MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sadadd2lem2 Structured version   Unicode version

Theorem sadadd2lem2 14309
Description: The core of the proof of sadadd2 14319. 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 9618 . . . . . . . . 9  |-  0  e.  CC
2 ifcl 3927 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  if ( ps ,  A ,  0 )  e.  CC )
31, 2mpan2 669 . . . . . . . 8  |-  ( A  e.  CC  ->  if ( ps ,  A , 
0 )  e.  CC )
43ad2antrr 724 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  if ( ps ,  A ,  0 )  e.  CC )
5 simpll 752 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  A  e.  CC )
64, 5, 5add12d 9837 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  ( if ( ps ,  A , 
0 )  +  ( A  +  A ) )  =  ( A  +  ( if ( ps ,  A , 
0 )  +  A
) ) )
75, 4, 5addassd 9648 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  ( ( A  +  if ( ps ,  A ,  0 ) )  +  A
)  =  ( A  +  ( if ( ps ,  A , 
0 )  +  A
) ) )
86, 7eqtr4d 2446 . . . . 5  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  ( if ( ps ,  A , 
0 )  +  ( A  +  A ) )  =  ( ( A  +  if ( ps ,  A , 
0 ) )  +  A ) )
9 pm5.501 339 . . . . . . . . 9  |-  ( ph  ->  ( ps  <->  ( ph  <->  ps ) ) )
109adantl 464 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  ( ps  <->  ( ph  <->  ps ) ) )
1110bicomd 201 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  ( ( ph  <->  ps )  <->  ps ) )
1211ifbid 3907 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  if ( (
ph 
<->  ps ) ,  A ,  0 )  =  if ( ps ,  A ,  0 ) )
13 simpr 459 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  ph )
1413orcd 390 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  ( ph  \/  ps ) )
15 iftrue 3891 . . . . . . . 8  |-  ( (
ph  \/  ps )  ->  if ( ( ph  \/  ps ) ,  ( 2  x.  A ) ,  0 )  =  ( 2  x.  A
) )
1614, 15syl 17 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  if ( (
ph  \/  ps ) ,  ( 2  x.  A ) ,  0 )  =  ( 2  x.  A ) )
1752timesd 10822 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  ( 2  x.  A )  =  ( A  +  A ) )
1816, 17eqtrd 2443 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  if ( (
ph  \/  ps ) ,  ( 2  x.  A ) ,  0 )  =  ( A  +  A ) )
1912, 18oveq12d 6296 . . . . 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 3891 . . . . . . . 8  |-  ( ph  ->  if ( ph ,  A ,  0 )  =  A )
2120adantl 464 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  if ( ph ,  A ,  0 )  =  A )
2221oveq1d 6293 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  ( if (
ph ,  A , 
0 )  +  if ( ps ,  A , 
0 ) )  =  ( A  +  if ( ps ,  A , 
0 ) ) )
2322oveq1d 6293 . . . . 5  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  ph )  ->  ( ( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) )  +  A )  =  ( ( A  +  if ( ps ,  A ,  0 ) )  +  A
) )
248, 19, 233eqtr4d 2453 . . . 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 3894 . . . . . . . . 9  |-  ( -. 
ph  ->  if ( ph ,  A ,  0 )  =  0 )
2625adantl 464 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  if ( ph ,  A , 
0 )  =  0 )
2726oveq1d 6293 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  ( if ( ph ,  A , 
0 )  +  if ( ps ,  A , 
0 ) )  =  ( 0  +  if ( ps ,  A , 
0 ) ) )
283ad2antrr 724 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  if ( ps ,  A , 
0 )  e.  CC )
2928addid2d 9815 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  ( 0  +  if ( ps ,  A ,  0 ) )  =  if ( ps ,  A ,  0 ) )
3027, 29eqtrd 2443 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  ( if ( ph ,  A , 
0 )  +  if ( ps ,  A , 
0 ) )  =  if ( ps ,  A ,  0 ) )
3130oveq1d 6293 . . . . 5  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  ( ( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) )  +  A )  =  ( if ( ps ,  A , 
0 )  +  A
) )
32 2cnd 10649 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  2  e.  CC )
33 id 22 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  A  e.  CC )
3432, 33mulcld 9646 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  (
2  x.  A )  e.  CC )
3534addid2d 9815 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
0  +  ( 2  x.  A ) )  =  ( 2  x.  A ) )
36 2times 10695 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
2  x.  A )  =  ( A  +  A ) )
3735, 36eqtrd 2443 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
0  +  ( 2  x.  A ) )  =  ( A  +  A ) )
3837adantr 463 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ps )  ->  ( 0  +  ( 2  x.  A ) )  =  ( A  +  A
) )
39 iftrue 3891 . . . . . . . . . 10  |-  ( ps 
->  if ( ps , 
0 ,  A )  =  0 )
4039adantl 464 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ps )  ->  if ( ps ,  0 ,  A )  =  0 )
41 iftrue 3891 . . . . . . . . . 10  |-  ( ps 
->  if ( ps , 
( 2  x.  A
) ,  0 )  =  ( 2  x.  A ) )
4241adantl 464 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ps )  ->  if ( ps ,  ( 2  x.  A ) ,  0 )  =  ( 2  x.  A ) )
4340, 42oveq12d 6296 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ps )  ->  ( if ( ps ,  0 ,  A )  +  if ( ps , 
( 2  x.  A
) ,  0 ) )  =  ( 0  +  ( 2  x.  A ) ) )
44 iftrue 3891 . . . . . . . . . 10  |-  ( ps 
->  if ( ps ,  A ,  0 )  =  A )
4544adantl 464 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  ps )  ->  if ( ps ,  A , 
0 )  =  A )
4645oveq1d 6293 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ps )  ->  ( if ( ps ,  A ,  0 )  +  A )  =  ( A  +  A ) )
4738, 43, 463eqtr4d 2453 . . . . . . 7  |-  ( ( A  e.  CC  /\  ps )  ->  ( if ( ps ,  0 ,  A )  +  if ( ps , 
( 2  x.  A
) ,  0 ) )  =  ( if ( ps ,  A ,  0 )  +  A ) )
48 simpl 455 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  -.  ps )  ->  A  e.  CC )
49 0cnd 9619 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  -.  ps )  ->  0  e.  CC )
5048, 49addcomd 9816 . . . . . . . 8  |-  ( ( A  e.  CC  /\  -.  ps )  ->  ( A  +  0 )  =  ( 0  +  A ) )
51 iffalse 3894 . . . . . . . . . 10  |-  ( -. 
ps  ->  if ( ps ,  0 ,  A
)  =  A )
5251adantl 464 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  -.  ps )  ->  if ( ps ,  0 ,  A )  =  A )
53 iffalse 3894 . . . . . . . . . 10  |-  ( -. 
ps  ->  if ( ps ,  ( 2  x.  A ) ,  0 )  =  0 )
5453adantl 464 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  -.  ps )  ->  if ( ps ,  ( 2  x.  A ) ,  0 )  =  0 )
5552, 54oveq12d 6296 . . . . . . . 8  |-  ( ( A  e.  CC  /\  -.  ps )  ->  ( if ( ps ,  0 ,  A )  +  if ( ps , 
( 2  x.  A
) ,  0 ) )  =  ( A  +  0 ) )
56 iffalse 3894 . . . . . . . . . 10  |-  ( -. 
ps  ->  if ( ps ,  A ,  0 )  =  0 )
5756adantl 464 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  -.  ps )  ->  if ( ps ,  A , 
0 )  =  0 )
5857oveq1d 6293 . . . . . . . 8  |-  ( ( A  e.  CC  /\  -.  ps )  ->  ( if ( ps ,  A ,  0 )  +  A )  =  ( 0  +  A ) )
5950, 55, 583eqtr4d 2453 . . . . . . 7  |-  ( ( A  e.  CC  /\  -.  ps )  ->  ( if ( ps ,  0 ,  A )  +  if ( ps , 
( 2  x.  A
) ,  0 ) )  =  ( if ( ps ,  A ,  0 )  +  A ) )
6047, 59pm2.61dan 792 . . . . . 6  |-  ( A  e.  CC  ->  ( if ( ps ,  0 ,  A )  +  if ( ps , 
( 2  x.  A
) ,  0 ) )  =  ( if ( ps ,  A ,  0 )  +  A ) )
6160ad2antrr 724 . . . . 5  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  ( if ( ps ,  0 ,  A )  +  if ( ps ,  ( 2  x.  A ) ,  0 ) )  =  ( if ( ps ,  A ,  0 )  +  A ) )
62 ifnot 3930 . . . . . . 7  |-  if ( -.  ps ,  A ,  0 )  =  if ( ps , 
0 ,  A )
63 nbn2 343 . . . . . . . . 9  |-  ( -. 
ph  ->  ( -.  ps  <->  (
ph 
<->  ps ) ) )
6463adantl 464 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  ( -.  ps 
<->  ( ph  <->  ps )
) )
6564ifbid 3907 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  if ( -.  ps ,  A , 
0 )  =  if ( ( ph  <->  ps ) ,  A ,  0 ) )
6662, 65syl5eqr 2457 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  if ( ps ,  0 ,  A )  =  if ( ( ph  <->  ps ) ,  A ,  0 ) )
67 biorf 403 . . . . . . . 8  |-  ( -. 
ph  ->  ( ps  <->  ( ph  \/  ps ) ) )
6867adantl 464 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  ( ps  <->  (
ph  \/  ps )
) )
6968ifbid 3907 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
ch )  /\  -.  ph )  ->  if ( ps ,  ( 2  x.  A ) ,  0 )  =  if ( ( ph  \/  ps ) ,  ( 2  x.  A ) ,  0 ) )
7066, 69oveq12d 6296 . . . . 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 2450 . . . 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 792 . . 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 1471 . . . . . . 7  |-  (hadd ( ch ,  ph ,  ps )  <-> hadd ( ph ,  ps ,  ch ) )
74 had1 1489 . . . . . . 7  |-  ( ch 
->  (hadd ( ch ,  ph ,  ps )  <->  (
ph 
<->  ps ) ) )
7573, 74syl5bbr 259 . . . . . 6  |-  ( ch 
->  (hadd ( ph ,  ps ,  ch )  <->  (
ph 
<->  ps ) ) )
7675adantl 464 . . . . 5  |-  ( ( A  e.  CC  /\  ch )  ->  (hadd (
ph ,  ps ,  ch )  <->  ( ph  <->  ps )
) )
7776ifbid 3907 . . . 4  |-  ( ( A  e.  CC  /\  ch )  ->  if (hadd ( ph ,  ps ,  ch ) ,  A ,  0 )  =  if ( ( ph  <->  ps ) ,  A , 
0 ) )
78 cad1 1483 . . . . . 6  |-  ( ch 
->  (cadd ( ph ,  ps ,  ch )  <->  (
ph  \/  ps )
) )
7978adantl 464 . . . . 5  |-  ( ( A  e.  CC  /\  ch )  ->  (cadd (
ph ,  ps ,  ch )  <->  ( ph  \/  ps ) ) )
8079ifbid 3907 . . . 4  |-  ( ( A  e.  CC  /\  ch )  ->  if (cadd ( ph ,  ps ,  ch ) ,  ( 2  x.  A ) ,  0 )  =  if ( ( ph  \/  ps ) ,  ( 2  x.  A ) ,  0 ) )
8177, 80oveq12d 6296 . . 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 3891 . . . . 5  |-  ( ch 
->  if ( ch ,  A ,  0 )  =  A )
8382adantl 464 . . . 4  |-  ( ( A  e.  CC  /\  ch )  ->  if ( ch ,  A , 
0 )  =  A )
8483oveq2d 6294 . . 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 2453 . 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 464 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  ph )  ->  if ( ph ,  A ,  0 )  =  A )
8786oveq1d 6293 . . . . 5  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  ph )  ->  ( if (
ph ,  A , 
0 )  +  if ( ps ,  A , 
0 ) )  =  ( A  +  if ( ps ,  A , 
0 ) ) )
8845oveq2d 6294 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ps )  ->  ( A  +  if ( ps ,  A ,  0 ) )  =  ( A  +  A ) )
8938, 43, 883eqtr4d 2453 . . . . . . 7  |-  ( ( A  e.  CC  /\  ps )  ->  ( if ( ps ,  0 ,  A )  +  if ( ps , 
( 2  x.  A
) ,  0 ) )  =  ( A  +  if ( ps ,  A ,  0 ) ) )
9054, 57eqtr4d 2446 . . . . . . . 8  |-  ( ( A  e.  CC  /\  -.  ps )  ->  if ( ps ,  ( 2  x.  A ) ,  0 )  =  if ( ps ,  A ,  0 ) )
9152, 90oveq12d 6296 . . . . . . 7  |-  ( ( A  e.  CC  /\  -.  ps )  ->  ( if ( ps ,  0 ,  A )  +  if ( ps , 
( 2  x.  A
) ,  0 ) )  =  ( A  +  if ( ps ,  A ,  0 ) ) )
9289, 91pm2.61dan 792 . . . . . 6  |-  ( A  e.  CC  ->  ( if ( ps ,  0 ,  A )  +  if ( ps , 
( 2  x.  A
) ,  0 ) )  =  ( A  +  if ( ps ,  A ,  0 ) ) )
9392ad2antrr 724 . . . . 5  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  ph )  ->  ( if ( ps ,  0 ,  A )  +  if ( ps ,  ( 2  x.  A ) ,  0 ) )  =  ( A  +  if ( ps ,  A , 
0 ) ) )
949adantl 464 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  ph )  ->  ( ps  <->  ( ph  <->  ps ) ) )
9594notbid 292 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  ph )  ->  ( -.  ps  <->  -.  ( ph  <->  ps )
) )
96 df-xor 1367 . . . . . . . . 9  |-  ( (
ph  \/_  ps )  <->  -.  ( ph  <->  ps )
)
9795, 96syl6bbr 263 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  ph )  ->  ( -.  ps  <->  (
ph  \/_  ps )
) )
9897ifbid 3907 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  ph )  ->  if ( -. 
ps ,  A , 
0 )  =  if ( ( ph  \/_  ps ) ,  A , 
0 ) )
9962, 98syl5eqr 2457 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  ph )  ->  if ( ps ,  0 ,  A
)  =  if ( ( ph  \/_  ps ) ,  A , 
0 ) )
100 ibar 502 . . . . . . . 8  |-  ( ph  ->  ( ps  <->  ( ph  /\ 
ps ) ) )
101100adantl 464 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  ph )  ->  ( ps  <->  ( ph  /\ 
ps ) ) )
102101ifbid 3907 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  ph )  ->  if ( ps ,  ( 2  x.  A ) ,  0 )  =  if ( ( ph  /\  ps ) ,  ( 2  x.  A ) ,  0 ) )
10399, 102oveq12d 6296 . . . . 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 2450 . . . 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 760 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  -.  ch )  /\  -.  ph )  /\  ps )  ->  A  e.  CC )
106 0cnd 9619 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  -.  ch )  /\  -.  ph )  /\  -.  ps )  -> 
0  e.  CC )
107105, 106ifclda 3917 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  if ( ps ,  A , 
0 )  e.  CC )
108 0cnd 9619 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  0  e.  CC )
109107, 108addcomd 9816 . . . . 5  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  ( if ( ps ,  A , 
0 )  +  0 )  =  ( 0  +  if ( ps ,  A ,  0 ) ) )
11063adantl 464 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  ( -.  ps 
<->  ( ph  <->  ps )
) )
111110con1bid 328 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  ( -.  ( ph  <->  ps )  <->  ps )
)
11296, 111syl5bb 257 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  ( ( ph  \/_  ps )  <->  ps )
)
113112ifbid 3907 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  if (
( ph  \/_  ps ) ,  A ,  0 )  =  if ( ps ,  A ,  0 ) )
114 simpr 459 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  -.  ph )
115114intnanrd 918 . . . . . . 7  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  -.  ( ph  /\  ps ) )
116 iffalse 3894 . . . . . . 7  |-  ( -.  ( ph  /\  ps )  ->  if ( (
ph  /\  ps ) ,  ( 2  x.  A ) ,  0 )  =  0 )
117115, 116syl 17 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  if (
( ph  /\  ps ) ,  ( 2  x.  A ) ,  0 )  =  0 )
118113, 117oveq12d 6296 . . . . 5  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  ( if ( ( ph  \/_  ps ) ,  A , 
0 )  +  if ( ( ph  /\  ps ) ,  ( 2  x.  A ) ,  0 ) )  =  ( if ( ps ,  A ,  0 )  +  0 ) )
11925adantl 464 . . . . . 6  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  if ( ph ,  A , 
0 )  =  0 )
120119oveq1d 6293 . . . . 5  |-  ( ( ( A  e.  CC  /\ 
-.  ch )  /\  -.  ph )  ->  ( if ( ph ,  A , 
0 )  +  if ( ps ,  A , 
0 ) )  =  ( 0  +  if ( ps ,  A , 
0 ) ) )
121109, 118, 1203eqtr4d 2453 . . . 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 792 . . 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 1491 . . . . . . 7  |-  ( -. 
ch  ->  (hadd ( ch ,  ph ,  ps ) 
<->  ( ph  \/_  ps ) ) )
12473, 123syl5bbr 259 . . . . . 6  |-  ( -. 
ch  ->  (hadd ( ph ,  ps ,  ch )  <->  (
ph  \/_  ps )
) )
125124adantl 464 . . . . 5  |-  ( ( A  e.  CC  /\  -.  ch )  ->  (hadd ( ph ,  ps ,  ch )  <->  ( ph  \/_  ps ) ) )
126125ifbid 3907 . . . 4  |-  ( ( A  e.  CC  /\  -.  ch )  ->  if (hadd ( ph ,  ps ,  ch ) ,  A ,  0 )  =  if ( ( ph  \/_ 
ps ) ,  A ,  0 ) )
127 cad0 1486 . . . . . 6  |-  ( -. 
ch  ->  (cadd ( ph ,  ps ,  ch )  <->  (
ph  /\  ps )
) )
128127adantl 464 . . . . 5  |-  ( ( A  e.  CC  /\  -.  ch )  ->  (cadd ( ph ,  ps ,  ch )  <->  ( ph  /\  ps ) ) )
129128ifbid 3907 . . . 4  |-  ( ( A  e.  CC  /\  -.  ch )  ->  if (cadd ( ph ,  ps ,  ch ) ,  ( 2  x.  A ) ,  0 )  =  if ( ( ph  /\ 
ps ) ,  ( 2  x.  A ) ,  0 ) )
130126, 129oveq12d 6296 . . 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 3894 . . . . 5  |-  ( -. 
ch  ->  if ( ch ,  A ,  0 )  =  0 )
132131oveq2d 6294 . . . 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 3927 . . . . . . 7  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  if ( ph ,  A ,  0 )  e.  CC )
1341, 133mpan2 669 . . . . . 6  |-  ( A  e.  CC  ->  if ( ph ,  A , 
0 )  e.  CC )
135134, 3addcld 9645 . . . . 5  |-  ( A  e.  CC  ->  ( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) )  e.  CC )
136135addid1d 9814 . . . 4  |-  ( A  e.  CC  ->  (
( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) )  +  0 )  =  ( if ( ph ,  A ,  0 )  +  if ( ps ,  A ,  0 ) ) )
137132, 136sylan9eqr 2465 . . 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 2453 . 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 792 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 366    /\ wa 367    \/_ wxo 1366    = wceq 1405  haddwhad 1459  caddwcad 1460    e. wcel 1842   ifcif 3885  (class class class)co 6278   CCcc 9520   0cc0 9522    + caddc 9525    x. cmul 9527   2c2 10626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-xor 1367  df-tru 1408  df-had 1461  df-cad 1462  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-ltxr 9663  df-2 10635
This theorem is referenced by:  sadadd2lem  14318
  Copyright terms: Public domain W3C validator