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Metamath Proof Explorer

Theorem mdi

Description: Consequence of the modular pair property. (Contributed by NM, 22-Jun-2004) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 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 ) ) ) ) )
2 1 biimpd 
 |-  ( ( 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 ) ) ) ) )
3 sseq1 
 |-  ( x = C -> ( x C_ B <-> C C_ B ) )
4 oveq1 
 |-  ( x = C -> ( x vH A ) = ( C vH A ) )
5 4 ineq1d 
 |-  ( x = C -> ( ( x vH A ) i^i B ) = ( ( C vH A ) i^i B ) )
6 oveq1 
 |-  ( x = C -> ( x vH ( A i^i B ) ) = ( C vH ( A i^i B ) ) )
7 5 6 eqeq12d 
 |-  ( x = C -> ( ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) <-> ( ( C vH A ) i^i B ) = ( C vH ( A i^i B ) ) ) )
8 3 7 imbi12d 
 |-  ( x = C -> ( ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) <-> ( C C_ B -> ( ( C vH A ) i^i B ) = ( C vH ( A i^i B ) ) ) ) )
9 8 rspcv 
 |-  ( C e. CH -> ( A. x e. CH ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) -> ( C C_ B -> ( ( C vH A ) i^i B ) = ( C vH ( A i^i B ) ) ) ) )
10 2 9 sylan9 
 |-  ( ( ( A e. CH /\ B e. CH ) /\ C e. CH ) -> ( A MH B -> ( C C_ B -> ( ( C vH A ) i^i B ) = ( C vH ( A i^i B ) ) ) ) )
11 10 3impa 
 |-  ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( A MH B -> ( C C_ B -> ( ( C vH A ) i^i B ) = ( C vH ( A i^i B ) ) ) ) )
12 11 imp32 
 |-  ( ( ( A e. CH /\ B e. CH /\ C e. CH ) /\ ( A MH B /\ C C_ B ) ) -> ( ( C vH A ) i^i B ) = ( C vH ( A i^i B ) ) )