorem is referenced by:  mdit 10222  mdbr2 10223  mdbr3 10224  dmdmdt 10227  mddmd 10236  mdsl1 10248

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3965  df-md 10207

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