Theorem joinval2lem 17008

Description: Lemma for joinval2 17009 and joineu 17010. (Contributed by NM, 12-Sep-2018.) TODO: combine this through joineu into joinlem?

Hypotheses

Ref	Expression
joinval2.b	|- B = ( Base ` K )
joinval2.l	|- .<_ = ( le ` K )
joinval2.j	|- .\/ = ( join ` K )
joinval2.k	|- ( ph -> K e. V )
joinval2.x	|- ( ph -> X e. B )
joinval2.y	|- ( ph -> Y e. B )

Assertion

Ref	Expression
joinval2lem	|- ( ( X e. B /\ Y e. B ) -> ( ( A. y e. { X , Y } y .<_ x /\ A. z e. B ( A. y e. { X , Y } y .<_ z -> x .<_ z ) ) <-> ( ( X .<_ x /\ Y .<_ x ) /\ A. z e. B ( ( X .<_ z /\ Y .<_ z ) -> x .<_ z ) ) ) )

Distinct variable groups:   x, z, B   x, .\/ , z   x, y, K, z   y, .<_   x, X, y, z   x, Y, y, z

Allowed substitution hints:   ph( x, y, z)   B( y)   .\/ ( y)   .<_ ( x, z)   V( x, y, z)

Proof of Theorem joinval2lem

Step	Hyp	Ref	Expression
1		breq1 4656	. . 3 |- ( y = X -> ( y .<_ x <-> X .<_ x ) )
2		breq1 4656	. . 3 |- ( y = Y -> ( y .<_ x <-> Y .<_ x ) )
3	1, 2	ralprg 4234	. 2 |- ( ( X e. B /\ Y e. B ) -> ( A. y e. { X , Y } y .<_ x <-> ( X .<_ x /\ Y .<_ x ) ) )
4		breq1 4656	. . . . 5 |- ( y = X -> ( y .<_ z <-> X .<_ z ) )
5		breq1 4656	. . . . 5 |- ( y = Y -> ( y .<_ z <-> Y .<_ z ) )
6	4, 5	ralprg 4234	. . . 4 |- ( ( X e. B /\ Y e. B ) -> ( A. y e. { X , Y } y .<_ z <-> ( X .<_ z /\ Y .<_ z ) ) )
7	6	imbi1d 331	. . 3 |- ( ( X e. B /\ Y e. B ) -> ( ( A. y e. { X , Y } y .<_ z -> x .<_ z ) <-> ( ( X .<_ z /\ Y .<_ z ) -> x .<_ z ) ) )
8	7	ralbidv 2986	. 2 |- ( ( X e. B /\ Y e. B ) -> ( A. z e. B ( A. y e. { X , Y } y .<_ z -> x .<_ z ) <-> A. z e. B ( ( X .<_ z /\ Y .<_ z ) -> x .<_ z ) ) )
9	3, 8	anbi12d 747	1 |- ( ( X e. B /\ Y e. B ) -> ( ( A. y e. { X , Y } y .<_ x /\ A. z e. B ( A. y e. { X , Y } y .<_ z -> x .<_ z ) ) <-> ( ( X .<_ x /\ Y .<_ x ) /\ A. z e. B ( ( X .<_ z /\ Y .<_ z ) -> x .<_ z ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {cpr 4179   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   joincjn 16944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654

This theorem is referenced by:  joinval2  17009  joineu  17010