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Theorem sadadd2lem2 13959
Description: The core of the proof of sadadd2 13969. The intuitive justification for this is that cadd is true if at least two arguments are true, and hadd is true if an odd number of arguments are true, so altogether the result is  n  x.  A where  n is the number of true arguments, which is equivalently obtained by adding together one  A for each true argument, on the right side. (Contributed by Mario Carneiro, 8-Sep-2016.)
Assertion
Ref Expression
sadadd2lem2  |-  ( A  e.  CC  ->  ( if (hadd ( ph ,  ps ,  ch ) ,  A ,  0 )  +  if (cadd (
ph ,  ps ,  ch ) ,  ( 2  x.  A ) ,  0 ) )  =  ( ( if (
ph ,  A , 
0 )  +  if ( ps ,  A , 
0 ) )  +  if ( ch ,  A ,  0 ) ) )

Proof of Theorem sadadd2lem2
StepHypRef Expression
1 0cn 9588 . . . . . . . . 9  |-  0  e.  CC
2 ifcl 3981 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  if ( ps ,  A ,  0 )  e.  CC )
31, 2mpan2 671 . . . . . . . 8  |-  ( A  e.  CC  ->  if ( ps ,  A , 
0 )  e.  CC )
43ad2antrr 725 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  if ( ps ,  A ,  0 )  e.  CC )
5 simpll 753 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  A  e.  CC )
64, 5, 5add12d 9801 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  ( if ( ps ,  A , 
0 )  +  ( A  +  A ) )  =  ( A  +  ( if ( ps ,  A , 
0 )  +  A
) ) )
75, 4, 5addassd 9618 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  ( ( A  +  if ( ps ,  A ,  0 ) )  +  A
)  =  ( A  +  ( if ( ps ,  A , 
0 )  +  A
) ) )
86, 7eqtr4d 2511 . . . . 5  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  ( if ( ps ,  A , 
0 )  +  ( A  +  A ) )  =  ( ( A  +  if ( ps ,  A , 
0 ) )  +  A ) )
9 pm5.501 341 . . . . . . . . 9  |-  ( ph  ->  ( ps  <->  ( ph  <->  ps ) ) )
109adantl 466 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  ( ps  <->  ( ph  <->  ps ) ) )
1110bicomd 201 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  ( ( ph  <->  ps )  <->  ps ) )
1211ifbid 3961 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  if ( (
ph 
<->  ps ) ,  A ,  0 )  =  if ( ps ,  A ,  0 ) )
13 simpr 461 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  ph )
1413orcd 392 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  ( ph  \/  ps ) )
15 iftrue 3945 . . . . . . . 8  |-  ( (
ph  \/  ps )  ->  if ( ( ph  \/  ps ) ,  ( 2  x.  A ) ,  0 )  =  ( 2  x.  A
) )
1614, 15syl 16 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  if ( (
ph  \/  ps ) ,  ( 2  x.  A ) ,  0 )  =  ( 2  x.  A ) )
1752timesd 10781 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  ( 2  x.  A )  =  ( A  +  A ) )
1816, 17eqtrd 2508 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  if ( (
ph  \/  ps ) ,  ( 2  x.  A ) ,  0 )  =  ( A  +  A ) )
1912, 18oveq12d 6302 . . . . 5  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  ( if ( ( ph  <->  ps ) ,  A ,  0 )  +  if ( (
ph  \/  ps ) ,  ( 2  x.  A ) ,  0 ) )  =  ( if ( ps ,  A ,  0 )  +  ( A  +  A ) ) )
20 iftrue 3945 . . . . . . . 8  |-  ( ph  ->  if ( ph ,  A ,  0 )  =  A )
2120adantl 466 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  if ( ph ,  A ,  0 )  =  A )
2221oveq1d 6299 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  ( if (
ph ,  A , 
0 )  +  if ( ps ,  A , 
0 ) )  =  ( A  +  if ( ps ,  A , 
0 ) ) )
2322oveq1d 6299 . . . . 5  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  ( ( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) )  +  A )  =  ( ( A  +  if ( ps ,  A ,  0 ) )  +  A
) )
248, 19, 233eqtr4d 2518 . . . 4  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  ( if ( ( ph  <->  ps ) ,  A ,  0 )  +  if ( (
ph  \/  ps ) ,  ( 2  x.  A ) ,  0 ) )  =  ( ( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) )  +  A
) )
25 iffalse 3948 . . . . . . . . 9  |-  ( -. 
ph  ->  if ( ph ,  A ,  0 )  =  0 )
2625adantl 466 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  if ( ph ,  A , 
0 )  =  0 )
2726oveq1d 6299 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  ( if ( ph ,  A , 
0 )  +  if ( ps ,  A , 
0 ) )  =  ( 0  +  if ( ps ,  A , 
0 ) ) )
283ad2antrr 725 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  if ( ps ,  A , 
0 )  e.  CC )
2928addid2d 9780 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  ( 0  +  if ( ps ,  A ,  0 ) )  =  if ( ps ,  A ,  0 ) )
3027, 29eqtrd 2508 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  ( if ( ph ,  A , 
0 )  +  if ( ps ,  A , 
0 ) )  =  if ( ps ,  A ,  0 ) )
3130oveq1d 6299 . . . . 5  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  ( ( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) )  +  A )  =  ( if ( ps ,  A , 
0 )  +  A
) )
32 2cnd 10608 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  2  e.  CC )
33 id 22 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  A  e.  CC )
3432, 33mulcld 9616 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  (
2  x.  A )  e.  CC )
3534addid2d 9780 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
0  +  ( 2  x.  A ) )  =  ( 2  x.  A ) )
36 2times 10654 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )
3735, 36eqtrd 2508 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
0  +  ( 2  x.  A ) )  =  ( A  +  A ) )
3837adantr 465 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ps )  ->  ( 0  +  ( 2  x.  A ) )  =  ( A  +  A
) )
39 iftrue 3945 . . . . . . . . . 10  |-  ( ps 
->  if ( ps , 
0 ,  A )  =  0 )
4039adantl 466 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ps )  ->  if ( ps ,  0 ,  A )  =  0 )
41 iftrue 3945 . . . . . . . . . 10  |-  ( ps 
->  if ( ps , 
( 2  x.  A
) ,  0 )  =  ( 2  x.  A ) )
4241adantl 466 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ps )  ->  if ( ps ,  ( 2  x.  A ) ,  0 )  =  ( 2  x.  A ) )
4340, 42oveq12d 6302 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ps )  ->  ( if ( ps ,  0 ,  A )  +  if ( ps , 
( 2  x.  A
) ,  0 ) )  =  ( 0  +  ( 2  x.  A ) ) )
44 iftrue 3945 . . . . . . . . . 10  |-  ( ps 
->  if ( ps ,  A ,  0 )  =  A )
4544adantl 466 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ps )  ->  if ( ps ,  A , 
0 )  =  A )
4645oveq1d 6299 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ps )  ->  ( if ( ps ,  A ,  0 )  +  A )  =  ( A  +  A ) )
4738, 43, 463eqtr4d 2518 . . . . . . 7  |-  ( ( A  e.  CC  /\  ps )  ->  ( if ( ps ,  0 ,  A )  +  if ( ps , 
( 2  x.  A
) ,  0 ) )  =  ( if ( ps ,  A ,  0 )  +  A ) )
48 simpl 457 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  -.  ps )  ->  A  e.  CC )
49 0cnd 9589 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  -.  ps )  ->  0  e.  CC )
5048, 49addcomd 9781 . . . . . . . 8  |-  ( ( A  e.  CC  /\  -.  ps )  ->  ( A  +  0 )  =  ( 0  +  A ) )
51 iffalse 3948 . . . . . . . . . 10  |-  ( -. 
ps  ->  if ( ps ,  0 ,  A
)  =  A )
5251adantl 466 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  -.  ps )  ->  if ( ps ,  0 ,  A )  =  A )
53 iffalse 3948 . . . . . . . . . 10  |-  ( -. 
ps  ->  if ( ps ,  ( 2  x.  A ) ,  0 )  =  0 )
5453adantl 466 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  -.  ps )  ->  if ( ps ,  ( 2  x.  A ) ,  0 )  =  0 )
5552, 54oveq12d 6302 . . . . . . . 8  |-  ( ( A  e.  CC  /\  -.  ps )  ->  ( if ( ps ,  0 ,  A )  +  if ( ps , 
( 2  x.  A
) ,  0 ) )  =  ( A  +  0 ) )
56 iffalse 3948 . . . . . . . . . 10  |-  ( -. 
ps  ->  if ( ps ,  A ,  0 )  =  0 )
5756adantl 466 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  -.  ps )  ->  if ( ps ,  A , 
0 )  =  0 )
5857oveq1d 6299 . . . . . . . 8  |-  ( ( A  e.  CC  /\  -.  ps )  ->  ( if ( ps ,  A ,  0 )  +  A )  =  ( 0  +  A ) )
5950, 55, 583eqtr4d 2518 . . . . . . 7  |-  ( ( A  e.  CC  /\  -.  ps )  ->  ( if ( ps ,  0 ,  A )  +  if ( ps , 
( 2  x.  A
) ,  0 ) )  =  ( if ( ps ,  A ,  0 )  +  A ) )
6047, 59pm2.61dan 789 . . . . . 6  |-  ( A  e.  CC  ->  ( if ( ps ,  0 ,  A )  +  if ( ps , 
( 2  x.  A
) ,  0 ) )  =  ( if ( ps ,  A ,  0 )  +  A ) )
6160ad2antrr 725 . . . . 5  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  ( if ( ps ,  0 ,  A )  +  if ( ps ,  ( 2  x.  A ) ,  0 ) )  =  ( if ( ps ,  A ,  0 )  +  A ) )
62 ifnot 3984 . . . . . . 7  |-  if ( -.  ps ,  A ,  0 )  =  if ( ps , 
0 ,  A )
63 nbn2 345 . . . . . . . . 9  |-  ( -. 
ph  ->  ( -.  ps  <->  (
ph 
<->  ps ) ) )
6463adantl 466 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  ( -.  ps 
<->  ( ph  <->  ps )
) )
6564ifbid 3961 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  if ( -.  ps ,  A , 
0 )  =  if ( ( ph  <->  ps ) ,  A ,  0 ) )
6662, 65syl5eqr 2522 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  if ( ps ,  0 ,  A )  =  if ( ( ph  <->  ps ) ,  A ,  0 ) )
67 biorf 405 . . . . . . . 8  |-  ( -. 
ph  ->  ( ps  <->  ( ph  \/  ps ) ) )
6867adantl 466 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  ( ps  <->  (
ph  \/  ps )
) )
6968ifbid 3961 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  if ( ps ,  ( 2  x.  A ) ,  0 )  =  if ( ( ph  \/  ps ) ,  ( 2  x.  A ) ,  0 ) )
7066, 69oveq12d 6302 . . . . 5  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  ( if ( ps ,  0 ,  A )  +  if ( ps ,  ( 2  x.  A ) ,  0 ) )  =  ( if ( (
ph 
<->  ps ) ,  A ,  0 )  +  if ( ( ph  \/  ps ) ,  ( 2  x.  A ) ,  0 ) ) )
7131, 61, 703eqtr2rd 2515 . . . 4  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  ( if ( ( ph  <->  ps ) ,  A ,  0 )  +  if ( (
ph  \/  ps ) ,  ( 2  x.  A ) ,  0 ) )  =  ( ( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) )  +  A
) )
7224, 71pm2.61dan 789 . . 3  |-  ( ( A  e.  CC  /\  ch )  ->  ( if ( ( ph  <->  ps ) ,  A ,  0 )  +  if ( (
ph  \/  ps ) ,  ( 2  x.  A ) ,  0 ) )  =  ( ( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) )  +  A
) )
73 hadrot 1441 . . . . . . 7  |-  (hadd ( ch ,  ph ,  ps )  <-> hadd ( ph ,  ps ,  ch ) )
74 had1 1454 . . . . . . 7  |-  ( ch 
->  (hadd ( ch ,  ph ,  ps )  <->  (
ph 
<->  ps ) ) )
7573, 74syl5bbr 259 . . . . . 6  |-  ( ch 
->  (hadd ( ph ,  ps ,  ch )  <->  (
ph 
<->  ps ) ) )
7675adantl 466 . . . . 5  |-  ( ( A  e.  CC  /\  ch )  ->  (hadd (
ph ,  ps ,  ch )  <->  ( ph  <->  ps )
) )
7776ifbid 3961 . . . 4  |-  ( ( A  e.  CC  /\  ch )  ->  if (hadd ( ph ,  ps ,  ch ) ,  A ,  0 )  =  if ( ( ph  <->  ps ) ,  A , 
0 ) )
78 cad1 1450 . . . . . 6  |-  ( ch 
->  (cadd ( ph ,  ps ,  ch )  <->  (
ph  \/  ps )
) )
7978adantl 466 . . . . 5  |-  ( ( A  e.  CC  /\  ch )  ->  (cadd (
ph ,  ps ,  ch )  <->  ( ph  \/  ps ) ) )
8079ifbid 3961 . . . 4  |-  ( ( A  e.  CC  /\  ch )  ->  if (cadd ( ph ,  ps ,  ch ) ,  ( 2  x.  A ) ,  0 )  =  if ( ( ph  \/  ps ) ,  ( 2  x.  A ) ,  0 ) )
8177, 80oveq12d 6302 . . 3  |-  ( ( A  e.  CC  /\  ch )  ->  ( if (hadd ( ph ,  ps ,  ch ) ,  A ,  0 )  +  if (cadd (
ph ,  ps ,  ch ) ,  ( 2  x.  A ) ,  0 ) )  =  ( if ( (
ph 
<->  ps ) ,  A ,  0 )  +  if ( ( ph  \/  ps ) ,  ( 2  x.  A ) ,  0 ) ) )
82 iftrue 3945 . . . . 5  |-  ( ch 
->  if ( ch ,  A ,  0 )  =  A )
8382adantl 466 . . . 4  |-  ( ( A  e.  CC  /\  ch )  ->  if ( ch ,  A , 
0 )  =  A )
8483oveq2d 6300 . . 3  |-  ( ( A  e.  CC  /\  ch )  ->  ( ( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) )  +  if ( ch ,  A , 
0 ) )  =  ( ( if (
ph ,  A , 
0 )  +  if ( ps ,  A , 
0 ) )  +  A ) )
8572, 81, 843eqtr4d 2518 . 2  |-  ( ( A  e.  CC  /\  ch )  ->  ( if (hadd ( ph ,  ps ,  ch ) ,  A ,  0 )  +  if (cadd (
ph ,  ps ,  ch ) ,  ( 2  x.  A ) ,  0 ) )  =  ( ( if (
ph ,  A , 
0 )  +  if ( ps ,  A , 
0 ) )  +  if ( ch ,  A ,  0 ) ) )
8620adantl 466 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  ph )  ->  if ( ph ,  A ,  0 )  =  A )
8786oveq1d 6299 . . . . 5  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  ph )  ->  ( if (
ph ,  A , 
0 )  +  if ( ps ,  A , 
0 ) )  =  ( A  +  if ( ps ,  A , 
0 ) ) )
8845oveq2d 6300 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ps )  ->  ( A  +  if ( ps ,  A ,  0 ) )  =  ( A  +  A ) )
8938, 43, 883eqtr4d 2518 . . . . . . 7  |-  ( ( A  e.  CC  /\  ps )  ->  ( if ( ps ,  0 ,  A )  +  if ( ps , 
( 2  x.  A
) ,  0 ) )  =  ( A  +  if ( ps ,  A ,  0 ) ) )
9054, 57eqtr4d 2511 . . . . . . . 8  |-  ( ( A  e.  CC  /\  -.  ps )  ->  if ( ps ,  ( 2  x.  A ) ,  0 )  =  if ( ps ,  A ,  0 ) )
9152, 90oveq12d 6302 . . . . . . 7  |-  ( ( A  e.  CC  /\  -.  ps )  ->  ( if ( ps ,  0 ,  A )  +  if ( ps , 
( 2  x.  A
) ,  0 ) )  =  ( A  +  if ( ps ,  A ,  0 ) ) )
9289, 91pm2.61dan 789 . . . . . 6  |-  ( A  e.  CC  ->  ( if ( ps ,  0 ,  A )  +  if ( ps , 
( 2  x.  A
) ,  0 ) )  =  ( A  +  if ( ps ,  A ,  0 ) ) )
9392ad2antrr 725 . . . . 5  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  ph )  ->  ( if ( ps ,  0 ,  A )  +  if ( ps ,  ( 2  x.  A ) ,  0 ) )  =  ( A  +  if ( ps ,  A , 
0 ) ) )
949adantl 466 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  ph )  ->  ( ps  <->  ( ph  <->  ps ) ) )
9594notbid 294 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  ph )  ->  ( -.  ps  <->  -.  ( ph  <->  ps )
) )
96 df-xor 1361 . . . . . . . . 9  |-  ( (
ph  \/_  ps )  <->  -.  ( ph  <->  ps )
)
9795, 96syl6bbr 263 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  ph )  ->  ( -.  ps  <->  (
ph  \/_  ps )
) )
9897ifbid 3961 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  ph )  ->  if ( -. 
ps ,  A , 
0 )  =  if ( ( ph  \/_  ps ) ,  A , 
0 ) )
9962, 98syl5eqr 2522 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  ph )  ->  if ( ps ,  0 ,  A
)  =  if ( ( ph  \/_  ps ) ,  A , 
0 ) )
100 ibar 504 . . . . . . . 8  |-  ( ph  ->  ( ps  <->  ( ph  /\ 
ps ) ) )
101100adantl 466 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  ph )  ->  ( ps  <->  ( ph  /\ 
ps ) ) )
102101ifbid 3961 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  ph )  ->  if ( ps ,  ( 2  x.  A ) ,  0 )  =  if ( ( ph  /\  ps ) ,  ( 2  x.  A ) ,  0 ) )
10399, 102oveq12d 6302 . . . . 5  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  ph )  ->  ( if ( ps ,  0 ,  A )  +  if ( ps ,  ( 2  x.  A ) ,  0 ) )  =  ( if ( (
ph  \/_  ps ) ,  A ,  0 )  +  if ( (
ph  /\  ps ) ,  ( 2  x.  A ) ,  0 ) ) )
10487, 93, 1033eqtr2rd 2515 . . . 4  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  ph )  ->  ( if ( ( ph  \/_  ps ) ,  A , 
0 )  +  if ( ( ph  /\  ps ) ,  ( 2  x.  A ) ,  0 ) )  =  ( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) ) )
105 simplll 757 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  -.  ch )  /\  -.  ph )  /\  ps )  ->  A  e.  CC )
106 0cnd 9589 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  -.  ch )  /\  -.  ph )  /\  -.  ps )  -> 
0  e.  CC )
107105, 106ifclda 3971 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  if ( ps ,  A , 
0 )  e.  CC )
108 0cnd 9589 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  0  e.  CC )
109107, 108addcomd 9781 . . . . 5  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  ( if ( ps ,  A , 
0 )  +  0 )  =  ( 0  +  if ( ps ,  A ,  0 ) ) )
11063adantl 466 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  ( -.  ps 
<->  ( ph  <->  ps )
) )
111110con1bid 330 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  ( -.  ( ph  <->  ps )  <->  ps )
)
11296, 111syl5bb 257 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  ( ( ph  \/_  ps )  <->  ps )
)
113112ifbid 3961 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  if (
( ph  \/_  ps ) ,  A ,  0 )  =  if ( ps ,  A ,  0 ) )
114 simpr 461 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  -.  ph )
115114intnanrd 915 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  -.  ( ph  /\  ps ) )
116 iffalse 3948 . . . . . . 7  |-  ( -.  ( ph  /\  ps )  ->  if ( (
ph  /\  ps ) ,  ( 2  x.  A ) ,  0 )  =  0 )
117115, 116syl 16 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  if (
( ph  /\  ps ) ,  ( 2  x.  A ) ,  0 )  =  0 )
118113, 117oveq12d 6302 . . . . 5  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  ( if ( ( ph  \/_  ps ) ,  A , 
0 )  +  if ( ( ph  /\  ps ) ,  ( 2  x.  A ) ,  0 ) )  =  ( if ( ps ,  A ,  0 )  +  0 ) )
11925adantl 466 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  if ( ph ,  A , 
0 )  =  0 )
120119oveq1d 6299 . . . . 5  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  ( if ( ph ,  A , 
0 )  +  if ( ps ,  A , 
0 ) )  =  ( 0  +  if ( ps ,  A , 
0 ) ) )
121109, 118, 1203eqtr4d 2518 . . . 4  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  ( if ( ( ph  \/_  ps ) ,  A , 
0 )  +  if ( ( ph  /\  ps ) ,  ( 2  x.  A ) ,  0 ) )  =  ( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) ) )
122104, 121pm2.61dan 789 . . 3  |-  ( ( A  e.  CC  /\  -.  ch )  ->  ( if ( ( ph  \/_  ps ) ,  A , 
0 )  +  if ( ( ph  /\  ps ) ,  ( 2  x.  A ) ,  0 ) )  =  ( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) ) )
123 had0 1455 . . . . . . 7  |-  ( -. 
ch  ->  (hadd ( ch ,  ph ,  ps ) 
<->  ( ph  \/_  ps ) ) )
12473, 123syl5bbr 259 . . . . . 6  |-  ( -. 
ch  ->  (hadd ( ph ,  ps ,  ch )  <->  (
ph  \/_  ps )
) )
125124adantl 466 . . . . 5  |-  ( ( A  e.  CC  /\  -.  ch )  ->  (hadd ( ph ,  ps ,  ch )  <->  ( ph  \/_  ps ) ) )
126125ifbid 3961 . . . 4  |-  ( ( A  e.  CC  /\  -.  ch )  ->  if (hadd ( ph ,  ps ,  ch ) ,  A ,  0 )  =  if ( ( ph  \/_ 
ps ) ,  A ,  0 ) )
127 cad0 1452 . . . . . 6  |-  ( -. 
ch  ->  (cadd ( ph ,  ps ,  ch )  <->  (
ph  /\  ps )
) )
128127adantl 466 . . . . 5  |-  ( ( A  e.  CC  /\  -.  ch )  ->  (cadd ( ph ,  ps ,  ch )  <->  ( ph  /\  ps ) ) )
129128ifbid 3961 . . . 4  |-  ( ( A  e.  CC  /\  -.  ch )  ->  if (cadd ( ph ,  ps ,  ch ) ,  ( 2  x.  A ) ,  0 )  =  if ( ( ph  /\ 
ps ) ,  ( 2  x.  A ) ,  0 ) )
130126, 129oveq12d 6302 . . 3  |-  ( ( A  e.  CC  /\  -.  ch )  ->  ( if (hadd ( ph ,  ps ,  ch ) ,  A ,  0 )  +  if (cadd (
ph ,  ps ,  ch ) ,  ( 2  x.  A ) ,  0 ) )  =  ( if ( (
ph  \/_  ps ) ,  A ,  0 )  +  if ( (
ph  /\  ps ) ,  ( 2  x.  A ) ,  0 ) ) )
131 iffalse 3948 . . . . 5  |-  ( -. 
ch  ->  if ( ch ,  A ,  0 )  =  0 )
132131oveq2d 6300 . . . 4  |-  ( -. 
ch  ->  ( ( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) )  +  if ( ch ,  A , 
0 ) )  =  ( ( if (
ph ,  A , 
0 )  +  if ( ps ,  A , 
0 ) )  +  0 ) )
133 ifcl 3981 . . . . . . 7  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  if ( ph ,  A ,  0 )  e.  CC )
1341, 133mpan2 671 . . . . . 6  |-  ( A  e.  CC  ->  if ( ph ,  A , 
0 )  e.  CC )
135134, 3addcld 9615 . . . . 5  |-  ( A  e.  CC  ->  ( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) )  e.  CC )
136135addid1d 9779 . . . 4  |-  ( A  e.  CC  ->  (
( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) )  +  0 )  =  ( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) ) )
137132, 136sylan9eqr 2530 . . 3  |-  ( ( A  e.  CC  /\  -.  ch )  ->  (
( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) )  +  if ( ch ,  A , 
0 ) )  =  ( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) ) )
138122, 130, 1373eqtr4d 2518 . 2  |-  ( ( A  e.  CC  /\  -.  ch )  ->  ( if (hadd ( ph ,  ps ,  ch ) ,  A ,  0 )  +  if (cadd (
ph ,  ps ,  ch ) ,  ( 2  x.  A ) ,  0 ) )  =  ( ( if (
ph ,  A , 
0 )  +  if ( ps ,  A , 
0 ) )  +  if ( ch ,  A ,  0 ) ) )
13985, 138pm2.61dan 789 1  |-  ( A  e.  CC  ->  ( if (hadd ( ph ,  ps ,  ch ) ,  A ,  0 )  +  if (cadd (
ph ,  ps ,  ch ) ,  ( 2  x.  A ) ,  0 ) )  =  ( ( if (
ph ,  A , 
0 )  +  if ( ps ,  A , 
0 ) )  +  if ( ch ,  A ,  0 ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    \/_ wxo 1360    = wceq 1379  haddwhad 1429  caddwcad 1430    e. wcel 1767   ifcif 3939  (class class class)co 6284   CCcc 9490   0cc0 9492    + caddc 9495    x. cmul 9497   2c2 10585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-xor 1361  df-tru 1382  df-had 1431  df-cad 1432  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-ltxr 9633  df-2 10594
This theorem is referenced by:  sadadd2lem  13968
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