Theorem dmdsl3 29174

Description: Sublattice mapping for a dual-modular pair. Part of Theorem 1.3 of [MaedaMaeda] p. 2. (Contributed by NM, 26-Apr-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
dmdsl3	|- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( B MH* A /\ A C_ C /\ C C_ ( A vH B ) ) ) -> ( ( C i^i B ) vH A ) = C )

Proof of Theorem dmdsl3

Step	Hyp	Ref	Expression
1		dmdi 29161	. . . . . 6 |- ( ( ( B e. CH /\ A e. CH /\ C e. CH ) /\ ( B MH* A /\ A C_ C ) ) -> ( ( C i^i B ) vH A ) = ( C i^i ( B vH A ) ) )
2	1	exp32 631	. . . . 5 |- ( ( B e. CH /\ A e. CH /\ C e. CH ) -> ( B MH* A -> ( A C_ C -> ( ( C i^i B ) vH A ) = ( C i^i ( B vH A ) ) ) ) )
3	2	3com12 1269	. . . 4 |- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( B MH* A -> ( A C_ C -> ( ( C i^i B ) vH A ) = ( C i^i ( B vH A ) ) ) ) )
4	3	imp32 449	. . 3 |- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( B MH* A /\ A C_ C ) ) -> ( ( C i^i B ) vH A ) = ( C i^i ( B vH A ) ) )
5	4	3adantr3 1222	. 2 |- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( B MH* A /\ A C_ C /\ C C_ ( A vH B ) ) ) -> ( ( C i^i B ) vH A ) = ( C i^i ( B vH A ) ) )
6		chjcom 28365	. . . . . 6 |- ( ( A e. CH /\ B e. CH ) -> ( A vH B ) = ( B vH A ) )
7	6	ineq2d 3814	. . . . 5 |- ( ( A e. CH /\ B e. CH ) -> ( C i^i ( A vH B ) ) = ( C i^i ( B vH A ) ) )
8	7	3adant3 1081	. . . 4 |- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( C i^i ( A vH B ) ) = ( C i^i ( B vH A ) ) )
9		df-ss 3588	. . . . 5 |- ( C C_ ( A vH B ) <-> ( C i^i ( A vH B ) ) = C )
10	9	biimpi 206	. . . 4 |- ( C C_ ( A vH B ) -> ( C i^i ( A vH B ) ) = C )
11	8, 10	sylan9req 2677	. . 3 |- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ C C_ ( A vH B ) ) -> ( C i^i ( B vH A ) ) = C )
12	11	3ad2antr3 1228	. 2 |- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( B MH* A /\ A C_ C /\ C C_ ( A vH B ) ) ) -> ( C i^i ( B vH A ) ) = C )
13	5, 12	eqtrd 2656	1 |- ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( B MH* A /\ A C_ C /\ C C_ ( A vH B ) ) ) -> ( ( C i^i B ) vH A ) = C )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   class class class wbr 4653  (class class class)co 6650   CHcch 27786   vH chj 27790   MH* cdmd 27824

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  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-hilex 27856

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-sh 28064  df-ch 28078  df-chj 28169  df-dmd 29140

This theorem is referenced by:  mdslle1i  29176  mdslj1i  29178  mdslj2i  29179  mdslmd1lem1  29184