Theorem sadadd2lem2 15172

Description: The core of the proof of sadadd2 15182. 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

Step	Hyp	Ref	Expression
1		0cn 10032	. . . . . . . . 9 |- 0 e. CC
2		ifcl 4130	. . . . . . . . 9 |- ( ( A e. CC /\ 0 e. CC ) -> if ( ps , A , 0 ) e. CC )
3	1, 2	mpan2 707	. . . . . . . 8 |- ( A e. CC -> if ( ps , A , 0 ) e. CC )
4	3	ad2antrr 762	. . . . . . 7 |- ( ( ( A e. CC /\ ch ) /\ ph ) -> if ( ps , A , 0 ) e. CC )
5		simpll 790	. . . . . . 7 |- ( ( ( A e. CC /\ ch ) /\ ph ) -> A e. CC )
6	4, 5, 5	add12d 10262	. . . . . 6 |- ( ( ( A e. CC /\ ch ) /\ ph ) -> ( if ( ps , A , 0 ) + ( A + A ) ) = ( A + ( if ( ps , A , 0 ) + A ) ) )
7	5, 4, 5	addassd 10062	. . . . . 6 |- ( ( ( A e. CC /\ ch ) /\ ph ) -> ( ( A + if ( ps , A , 0 ) ) + A ) = ( A + ( if ( ps , A , 0 ) + A ) ) )
8	6, 7	eqtr4d 2659	. . . . 5 |- ( ( ( A e. CC /\ ch ) /\ ph ) -> ( if ( ps , A , 0 ) + ( A + A ) ) = ( ( A + if ( ps , A , 0 ) ) + A ) )
9		pm5.501 356	. . . . . . . . 9 |- ( ph -> ( ps <-> ( ph <-> ps ) ) )
10	9	adantl 482	. . . . . . . 8 |- ( ( ( A e. CC /\ ch ) /\ ph ) -> ( ps <-> ( ph <-> ps ) ) )
11	10	bicomd 213	. . . . . . 7 |- ( ( ( A e. CC /\ ch ) /\ ph ) -> ( ( ph <-> ps ) <-> ps ) )
12	11	ifbid 4108	. . . . . 6 |- ( ( ( A e. CC /\ ch ) /\ ph ) -> if ( ( ph <-> ps ) , A , 0 ) = if ( ps , A , 0 ) )
13		simpr 477	. . . . . . . . 9 |- ( ( ( A e. CC /\ ch ) /\ ph ) -> ph )
14	13	orcd 407	. . . . . . . 8 |- ( ( ( A e. CC /\ ch ) /\ ph ) -> ( ph \/ ps ) )
15		iftrue 4092	. . . . . . . 8 |- ( ( ph \/ ps ) -> if ( ( ph \/ ps ) , ( 2 x. A ) , 0 ) = ( 2 x. A ) )
16	14, 15	syl 17	. . . . . . 7 |- ( ( ( A e. CC /\ ch ) /\ ph ) -> if ( ( ph \/ ps ) , ( 2 x. A ) , 0 ) = ( 2 x. A ) )
17	5	2timesd 11275	. . . . . . 7 |- ( ( ( A e. CC /\ ch ) /\ ph ) -> ( 2 x. A ) = ( A + A ) )
18	16, 17	eqtrd 2656	. . . . . 6 |- ( ( ( A e. CC /\ ch ) /\ ph ) -> if ( ( ph \/ ps ) , ( 2 x. A ) , 0 ) = ( A + A ) )
19	12, 18	oveq12d 6668	. . . . 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 4092	. . . . . . . 8 |- ( ph -> if ( ph , A , 0 ) = A )
21	20	adantl 482	. . . . . . 7 |- ( ( ( A e. CC /\ ch ) /\ ph ) -> if ( ph , A , 0 ) = A )
22	21	oveq1d 6665	. . . . . 6 |- ( ( ( A e. CC /\ ch ) /\ ph ) -> ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) = ( A + if ( ps , A , 0 ) ) )
23	22	oveq1d 6665	. . . . 5 |- ( ( ( A e. CC /\ ch ) /\ ph ) -> ( ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) + A ) = ( ( A + if ( ps , A , 0 ) ) + A ) )
24	8, 19, 23	3eqtr4d 2666	. . . 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 4095	. . . . . . . . 9 |- ( -. ph -> if ( ph , A , 0 ) = 0 )
26	25	adantl 482	. . . . . . . 8 |- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> if ( ph , A , 0 ) = 0 )
27	26	oveq1d 6665	. . . . . . 7 |- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) = ( 0 + if ( ps , A , 0 ) ) )
28	3	ad2antrr 762	. . . . . . . 8 |- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> if ( ps , A , 0 ) e. CC )
29	28	addid2d 10237	. . . . . . 7 |- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> ( 0 + if ( ps , A , 0 ) ) = if ( ps , A , 0 ) )
30	27, 29	eqtrd 2656	. . . . . 6 |- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) = if ( ps , A , 0 ) )
31	30	oveq1d 6665	. . . . 5 |- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> ( ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) + A ) = ( if ( ps , A , 0 ) + A ) )
32		2cnd 11093	. . . . . . . . . . . 12 |- ( A e. CC -> 2 e. CC )
33		id 22	. . . . . . . . . . . 12 |- ( A e. CC -> A e. CC )
34	32, 33	mulcld 10060	. . . . . . . . . . 11 |- ( A e. CC -> ( 2 x. A ) e. CC )
35	34	addid2d 10237	. . . . . . . . . 10 |- ( A e. CC -> ( 0 + ( 2 x. A ) ) = ( 2 x. A ) )
36		2times 11145	. . . . . . . . . 10 |- ( A e. CC -> ( 2 x. A ) = ( A + A ) )
37	35, 36	eqtrd 2656	. . . . . . . . 9 |- ( A e. CC -> ( 0 + ( 2 x. A ) ) = ( A + A ) )
38	37	adantr 481	. . . . . . . 8 |- ( ( A e. CC /\ ps ) -> ( 0 + ( 2 x. A ) ) = ( A + A ) )
39		iftrue 4092	. . . . . . . . . 10 |- ( ps -> if ( ps , 0 , A ) = 0 )
40	39	adantl 482	. . . . . . . . 9 |- ( ( A e. CC /\ ps ) -> if ( ps , 0 , A ) = 0 )
41		iftrue 4092	. . . . . . . . . 10 |- ( ps -> if ( ps , ( 2 x. A ) , 0 ) = ( 2 x. A ) )
42	41	adantl 482	. . . . . . . . 9 |- ( ( A e. CC /\ ps ) -> if ( ps , ( 2 x. A ) , 0 ) = ( 2 x. A ) )
43	40, 42	oveq12d 6668	. . . . . . . 8 |- ( ( A e. CC /\ ps ) -> ( if ( ps , 0 , A ) + if ( ps , ( 2 x. A ) , 0 ) ) = ( 0 + ( 2 x. A ) ) )
44		iftrue 4092	. . . . . . . . . 10 |- ( ps -> if ( ps , A , 0 ) = A )
45	44	adantl 482	. . . . . . . . 9 |- ( ( A e. CC /\ ps ) -> if ( ps , A , 0 ) = A )
46	45	oveq1d 6665	. . . . . . . 8 |- ( ( A e. CC /\ ps ) -> ( if ( ps , A , 0 ) + A ) = ( A + A ) )
47	38, 43, 46	3eqtr4d 2666	. . . . . . 7 |- ( ( A e. CC /\ ps ) -> ( if ( ps , 0 , A ) + if ( ps , ( 2 x. A ) , 0 ) ) = ( if ( ps , A , 0 ) + A ) )
48		simpl 473	. . . . . . . . 9 |- ( ( A e. CC /\ -. ps ) -> A e. CC )
49		0cnd 10033	. . . . . . . . 9 |- ( ( A e. CC /\ -. ps ) -> 0 e. CC )
50	48, 49	addcomd 10238	. . . . . . . 8 |- ( ( A e. CC /\ -. ps ) -> ( A + 0 ) = ( 0 + A ) )
51		iffalse 4095	. . . . . . . . . 10 |- ( -. ps -> if ( ps , 0 , A ) = A )
52	51	adantl 482	. . . . . . . . 9 |- ( ( A e. CC /\ -. ps ) -> if ( ps , 0 , A ) = A )
53		iffalse 4095	. . . . . . . . . 10 |- ( -. ps -> if ( ps , ( 2 x. A ) , 0 ) = 0 )
54	53	adantl 482	. . . . . . . . 9 |- ( ( A e. CC /\ -. ps ) -> if ( ps , ( 2 x. A ) , 0 ) = 0 )
55	52, 54	oveq12d 6668	. . . . . . . 8 |- ( ( A e. CC /\ -. ps ) -> ( if ( ps , 0 , A ) + if ( ps , ( 2 x. A ) , 0 ) ) = ( A + 0 ) )
56		iffalse 4095	. . . . . . . . . 10 |- ( -. ps -> if ( ps , A , 0 ) = 0 )
57	56	adantl 482	. . . . . . . . 9 |- ( ( A e. CC /\ -. ps ) -> if ( ps , A , 0 ) = 0 )
58	57	oveq1d 6665	. . . . . . . 8 |- ( ( A e. CC /\ -. ps ) -> ( if ( ps , A , 0 ) + A ) = ( 0 + A ) )
59	50, 55, 58	3eqtr4d 2666	. . . . . . 7 |- ( ( A e. CC /\ -. ps ) -> ( if ( ps , 0 , A ) + if ( ps , ( 2 x. A ) , 0 ) ) = ( if ( ps , A , 0 ) + A ) )
60	47, 59	pm2.61dan 832	. . . . . 6 |- ( A e. CC -> ( if ( ps , 0 , A ) + if ( ps , ( 2 x. A ) , 0 ) ) = ( if ( ps , A , 0 ) + A ) )
61	60	ad2antrr 762	. . . . 5 |- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> ( if ( ps , 0 , A ) + if ( ps , ( 2 x. A ) , 0 ) ) = ( if ( ps , A , 0 ) + A ) )
62		ifnot 4133	. . . . . . 7 |- if ( -. ps , A , 0 ) = if ( ps , 0 , A )
63		nbn2 360	. . . . . . . . 9 |- ( -. ph -> ( -. ps <-> ( ph <-> ps ) ) )
64	63	adantl 482	. . . . . . . 8 |- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> ( -. ps <-> ( ph <-> ps ) ) )
65	64	ifbid 4108	. . . . . . 7 |- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> if ( -. ps , A , 0 ) = if ( ( ph <-> ps ) , A , 0 ) )
66	62, 65	syl5eqr 2670	. . . . . 6 |- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> if ( ps , 0 , A ) = if ( ( ph <-> ps ) , A , 0 ) )
67		biorf 420	. . . . . . . 8 |- ( -. ph -> ( ps <-> ( ph \/ ps ) ) )
68	67	adantl 482	. . . . . . 7 |- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> ( ps <-> ( ph \/ ps ) ) )
69	68	ifbid 4108	. . . . . 6 |- ( ( ( A e. CC /\ ch ) /\ -. ph ) -> if ( ps , ( 2 x. A ) , 0 ) = if ( ( ph \/ ps ) , ( 2 x. A ) , 0 ) )
70	66, 69	oveq12d 6668	. . . . 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 ) ) )
71	31, 61, 70	3eqtr2rd 2663	. . . 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 ) )
72	24, 71	pm2.61dan 832	. . 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 1540	. . . . . . 7 |- (hadd ( ch , ph , ps ) <-> hadd ( ph , ps , ch ) )
74		had1 1542	. . . . . . 7 |- ( ch -> (hadd ( ch , ph , ps ) <-> ( ph <-> ps ) ) )
75	73, 74	syl5bbr 274	. . . . . 6 |- ( ch -> (hadd ( ph , ps , ch ) <-> ( ph <-> ps ) ) )
76	75	adantl 482	. . . . 5 |- ( ( A e. CC /\ ch ) -> (hadd ( ph , ps , ch ) <-> ( ph <-> ps ) ) )
77	76	ifbid 4108	. . . 4 |- ( ( A e. CC /\ ch ) -> if (hadd ( ph , ps , ch ) , A , 0 ) = if ( ( ph <-> ps ) , A , 0 ) )
78		cad1 1555	. . . . . 6 |- ( ch -> (cadd ( ph , ps , ch ) <-> ( ph \/ ps ) ) )
79	78	adantl 482	. . . . 5 |- ( ( A e. CC /\ ch ) -> (cadd ( ph , ps , ch ) <-> ( ph \/ ps ) ) )
80	79	ifbid 4108	. . . 4 |- ( ( A e. CC /\ ch ) -> if (cadd ( ph , ps , ch ) , ( 2 x. A ) , 0 ) = if ( ( ph \/ ps ) , ( 2 x. A ) , 0 ) )
81	77, 80	oveq12d 6668	. . 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 4092	. . . . 5 |- ( ch -> if ( ch , A , 0 ) = A )
83	82	adantl 482	. . . 4 |- ( ( A e. CC /\ ch ) -> if ( ch , A , 0 ) = A )
84	83	oveq2d 6666	. . 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 ) )
85	72, 81, 84	3eqtr4d 2666	. 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 ) ) )
86	20	adantl 482	. . . . . 6 |- ( ( ( A e. CC /\ -. ch ) /\ ph ) -> if ( ph , A , 0 ) = A )
87	86	oveq1d 6665	. . . . 5 |- ( ( ( A e. CC /\ -. ch ) /\ ph ) -> ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) = ( A + if ( ps , A , 0 ) ) )
88	45	oveq2d 6666	. . . . . . . 8 |- ( ( A e. CC /\ ps ) -> ( A + if ( ps , A , 0 ) ) = ( A + A ) )
89	38, 43, 88	3eqtr4d 2666	. . . . . . 7 |- ( ( A e. CC /\ ps ) -> ( if ( ps , 0 , A ) + if ( ps , ( 2 x. A ) , 0 ) ) = ( A + if ( ps , A , 0 ) ) )
90	54, 57	eqtr4d 2659	. . . . . . . 8 |- ( ( A e. CC /\ -. ps ) -> if ( ps , ( 2 x. A ) , 0 ) = if ( ps , A , 0 ) )
91	52, 90	oveq12d 6668	. . . . . . 7 |- ( ( A e. CC /\ -. ps ) -> ( if ( ps , 0 , A ) + if ( ps , ( 2 x. A ) , 0 ) ) = ( A + if ( ps , A , 0 ) ) )
92	89, 91	pm2.61dan 832	. . . . . 6 |- ( A e. CC -> ( if ( ps , 0 , A ) + if ( ps , ( 2 x. A ) , 0 ) ) = ( A + if ( ps , A , 0 ) ) )
93	92	ad2antrr 762	. . . . 5 |- ( ( ( A e. CC /\ -. ch ) /\ ph ) -> ( if ( ps , 0 , A ) + if ( ps , ( 2 x. A ) , 0 ) ) = ( A + if ( ps , A , 0 ) ) )
94	9	adantl 482	. . . . . . . . . 10 |- ( ( ( A e. CC /\ -. ch ) /\ ph ) -> ( ps <-> ( ph <-> ps ) ) )
95	94	notbid 308	. . . . . . . . 9 |- ( ( ( A e. CC /\ -. ch ) /\ ph ) -> ( -. ps <-> -. ( ph <-> ps ) ) )
96		df-xor 1465	. . . . . . . . 9 |- ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) )
97	95, 96	syl6bbr 278	. . . . . . . 8 |- ( ( ( A e. CC /\ -. ch ) /\ ph ) -> ( -. ps <-> ( ph \/_ ps ) ) )
98	97	ifbid 4108	. . . . . . 7 |- ( ( ( A e. CC /\ -. ch ) /\ ph ) -> if ( -. ps , A , 0 ) = if ( ( ph \/_ ps ) , A , 0 ) )
99	62, 98	syl5eqr 2670	. . . . . 6 |- ( ( ( A e. CC /\ -. ch ) /\ ph ) -> if ( ps , 0 , A ) = if ( ( ph \/_ ps ) , A , 0 ) )
100		ibar 525	. . . . . . . 8 |- ( ph -> ( ps <-> ( ph /\ ps ) ) )
101	100	adantl 482	. . . . . . 7 |- ( ( ( A e. CC /\ -. ch ) /\ ph ) -> ( ps <-> ( ph /\ ps ) ) )
102	101	ifbid 4108	. . . . . 6 |- ( ( ( A e. CC /\ -. ch ) /\ ph ) -> if ( ps , ( 2 x. A ) , 0 ) = if ( ( ph /\ ps ) , ( 2 x. A ) , 0 ) )
103	99, 102	oveq12d 6668	. . . . 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 ) ) )
104	87, 93, 103	3eqtr2rd 2663	. . . 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 798	. . . . . . 7 |- ( ( ( ( A e. CC /\ -. ch ) /\ -. ph ) /\ ps ) -> A e. CC )
106		0cnd 10033	. . . . . . 7 |- ( ( ( ( A e. CC /\ -. ch ) /\ -. ph ) /\ -. ps ) -> 0 e. CC )
107	105, 106	ifclda 4120	. . . . . 6 |- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> if ( ps , A , 0 ) e. CC )
108		0cnd 10033	. . . . . 6 |- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> 0 e. CC )
109	107, 108	addcomd 10238	. . . . 5 |- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> ( if ( ps , A , 0 ) + 0 ) = ( 0 + if ( ps , A , 0 ) ) )
110	63	adantl 482	. . . . . . . . 9 |- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> ( -. ps <-> ( ph <-> ps ) ) )
111	110	con1bid 345	. . . . . . . 8 |- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> ( -. ( ph <-> ps ) <-> ps ) )
112	96, 111	syl5bb 272	. . . . . . 7 |- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> ( ( ph \/_ ps ) <-> ps ) )
113	112	ifbid 4108	. . . . . 6 |- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> if ( ( ph \/_ ps ) , A , 0 ) = if ( ps , A , 0 ) )
114		simpr 477	. . . . . . . 8 |- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> -. ph )
115	114	intnanrd 963	. . . . . . 7 |- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> -. ( ph /\ ps ) )
116		iffalse 4095	. . . . . . 7 |- ( -. ( ph /\ ps ) -> if ( ( ph /\ ps ) , ( 2 x. A ) , 0 ) = 0 )
117	115, 116	syl 17	. . . . . 6 |- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> if ( ( ph /\ ps ) , ( 2 x. A ) , 0 ) = 0 )
118	113, 117	oveq12d 6668	. . . . 5 |- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> ( if ( ( ph \/_ ps ) , A , 0 ) + if ( ( ph /\ ps ) , ( 2 x. A ) , 0 ) ) = ( if ( ps , A , 0 ) + 0 ) )
119	25	adantl 482	. . . . . 6 |- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> if ( ph , A , 0 ) = 0 )
120	119	oveq1d 6665	. . . . 5 |- ( ( ( A e. CC /\ -. ch ) /\ -. ph ) -> ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) = ( 0 + if ( ps , A , 0 ) ) )
121	109, 118, 120	3eqtr4d 2666	. . . 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 ) ) )
122	104, 121	pm2.61dan 832	. . 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 1543	. . . . . . 7 |- ( -. ch -> (hadd ( ch , ph , ps ) <-> ( ph \/_ ps ) ) )
124	73, 123	syl5bbr 274	. . . . . 6 |- ( -. ch -> (hadd ( ph , ps , ch ) <-> ( ph \/_ ps ) ) )
125	124	adantl 482	. . . . 5 |- ( ( A e. CC /\ -. ch ) -> (hadd ( ph , ps , ch ) <-> ( ph \/_ ps ) ) )
126	125	ifbid 4108	. . . 4 |- ( ( A e. CC /\ -. ch ) -> if (hadd ( ph , ps , ch ) , A , 0 ) = if ( ( ph \/_ ps ) , A , 0 ) )
127		cad0 1556	. . . . . 6 |- ( -. ch -> (cadd ( ph , ps , ch ) <-> ( ph /\ ps ) ) )
128	127	adantl 482	. . . . 5 |- ( ( A e. CC /\ -. ch ) -> (cadd ( ph , ps , ch ) <-> ( ph /\ ps ) ) )
129	128	ifbid 4108	. . . 4 |- ( ( A e. CC /\ -. ch ) -> if (cadd ( ph , ps , ch ) , ( 2 x. A ) , 0 ) = if ( ( ph /\ ps ) , ( 2 x. A ) , 0 ) )
130	126, 129	oveq12d 6668	. . 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 4095	. . . . 5 |- ( -. ch -> if ( ch , A , 0 ) = 0 )
132	131	oveq2d 6666	. . . 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 4130	. . . . . . 7 |- ( ( A e. CC /\ 0 e. CC ) -> if ( ph , A , 0 ) e. CC )
134	1, 133	mpan2 707	. . . . . 6 |- ( A e. CC -> if ( ph , A , 0 ) e. CC )
135	134, 3	addcld 10059	. . . . 5 |- ( A e. CC -> ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) e. CC )
136	135	addid1d 10236	. . . 4 |- ( A e. CC -> ( ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) + 0 ) = ( if ( ph , A , 0 ) + if ( ps , A , 0 ) ) )
137	132, 136	sylan9eqr 2678	. . 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 ) ) )
138	122, 130, 137	3eqtr4d 2666	. 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 ) ) )
139	85, 138	pm2.61dan 832	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 ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/_ wxo 1464   = wceq 1483  haddwhad 1532  caddwcad 1545   e. wcel 1990   ifcif 4086  (class class class)co 6650   CCcc 9934   0cc0 9936   + caddc 9939   x. cmul 9941   2c2 11070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-xor 1465  df-tru 1486  df-had 1533  df-cad 1546  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-2 11079

This theorem is referenced by:  sadadd2lem  15181