Theorem dmdbrt 10182

Description: Binary relation expressing the dual modular pair property.

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem dmdbrt

Step	Hyp	Ref	Expression
1		eleq1 1532	. . . . 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		ineq2 2208	. . . . . . . 8 |- (y = A -> (x i^i y) = (x i^i A))
4	3	opreq1d 3970	. . . . . . 7 |- (y = A -> ((x i^i y) vH z) = ((x i^i A) vH z))
5		opreq1 3963	. . . . . . . 8 |- (y = A -> (y vH z) = (A vH z))
6	5	ineq2d 2214	. . . . . . 7 |- (y = A -> (x i^i (y vH z)) = (x i^i (A vH z)))
7	4, 6	eqeq12d 1487	. . . . . 6 |- (y = A -> (((x i^i y) vH z) = (x i^i (y vH z)) <-> ((x i^i A) vH z) = (x i^i (A vH z))))
8	7	imbi2d 611	. . . . 5 |- (y = A -> ((z (_ x -> ((x i^i y) vH z) = (x i^i (y vH z))) <-> (z (_ x -> ((x i^i A) vH z) = (x i^i (A vH z)))))
9	8	ralbidv 1661	. . . 4 |- (y = A -> (A.x e. CH (z (_ x -> ((x i^i y) vH z) = (x i^i (y vH z))) <-> A.x e. CH (z (_ x -> ((x i^i A) vH z) = (x i^i (A vH z)))))
10	2, 9	anbi12d 627	. . 3 |- (y = A -> (((y e. CH /\ z e. CH) /\ A.x e. CH (z (_ x -> ((x i^i y) vH z) = (x i^i (y vH z)))) <-> ((A e. CH /\ z e. CH) /\ A.x e. CH (z (_ x -> ((x i^i A) vH z) = (x i^i (A vH z))))))
11		eleq1 1532	. . . . 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		sseq1 2079	. . . . . 6 |- (z = B -> (z (_ x <-> B (_ x))
14		opreq2 3964	. . . . . . 7 |- (z = B -> ((x i^i A) vH z) = ((x i^i A) vH B))
15		opreq2 3964	. . . . . . . 8 |- (z = B -> (A vH z) = (A vH B))
16	15	ineq2d 2214	. . . . . . 7 |- (z = B -> (x i^i (A vH z)) = (x i^i (A vH B)))
17	14, 16	eqeq12d 1487	. . . . . 6 |- (z = B -> (((x i^i A) vH z) = (x i^i (A vH z)) <-> ((x i^i A) vH B) = (x i^i (A vH B))))
18	13, 17	imbi12d 625	. . . . 5 |- (z = B -> ((z (_ x -> ((x i^i A) vH z) = (x i^i (A vH z))) <-> (B (_ x -> ((x i^i A) vH B) = (x i^i (A vH B)))))
19	18	ralbidv 1661	. . . 4 |- (z = B -> (A.x e. CH (z (_ x -> ((x i^i A) vH z) = (x i^i (A vH z))) <-> A.x e. CH (B (_ x -> ((x i^i A) vH B) = (x i^i (A vH B)))))
20	12, 19	anbi12d 627	. . 3 |- (z = B -> (((A e. CH /\ z e. CH) /\ A.x e. CH (z (_ x -> ((x i^i A) vH z) = (x i^i (A vH z)))) <-> ((A e. CH /\ B e. CH) /\ A.x e. CH (B (_ x -> ((x i^i A) vH B) = (x i^i (A vH B))))))
21		df-dmd 10164	. . 3 |- MH* = {<.y, z>. | ((y e. CH /\ z e. CH) /\ A.x e. CH (z (_ x -> ((x i^i y) vH z) = (x i^i (y vH z))))}
22	10, 20, 21	brabg 2814	. 2 |- ((A e. CH /\ B e. CH) -> (A MH* B <-> ((A e. CH /\ B e. CH) /\ A.x e. CH (B (_ x -> ((x i^i A) vH B) = (x i^i (A vH B))))))
23	22	bianabs 652	1 |- ((A e. CH /\ B e. CH) -> (A MH* B <-> A.x e. CH (B (_ x -> ((x i^i A) vH B) = (x i^i (A vH B)))))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957  A.wral 1643   i^i cin 2043   (_ wss 2044   class class class wbr 2615  (class class class)co 3958  CHcch 8753   vH chj 8757   MH* cdmd 8791

This theorem is referenced by:  dmdmdt 10183  dmdit 10185  dmdbr2 10186  dmdbr3 10188  mddmd 10192

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-ral 1647  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-dmd 10164

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