Metamath Proof Explorer


Theorem addsdilem3

Description: Lemma for addsdi . Show one of the equalities involved in the final expression. (Contributed by Scott Fenton, 9-Mar-2025)

Ref Expression
Hypotheses addsdilem3.1 φ A No
addsdilem3.2 φ B No
addsdilem3.3 φ C No
addsdilem3.4 No typesetting found for |- ( ph -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) ( xO x.s ( B +s C ) ) = ( ( xO x.s B ) +s ( xO x.s C ) ) ) with typecode |-
addsdilem3.5 No typesetting found for |- ( ph -> A. yO e. ( ( _Left ` B ) u. ( _Right ` B ) ) ( A x.s ( yO +s C ) ) = ( ( A x.s yO ) +s ( A x.s C ) ) ) with typecode |-
addsdilem3.6 No typesetting found for |- ( ph -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) A. yO e. ( ( _Left ` B ) u. ( _Right ` B ) ) ( xO x.s ( yO +s C ) ) = ( ( xO x.s yO ) +s ( xO x.s C ) ) ) with typecode |-
addsdilem3.7 ψ X L A R A
addsdilem3.8 ψ Y L B R B
Assertion addsdilem3 φ ψ X s B + s C + s A s Y + s C - s X s Y + s C = X s B + s A s Y - s X s Y + s A s C

Proof

Step Hyp Ref Expression
1 addsdilem3.1 φ A No
2 addsdilem3.2 φ B No
3 addsdilem3.3 φ C No
4 addsdilem3.4 Could not format ( ph -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) ( xO x.s ( B +s C ) ) = ( ( xO x.s B ) +s ( xO x.s C ) ) ) : No typesetting found for |- ( ph -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) ( xO x.s ( B +s C ) ) = ( ( xO x.s B ) +s ( xO x.s C ) ) ) with typecode |-
5 addsdilem3.5 Could not format ( ph -> A. yO e. ( ( _Left ` B ) u. ( _Right ` B ) ) ( A x.s ( yO +s C ) ) = ( ( A x.s yO ) +s ( A x.s C ) ) ) : No typesetting found for |- ( ph -> A. yO e. ( ( _Left ` B ) u. ( _Right ` B ) ) ( A x.s ( yO +s C ) ) = ( ( A x.s yO ) +s ( A x.s C ) ) ) with typecode |-
6 addsdilem3.6 Could not format ( ph -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) A. yO e. ( ( _Left ` B ) u. ( _Right ` B ) ) ( xO x.s ( yO +s C ) ) = ( ( xO x.s yO ) +s ( xO x.s C ) ) ) : No typesetting found for |- ( ph -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) A. yO e. ( ( _Left ` B ) u. ( _Right ` B ) ) ( xO x.s ( yO +s C ) ) = ( ( xO x.s yO ) +s ( xO x.s C ) ) ) with typecode |-
7 addsdilem3.7 ψ X L A R A
8 addsdilem3.8 ψ Y L B R B
9 oveq1 Could not format ( xO = X -> ( xO x.s ( B +s C ) ) = ( X x.s ( B +s C ) ) ) : No typesetting found for |- ( xO = X -> ( xO x.s ( B +s C ) ) = ( X x.s ( B +s C ) ) ) with typecode |-
10 oveq1 Could not format ( xO = X -> ( xO x.s B ) = ( X x.s B ) ) : No typesetting found for |- ( xO = X -> ( xO x.s B ) = ( X x.s B ) ) with typecode |-
11 oveq1 Could not format ( xO = X -> ( xO x.s C ) = ( X x.s C ) ) : No typesetting found for |- ( xO = X -> ( xO x.s C ) = ( X x.s C ) ) with typecode |-
12 10 11 oveq12d Could not format ( xO = X -> ( ( xO x.s B ) +s ( xO x.s C ) ) = ( ( X x.s B ) +s ( X x.s C ) ) ) : No typesetting found for |- ( xO = X -> ( ( xO x.s B ) +s ( xO x.s C ) ) = ( ( X x.s B ) +s ( X x.s C ) ) ) with typecode |-
13 9 12 eqeq12d Could not format ( xO = X -> ( ( xO x.s ( B +s C ) ) = ( ( xO x.s B ) +s ( xO x.s C ) ) <-> ( X x.s ( B +s C ) ) = ( ( X x.s B ) +s ( X x.s C ) ) ) ) : No typesetting found for |- ( xO = X -> ( ( xO x.s ( B +s C ) ) = ( ( xO x.s B ) +s ( xO x.s C ) ) <-> ( X x.s ( B +s C ) ) = ( ( X x.s B ) +s ( X x.s C ) ) ) ) with typecode |-
14 4 adantr Could not format ( ( ph /\ ps ) -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) ( xO x.s ( B +s C ) ) = ( ( xO x.s B ) +s ( xO x.s C ) ) ) : No typesetting found for |- ( ( ph /\ ps ) -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) ( xO x.s ( B +s C ) ) = ( ( xO x.s B ) +s ( xO x.s C ) ) ) with typecode |-
15 7 adantl φ ψ X L A R A
16 13 14 15 rspcdva φ ψ X s B + s C = X s B + s X s C
17 oveq1 Could not format ( yO = Y -> ( yO +s C ) = ( Y +s C ) ) : No typesetting found for |- ( yO = Y -> ( yO +s C ) = ( Y +s C ) ) with typecode |-
18 17 oveq2d Could not format ( yO = Y -> ( A x.s ( yO +s C ) ) = ( A x.s ( Y +s C ) ) ) : No typesetting found for |- ( yO = Y -> ( A x.s ( yO +s C ) ) = ( A x.s ( Y +s C ) ) ) with typecode |-
19 oveq2 Could not format ( yO = Y -> ( A x.s yO ) = ( A x.s Y ) ) : No typesetting found for |- ( yO = Y -> ( A x.s yO ) = ( A x.s Y ) ) with typecode |-
20 19 oveq1d Could not format ( yO = Y -> ( ( A x.s yO ) +s ( A x.s C ) ) = ( ( A x.s Y ) +s ( A x.s C ) ) ) : No typesetting found for |- ( yO = Y -> ( ( A x.s yO ) +s ( A x.s C ) ) = ( ( A x.s Y ) +s ( A x.s C ) ) ) with typecode |-
21 18 20 eqeq12d Could not format ( yO = Y -> ( ( A x.s ( yO +s C ) ) = ( ( A x.s yO ) +s ( A x.s C ) ) <-> ( A x.s ( Y +s C ) ) = ( ( A x.s Y ) +s ( A x.s C ) ) ) ) : No typesetting found for |- ( yO = Y -> ( ( A x.s ( yO +s C ) ) = ( ( A x.s yO ) +s ( A x.s C ) ) <-> ( A x.s ( Y +s C ) ) = ( ( A x.s Y ) +s ( A x.s C ) ) ) ) with typecode |-
22 5 adantr Could not format ( ( ph /\ ps ) -> A. yO e. ( ( _Left ` B ) u. ( _Right ` B ) ) ( A x.s ( yO +s C ) ) = ( ( A x.s yO ) +s ( A x.s C ) ) ) : No typesetting found for |- ( ( ph /\ ps ) -> A. yO e. ( ( _Left ` B ) u. ( _Right ` B ) ) ( A x.s ( yO +s C ) ) = ( ( A x.s yO ) +s ( A x.s C ) ) ) with typecode |-
23 8 adantl φ ψ Y L B R B
24 21 22 23 rspcdva φ ψ A s Y + s C = A s Y + s A s C
25 16 24 oveq12d φ ψ X s B + s C + s A s Y + s C = X s B + s X s C + s A s Y + s A s C
26 oveq1 Could not format ( xO = X -> ( xO x.s ( yO +s C ) ) = ( X x.s ( yO +s C ) ) ) : No typesetting found for |- ( xO = X -> ( xO x.s ( yO +s C ) ) = ( X x.s ( yO +s C ) ) ) with typecode |-
27 oveq1 Could not format ( xO = X -> ( xO x.s yO ) = ( X x.s yO ) ) : No typesetting found for |- ( xO = X -> ( xO x.s yO ) = ( X x.s yO ) ) with typecode |-
28 27 11 oveq12d Could not format ( xO = X -> ( ( xO x.s yO ) +s ( xO x.s C ) ) = ( ( X x.s yO ) +s ( X x.s C ) ) ) : No typesetting found for |- ( xO = X -> ( ( xO x.s yO ) +s ( xO x.s C ) ) = ( ( X x.s yO ) +s ( X x.s C ) ) ) with typecode |-
29 26 28 eqeq12d Could not format ( xO = X -> ( ( xO x.s ( yO +s C ) ) = ( ( xO x.s yO ) +s ( xO x.s C ) ) <-> ( X x.s ( yO +s C ) ) = ( ( X x.s yO ) +s ( X x.s C ) ) ) ) : No typesetting found for |- ( xO = X -> ( ( xO x.s ( yO +s C ) ) = ( ( xO x.s yO ) +s ( xO x.s C ) ) <-> ( X x.s ( yO +s C ) ) = ( ( X x.s yO ) +s ( X x.s C ) ) ) ) with typecode |-
30 17 oveq2d Could not format ( yO = Y -> ( X x.s ( yO +s C ) ) = ( X x.s ( Y +s C ) ) ) : No typesetting found for |- ( yO = Y -> ( X x.s ( yO +s C ) ) = ( X x.s ( Y +s C ) ) ) with typecode |-
31 oveq2 Could not format ( yO = Y -> ( X x.s yO ) = ( X x.s Y ) ) : No typesetting found for |- ( yO = Y -> ( X x.s yO ) = ( X x.s Y ) ) with typecode |-
32 31 oveq1d Could not format ( yO = Y -> ( ( X x.s yO ) +s ( X x.s C ) ) = ( ( X x.s Y ) +s ( X x.s C ) ) ) : No typesetting found for |- ( yO = Y -> ( ( X x.s yO ) +s ( X x.s C ) ) = ( ( X x.s Y ) +s ( X x.s C ) ) ) with typecode |-
33 30 32 eqeq12d Could not format ( yO = Y -> ( ( X x.s ( yO +s C ) ) = ( ( X x.s yO ) +s ( X x.s C ) ) <-> ( X x.s ( Y +s C ) ) = ( ( X x.s Y ) +s ( X x.s C ) ) ) ) : No typesetting found for |- ( yO = Y -> ( ( X x.s ( yO +s C ) ) = ( ( X x.s yO ) +s ( X x.s C ) ) <-> ( X x.s ( Y +s C ) ) = ( ( X x.s Y ) +s ( X x.s C ) ) ) ) with typecode |-
34 6 adantr Could not format ( ( ph /\ ps ) -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) A. yO e. ( ( _Left ` B ) u. ( _Right ` B ) ) ( xO x.s ( yO +s C ) ) = ( ( xO x.s yO ) +s ( xO x.s C ) ) ) : No typesetting found for |- ( ( ph /\ ps ) -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) A. yO e. ( ( _Left ` B ) u. ( _Right ` B ) ) ( xO x.s ( yO +s C ) ) = ( ( xO x.s yO ) +s ( xO x.s C ) ) ) with typecode |-
35 29 33 34 15 23 rspc2dv φ ψ X s Y + s C = X s Y + s X s C
36 25 35 oveq12d φ ψ X s B + s C + s A s Y + s C - s X s Y + s C = X s B + s X s C + s A s Y + s A s C - s X s Y + s X s C
37 leftssno L A No
38 rightssno R A No
39 37 38 unssi L A R A No
40 39 7 sselid ψ X No
41 40 adantl φ ψ X No
42 2 adantr φ ψ B No
43 41 42 mulscld φ ψ X s B No
44 3 adantr φ ψ C No
45 41 44 mulscld φ ψ X s C No
46 pncans X s B No X s C No X s B + s X s C - s X s C = X s B
47 43 45 46 syl2anc φ ψ X s B + s X s C - s X s C = X s B
48 47 oveq1d φ ψ X s B + s X s C - s X s C + s A s Y + s A s C = X s B + s A s Y + s A s C
49 43 45 addscld φ ψ X s B + s X s C No
50 1 adantr φ ψ A No
51 leftssno L B No
52 rightssno R B No
53 51 52 unssi L B R B No
54 53 8 sselid ψ Y No
55 54 adantl φ ψ Y No
56 50 55 mulscld φ ψ A s Y No
57 1 3 mulscld φ A s C No
58 57 adantr φ ψ A s C No
59 56 58 addscld φ ψ A s Y + s A s C No
60 49 59 45 addsubsd φ ψ X s B + s X s C + s A s Y + s A s C - s X s C = X s B + s X s C - s X s C + s A s Y + s A s C
61 43 56 58 addsassd φ ψ X s B + s A s Y + s A s C = X s B + s A s Y + s A s C
62 48 60 61 3eqtr4d φ ψ X s B + s X s C + s A s Y + s A s C - s X s C = X s B + s A s Y + s A s C
63 62 oveq1d φ ψ X s B + s X s C + s A s Y + s A s C - s X s C - s X s Y = X s B + s A s Y + s A s C - s X s Y
64 49 59 addscld φ ψ X s B + s X s C + s A s Y + s A s C No
65 40 54 mulscld ψ X s Y No
66 65 adantl φ ψ X s Y No
67 64 45 66 subsubs4d φ ψ X s B + s X s C + s A s Y + s A s C - s X s C - s X s Y = X s B + s X s C + s A s Y + s A s C - s X s C + s X s Y
68 45 66 addscomd φ ψ X s C + s X s Y = X s Y + s X s C
69 68 oveq2d φ ψ X s B + s X s C + s A s Y + s A s C - s X s C + s X s Y = X s B + s X s C + s A s Y + s A s C - s X s Y + s X s C
70 67 69 eqtrd φ ψ X s B + s X s C + s A s Y + s A s C - s X s C - s X s Y = X s B + s X s C + s A s Y + s A s C - s X s Y + s X s C
71 43 56 addscld φ ψ X s B + s A s Y No
72 71 58 66 addsubsd φ ψ X s B + s A s Y + s A s C - s X s Y = X s B + s A s Y - s X s Y + s A s C
73 63 70 72 3eqtr3d φ ψ X s B + s X s C + s A s Y + s A s C - s X s Y + s X s C = X s B + s A s Y - s X s Y + s A s C
74 36 73 eqtrd φ ψ X s B + s C + s A s Y + s C - s X s Y + s C = X s B + s A s Y - s X s Y + s A s C