Theorem mdbr 11866

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 C_ B -> ((x vH A) i^i B) = (x vH (A i^i B)))))

Distinct variable groups:   x,A   x,B

Proof of Theorem mdbr

Step	Hyp	Ref	Expression
1		eleq1 1957	. . . . 5 |- (y = A -> (y e. CH <-> A e. CH))
2	1	anbi1d 679	. . . 4 |- (y = A -> ((y e. CH /\ z e. CH) <-> (A e. CH /\ z e. CH)))
3		opreq2 4890	. . . . . . . 8 |- (y = A -> (x vH y) = (x vH A))
4	3	ineq1d 2795	. . . . . . 7 |- (y = A -> ((x vH y) i^i z) = ((x vH A) i^i z))
5		ineq1 2789	. . . . . . . 8 |- (y = A -> (y i^i z) = (A i^i z))
6	5	opreq2d 4898	. . . . . . 7 |- (y = A -> (x vH (y i^i z)) = (x vH (A i^i z)))
7	4, 6	eqeq12d 1899	. . . . . 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 674	. . . . 5 |- (y = A -> ((x C_ z -> ((x vH y) i^i z) = (x vH (y i^i z))) <-> (x C_ z -> ((x vH A) i^i z) = (x vH (A i^i z)))))
9	8	ralbidv 2123	. . . 4 |- (y = A -> (A.x e. CH (x C_ z -> ((x vH y) i^i z) = (x vH (y i^i z))) <-> A.x e. CH (x C_ z -> ((x vH A) i^i z) = (x vH (A i^i z)))))
10	2, 9	anbi12d 690	. . 3 |- (y = A -> (((y e. CH /\ z e. CH) /\ A.x e. CH (x C_ z -> ((x vH y) i^i z) = (x vH (y i^i z)))) <-> ((A e. CH /\ z e. CH) /\ A.x e. CH (x C_ z -> ((x vH A) i^i z) = (x vH (A i^i z))))))
11		eleq1 1957	. . . . 5 |- (z = B -> (z e. CH <-> B e. CH))
12	11	anbi2d 678	. . . 4 |- (z = B -> ((A e. CH /\ z e. CH) <-> (A e. CH /\ B e. CH)))
13		sseq2 2639	. . . . . 6 |- (z = B -> (x C_ z <-> x C_ B))
14		ineq2 2790	. . . . . . 7 |- (z = B -> ((x vH A) i^i z) = ((x vH A) i^i B))
15		ineq2 2790	. . . . . . . 8 |- (z = B -> (A i^i z) = (A i^i B))
16	15	opreq2d 4898	. . . . . . 7 |- (z = B -> (x vH (A i^i z)) = (x vH (A i^i B)))
17	14, 16	eqeq12d 1899	. . . . . 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 688	. . . . 5 |- (z = B -> ((x C_ z -> ((x vH A) i^i z) = (x vH (A i^i z))) <-> (x C_ B -> ((x vH A) i^i B) = (x vH (A i^i B)))))
19	18	ralbidv 2123	. . . 4 |- (z = B -> (A.x e. CH (x C_ z -> ((x vH A) i^i z) = (x vH (A i^i z))) <-> A.x e. CH (x C_ B -> ((x vH A) i^i B) = (x vH (A i^i B)))))
20	12, 19	anbi12d 690	. . 3 |- (z = B -> (((A e. CH /\ z e. CH) /\ A.x e. CH (x C_ z -> ((x vH A) i^i z) = (x vH (A i^i z)))) <-> ((A e. CH /\ B e. CH) /\ A.x e. CH (x C_ B -> ((x vH A) i^i B) = (x vH (A i^i B))))))
21		df-md 11852	. . 3 |- MH = {<.y, z>. | ((y e. CH /\ z e. CH) /\ A.x e. CH (x C_ z -> ((x vH y) i^i z) = (x vH (y i^i z))))}
22	10, 20, 21	brabg 3568	. 2 |- ((A e. CH /\ B e. CH) -> (A MH B <-> ((A e. CH /\ B e. CH) /\ A.x e. CH (x C_ B -> ((x vH A) i^i B) = (x vH (A i^i B))))))
23	22	bianabs 715	1 |- ((A e. CH /\ B e. CH) -> (A MH B <-> A.x e. CH (x C_ B -> ((x vH A) i^i B) = (x vH (A i^i B)))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105   i^i cin 2592   C_ wss 2593   class class class wbr 3338  (class class class)co 4884  CHcch 10430   vH chj 10434   MH cmd 10467

This theorem is referenced by:  mdi 11867  mdbr2 11868  mdbr3 11869  dmdmd 11872  mddmd2 11881  mdsl1i 11893

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886  df-md 11852

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