Metamath Proof Explorer


Theorem addsdilem4

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

Ref Expression
Hypotheses addsdilem4.1 φ A No
addsdilem4.2 φ B No
addsdilem4.3 φ C No
addsdilem4.4 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 |-
addsdilem4.5 No typesetting found for |- ( ph -> A. zO e. ( ( _Left ` C ) u. ( _Right ` C ) ) ( A x.s ( B +s zO ) ) = ( ( A x.s B ) +s ( A x.s zO ) ) ) with typecode |-
addsdilem4.6 No typesetting found for |- ( ph -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) A. zO e. ( ( _Left ` C ) u. ( _Right ` C ) ) ( xO x.s ( B +s zO ) ) = ( ( xO x.s B ) +s ( xO x.s zO ) ) ) with typecode |-
addsdilem4.7 ψ X L A R A
addsdilem4.8 ψ Z L C R C
Assertion addsdilem4 φ ψ X s B + s C + s A s B + s Z - s X s B + s Z = A s B + s X s C + s A s Z - s X s Z

Proof

Step Hyp Ref Expression
1 addsdilem4.1 φ A No
2 addsdilem4.2 φ B No
3 addsdilem4.3 φ C No
4 addsdilem4.4 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 addsdilem4.5 Could not format ( ph -> A. zO e. ( ( _Left ` C ) u. ( _Right ` C ) ) ( A x.s ( B +s zO ) ) = ( ( A x.s B ) +s ( A x.s zO ) ) ) : No typesetting found for |- ( ph -> A. zO e. ( ( _Left ` C ) u. ( _Right ` C ) ) ( A x.s ( B +s zO ) ) = ( ( A x.s B ) +s ( A x.s zO ) ) ) with typecode |-
6 addsdilem4.6 Could not format ( ph -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) A. zO e. ( ( _Left ` C ) u. ( _Right ` C ) ) ( xO x.s ( B +s zO ) ) = ( ( xO x.s B ) +s ( xO x.s zO ) ) ) : No typesetting found for |- ( ph -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) A. zO e. ( ( _Left ` C ) u. ( _Right ` C ) ) ( xO x.s ( B +s zO ) ) = ( ( xO x.s B ) +s ( xO x.s zO ) ) ) with typecode |-
7 addsdilem4.7 ψ X L A R A
8 addsdilem4.8 ψ Z L C R C
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 oveq2 Could not format ( zO = Z -> ( B +s zO ) = ( B +s Z ) ) : No typesetting found for |- ( zO = Z -> ( B +s zO ) = ( B +s Z ) ) with typecode |-
18 17 oveq2d Could not format ( zO = Z -> ( A x.s ( B +s zO ) ) = ( A x.s ( B +s Z ) ) ) : No typesetting found for |- ( zO = Z -> ( A x.s ( B +s zO ) ) = ( A x.s ( B +s Z ) ) ) with typecode |-
19 oveq2 Could not format ( zO = Z -> ( A x.s zO ) = ( A x.s Z ) ) : No typesetting found for |- ( zO = Z -> ( A x.s zO ) = ( A x.s Z ) ) with typecode |-
20 19 oveq2d Could not format ( zO = Z -> ( ( A x.s B ) +s ( A x.s zO ) ) = ( ( A x.s B ) +s ( A x.s Z ) ) ) : No typesetting found for |- ( zO = Z -> ( ( A x.s B ) +s ( A x.s zO ) ) = ( ( A x.s B ) +s ( A x.s Z ) ) ) with typecode |-
21 18 20 eqeq12d Could not format ( zO = Z -> ( ( A x.s ( B +s zO ) ) = ( ( A x.s B ) +s ( A x.s zO ) ) <-> ( A x.s ( B +s Z ) ) = ( ( A x.s B ) +s ( A x.s Z ) ) ) ) : No typesetting found for |- ( zO = Z -> ( ( A x.s ( B +s zO ) ) = ( ( A x.s B ) +s ( A x.s zO ) ) <-> ( A x.s ( B +s Z ) ) = ( ( A x.s B ) +s ( A x.s Z ) ) ) ) with typecode |-
22 5 adantr Could not format ( ( ph /\ ps ) -> A. zO e. ( ( _Left ` C ) u. ( _Right ` C ) ) ( A x.s ( B +s zO ) ) = ( ( A x.s B ) +s ( A x.s zO ) ) ) : No typesetting found for |- ( ( ph /\ ps ) -> A. zO e. ( ( _Left ` C ) u. ( _Right ` C ) ) ( A x.s ( B +s zO ) ) = ( ( A x.s B ) +s ( A x.s zO ) ) ) with typecode |-
23 8 adantl φ ψ Z L C R C
24 21 22 23 rspcdva φ ψ A s B + s Z = A s B + s A s Z
25 16 24 oveq12d φ ψ X s B + s C + s A s B + s Z = X s B + s X s C + s A s B + s A s Z
26 oveq1 Could not format ( xO = X -> ( xO x.s ( B +s zO ) ) = ( X x.s ( B +s zO ) ) ) : No typesetting found for |- ( xO = X -> ( xO x.s ( B +s zO ) ) = ( X x.s ( B +s zO ) ) ) with typecode |-
27 oveq1 Could not format ( xO = X -> ( xO x.s zO ) = ( X x.s zO ) ) : No typesetting found for |- ( xO = X -> ( xO x.s zO ) = ( X x.s zO ) ) with typecode |-
28 10 27 oveq12d Could not format ( xO = X -> ( ( xO x.s B ) +s ( xO x.s zO ) ) = ( ( X x.s B ) +s ( X x.s zO ) ) ) : No typesetting found for |- ( xO = X -> ( ( xO x.s B ) +s ( xO x.s zO ) ) = ( ( X x.s B ) +s ( X x.s zO ) ) ) with typecode |-
29 26 28 eqeq12d Could not format ( xO = X -> ( ( xO x.s ( B +s zO ) ) = ( ( xO x.s B ) +s ( xO x.s zO ) ) <-> ( X x.s ( B +s zO ) ) = ( ( X x.s B ) +s ( X x.s zO ) ) ) ) : No typesetting found for |- ( xO = X -> ( ( xO x.s ( B +s zO ) ) = ( ( xO x.s B ) +s ( xO x.s zO ) ) <-> ( X x.s ( B +s zO ) ) = ( ( X x.s B ) +s ( X x.s zO ) ) ) ) with typecode |-
30 17 oveq2d Could not format ( zO = Z -> ( X x.s ( B +s zO ) ) = ( X x.s ( B +s Z ) ) ) : No typesetting found for |- ( zO = Z -> ( X x.s ( B +s zO ) ) = ( X x.s ( B +s Z ) ) ) with typecode |-
31 oveq2 Could not format ( zO = Z -> ( X x.s zO ) = ( X x.s Z ) ) : No typesetting found for |- ( zO = Z -> ( X x.s zO ) = ( X x.s Z ) ) with typecode |-
32 31 oveq2d Could not format ( zO = Z -> ( ( X x.s B ) +s ( X x.s zO ) ) = ( ( X x.s B ) +s ( X x.s Z ) ) ) : No typesetting found for |- ( zO = Z -> ( ( X x.s B ) +s ( X x.s zO ) ) = ( ( X x.s B ) +s ( X x.s Z ) ) ) with typecode |-
33 30 32 eqeq12d Could not format ( zO = Z -> ( ( X x.s ( B +s zO ) ) = ( ( X x.s B ) +s ( X x.s zO ) ) <-> ( X x.s ( B +s Z ) ) = ( ( X x.s B ) +s ( X x.s Z ) ) ) ) : No typesetting found for |- ( zO = Z -> ( ( X x.s ( B +s zO ) ) = ( ( X x.s B ) +s ( X x.s zO ) ) <-> ( X x.s ( B +s Z ) ) = ( ( X x.s B ) +s ( X x.s Z ) ) ) ) with typecode |-
34 6 adantr Could not format ( ( ph /\ ps ) -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) A. zO e. ( ( _Left ` C ) u. ( _Right ` C ) ) ( xO x.s ( B +s zO ) ) = ( ( xO x.s B ) +s ( xO x.s zO ) ) ) : No typesetting found for |- ( ( ph /\ ps ) -> A. xO e. ( ( _Left ` A ) u. ( _Right ` A ) ) A. zO e. ( ( _Left ` C ) u. ( _Right ` C ) ) ( xO x.s ( B +s zO ) ) = ( ( xO x.s B ) +s ( xO x.s zO ) ) ) with typecode |-
35 29 33 34 15 23 rspc2dv φ ψ X s B + s Z = X s B + s X s Z
36 25 35 oveq12d φ ψ X s B + s C + s A s B + s Z - s X s B + s Z = X s B + s X s C + s A s B + s A s Z - s X s B + s X s Z
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 43 45 addscld φ ψ X s B + s X s C No
47 1 2 mulscld φ A s B No
48 47 adantr φ ψ A s B No
49 1 adantr φ ψ A No
50 leftssno L C No
51 rightssno R C No
52 50 51 unssi L C R C No
53 52 8 sselid ψ Z No
54 53 adantl φ ψ Z No
55 49 54 mulscld φ ψ A s Z No
56 48 55 addscld φ ψ A s B + s A s Z No
57 46 56 addscld φ ψ X s B + s X s C + s A s B + s A s Z No
58 41 54 mulscld φ ψ X s Z No
59 57 43 58 subsubs4d φ ψ X s B + s X s C + s A s B + s A s Z - s X s B - s X s Z = X s B + s X s C + s A s B + s A s Z - s X s B + s X s Z
60 46 56 43 addsubsd φ ψ X s B + s X s C + s A s B + s A s Z - s X s B = X s B + s X s C - s X s B + s A s B + s A s Z
61 43 45 addscomd φ ψ X s B + s X s C = X s C + s X s B
62 61 oveq1d φ ψ X s B + s X s C - s X s B = X s C + s X s B - s X s B
63 pncans X s C No X s B No X s C + s X s B - s X s B = X s C
64 45 43 63 syl2anc φ ψ X s C + s X s B - s X s B = X s C
65 62 64 eqtrd φ ψ X s B + s X s C - s X s B = X s C
66 65 oveq1d φ ψ X s B + s X s C - s X s B + s A s B + s A s Z = X s C + s A s B + s A s Z
67 45 48 55 adds12d φ ψ X s C + s A s B + s A s Z = A s B + s X s C + s A s Z
68 60 66 67 3eqtrd φ ψ X s B + s X s C + s A s B + s A s Z - s X s B = A s B + s X s C + s A s Z
69 68 oveq1d φ ψ X s B + s X s C + s A s B + s A s Z - s X s B - s X s Z = A s B + s X s C + s A s Z - s X s Z
70 45 55 addscld φ ψ X s C + s A s Z No
71 48 70 58 addsubsassd φ ψ A s B + s X s C + s A s Z - s X s Z = A s B + s X s C + s A s Z - s X s Z
72 69 71 eqtrd φ ψ X s B + s X s C + s A s B + s A s Z - s X s B - s X s Z = A s B + s X s C + s A s Z - s X s Z
73 36 59 72 3eqtr2d φ ψ X s B + s C + s A s B + s Z - s X s B + s Z = A s B + s X s C + s A s Z - s X s Z