Theorem mogoldbblem 41629

Description: Lemma for mogoldbb 41673. (Contributed by AV, 26-Dec-2021.)

Assertion

Ref	Expression
mogoldbblem	|- ( ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N e. Even /\ ( N + 2 ) = ( ( P + Q ) + R ) ) -> E. p e. Prime E. q e. Prime N = ( p + q ) )

Distinct variable groups:   N, p, q   P, p, q   Q, p, q   R, p, q

Proof of Theorem mogoldbblem

Step	Hyp	Ref	Expression
1		2evenALTV 41603	. . . . 5 |- 2 e. Even
2		epee 41614	. . . . 5 |- ( ( N e. Even /\ 2 e. Even ) -> ( N + 2 ) e. Even )
3	1, 2	mpan2 707	. . . 4 |- ( N e. Even -> ( N + 2 ) e. Even )
4	3	3ad2ant2 1083	. . 3 |- ( ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N e. Even /\ ( N + 2 ) = ( ( P + Q ) + R ) ) -> ( N + 2 ) e. Even )
5		simp1 1061	. . 3 |- ( ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N e. Even /\ ( N + 2 ) = ( ( P + Q ) + R ) ) -> ( P e. Prime /\ Q e. Prime /\ R e. Prime ) )
6		simp3 1063	. . 3 |- ( ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N e. Even /\ ( N + 2 ) = ( ( P + Q ) + R ) ) -> ( N + 2 ) = ( ( P + Q ) + R ) )
7		even3prm2 41628	. . 3 |- ( ( ( N + 2 ) e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ ( N + 2 ) = ( ( P + Q ) + R ) ) -> ( P = 2 \/ Q = 2 \/ R = 2 ) )
8	4, 5, 6, 7	syl3anc 1326	. 2 |- ( ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N e. Even /\ ( N + 2 ) = ( ( P + Q ) + R ) ) -> ( P = 2 \/ Q = 2 \/ R = 2 ) )
9		oveq1 6657	. . . . . . . . . . 11 |- ( P = 2 -> ( P + Q ) = ( 2 + Q ) )
10	9	oveq1d 6665	. . . . . . . . . 10 |- ( P = 2 -> ( ( P + Q ) + R ) = ( ( 2 + Q ) + R ) )
11	10	eqeq2d 2632	. . . . . . . . 9 |- ( P = 2 -> ( ( N + 2 ) = ( ( P + Q ) + R ) <-> ( N + 2 ) = ( ( 2 + Q ) + R ) ) )
12		2cnd 11093	. . . . . . . . . . . . . . 15 |- ( ( Q e. Prime /\ R e. Prime ) -> 2 e. CC )
13		prmz 15389	. . . . . . . . . . . . . . . . 17 |- ( Q e. Prime -> Q e. ZZ )
14	13	zcnd 11483	. . . . . . . . . . . . . . . 16 |- ( Q e. Prime -> Q e. CC )
15	14	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( Q e. Prime /\ R e. Prime ) -> Q e. CC )
16		prmz 15389	. . . . . . . . . . . . . . . . 17 |- ( R e. Prime -> R e. ZZ )
17	16	zcnd 11483	. . . . . . . . . . . . . . . 16 |- ( R e. Prime -> R e. CC )
18	17	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( Q e. Prime /\ R e. Prime ) -> R e. CC )
19		simp1 1061	. . . . . . . . . . . . . . . 16 |- ( ( 2 e. CC /\ Q e. CC /\ R e. CC ) -> 2 e. CC )
20		addcl 10018	. . . . . . . . . . . . . . . . 17 |- ( ( Q e. CC /\ R e. CC ) -> ( Q + R ) e. CC )
21	20	3adant1 1079	. . . . . . . . . . . . . . . 16 |- ( ( 2 e. CC /\ Q e. CC /\ R e. CC ) -> ( Q + R ) e. CC )
22		addass 10023	. . . . . . . . . . . . . . . 16 |- ( ( 2 e. CC /\ Q e. CC /\ R e. CC ) -> ( ( 2 + Q ) + R ) = ( 2 + ( Q + R ) ) )
23	19, 21, 22	comraddd 10250	. . . . . . . . . . . . . . 15 |- ( ( 2 e. CC /\ Q e. CC /\ R e. CC ) -> ( ( 2 + Q ) + R ) = ( ( Q + R ) + 2 ) )
24	12, 15, 18, 23	syl3anc 1326	. . . . . . . . . . . . . 14 |- ( ( Q e. Prime /\ R e. Prime ) -> ( ( 2 + Q ) + R ) = ( ( Q + R ) + 2 ) )
25	24	eqeq2d 2632	. . . . . . . . . . . . 13 |- ( ( Q e. Prime /\ R e. Prime ) -> ( ( N + 2 ) = ( ( 2 + Q ) + R ) <-> ( N + 2 ) = ( ( Q + R ) + 2 ) ) )
26	25	adantr 481	. . . . . . . . . . . 12 |- ( ( ( Q e. Prime /\ R e. Prime ) /\ N e. Even ) -> ( ( N + 2 ) = ( ( 2 + Q ) + R ) <-> ( N + 2 ) = ( ( Q + R ) + 2 ) ) )
27		evenz 41543	. . . . . . . . . . . . . . 15 |- ( N e. Even -> N e. ZZ )
28	27	zcnd 11483	. . . . . . . . . . . . . 14 |- ( N e. Even -> N e. CC )
29	28	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( Q e. Prime /\ R e. Prime ) /\ N e. Even ) -> N e. CC )
30		zaddcl 11417	. . . . . . . . . . . . . . . 16 |- ( ( Q e. ZZ /\ R e. ZZ ) -> ( Q + R ) e. ZZ )
31	13, 16, 30	syl2an 494	. . . . . . . . . . . . . . 15 |- ( ( Q e. Prime /\ R e. Prime ) -> ( Q + R ) e. ZZ )
32	31	zcnd 11483	. . . . . . . . . . . . . 14 |- ( ( Q e. Prime /\ R e. Prime ) -> ( Q + R ) e. CC )
33	32	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( Q e. Prime /\ R e. Prime ) /\ N e. Even ) -> ( Q + R ) e. CC )
34		2cnd 11093	. . . . . . . . . . . . 13 |- ( ( ( Q e. Prime /\ R e. Prime ) /\ N e. Even ) -> 2 e. CC )
35	29, 33, 34	addcan2d 10240	. . . . . . . . . . . 12 |- ( ( ( Q e. Prime /\ R e. Prime ) /\ N e. Even ) -> ( ( N + 2 ) = ( ( Q + R ) + 2 ) <-> N = ( Q + R ) ) )
36	26, 35	bitrd 268	. . . . . . . . . . 11 |- ( ( ( Q e. Prime /\ R e. Prime ) /\ N e. Even ) -> ( ( N + 2 ) = ( ( 2 + Q ) + R ) <-> N = ( Q + R ) ) )
37		simpll 790	. . . . . . . . . . . . . 14 |- ( ( ( Q e. Prime /\ R e. Prime ) /\ N = ( Q + R ) ) -> Q e. Prime )
38		oveq1 6657	. . . . . . . . . . . . . . . . 17 |- ( p = Q -> ( p + q ) = ( Q + q ) )
39	38	eqeq2d 2632	. . . . . . . . . . . . . . . 16 |- ( p = Q -> ( N = ( p + q ) <-> N = ( Q + q ) ) )
40	39	rexbidv 3052	. . . . . . . . . . . . . . 15 |- ( p = Q -> ( E. q e. Prime N = ( p + q ) <-> E. q e. Prime N = ( Q + q ) ) )
41	40	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( ( Q e. Prime /\ R e. Prime ) /\ N = ( Q + R ) ) /\ p = Q ) -> ( E. q e. Prime N = ( p + q ) <-> E. q e. Prime N = ( Q + q ) ) )
42		simplr 792	. . . . . . . . . . . . . . 15 |- ( ( ( Q e. Prime /\ R e. Prime ) /\ N = ( Q + R ) ) -> R e. Prime )
43		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( ( Q e. Prime /\ R e. Prime ) /\ N = ( Q + R ) ) -> N = ( Q + R ) )
44		oveq2 6658	. . . . . . . . . . . . . . . . 17 |- ( q = R -> ( Q + q ) = ( Q + R ) )
45	44	eqcomd 2628	. . . . . . . . . . . . . . . 16 |- ( q = R -> ( Q + R ) = ( Q + q ) )
46	43, 45	sylan9eq 2676	. . . . . . . . . . . . . . 15 |- ( ( ( ( Q e. Prime /\ R e. Prime ) /\ N = ( Q + R ) ) /\ q = R ) -> N = ( Q + q ) )
47	42, 46	rspcedeq2vd 3319	. . . . . . . . . . . . . 14 |- ( ( ( Q e. Prime /\ R e. Prime ) /\ N = ( Q + R ) ) -> E. q e. Prime N = ( Q + q ) )
48	37, 41, 47	rspcedvd 3317	. . . . . . . . . . . . 13 |- ( ( ( Q e. Prime /\ R e. Prime ) /\ N = ( Q + R ) ) -> E. p e. Prime E. q e. Prime N = ( p + q ) )
49	48	ex 450	. . . . . . . . . . . 12 |- ( ( Q e. Prime /\ R e. Prime ) -> ( N = ( Q + R ) -> E. p e. Prime E. q e. Prime N = ( p + q ) ) )
50	49	adantr 481	. . . . . . . . . . 11 |- ( ( ( Q e. Prime /\ R e. Prime ) /\ N e. Even ) -> ( N = ( Q + R ) -> E. p e. Prime E. q e. Prime N = ( p + q ) ) )
51	36, 50	sylbid 230	. . . . . . . . . 10 |- ( ( ( Q e. Prime /\ R e. Prime ) /\ N e. Even ) -> ( ( N + 2 ) = ( ( 2 + Q ) + R ) -> E. p e. Prime E. q e. Prime N = ( p + q ) ) )
52	51	com12 32	. . . . . . . . 9 |- ( ( N + 2 ) = ( ( 2 + Q ) + R ) -> ( ( ( Q e. Prime /\ R e. Prime ) /\ N e. Even ) -> E. p e. Prime E. q e. Prime N = ( p + q ) ) )
53	11, 52	syl6bi 243	. . . . . . . 8 |- ( P = 2 -> ( ( N + 2 ) = ( ( P + Q ) + R ) -> ( ( ( Q e. Prime /\ R e. Prime ) /\ N e. Even ) -> E. p e. Prime E. q e. Prime N = ( p + q ) ) ) )
54	53	com13 88	. . . . . . 7 |- ( ( ( Q e. Prime /\ R e. Prime ) /\ N e. Even ) -> ( ( N + 2 ) = ( ( P + Q ) + R ) -> ( P = 2 -> E. p e. Prime E. q e. Prime N = ( p + q ) ) ) )
55	54	ex 450	. . . . . 6 |- ( ( Q e. Prime /\ R e. Prime ) -> ( N e. Even -> ( ( N + 2 ) = ( ( P + Q ) + R ) -> ( P = 2 -> E. p e. Prime E. q e. Prime N = ( p + q ) ) ) ) )
56	55	3adant1 1079	. . . . 5 |- ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) -> ( N e. Even -> ( ( N + 2 ) = ( ( P + Q ) + R ) -> ( P = 2 -> E. p e. Prime E. q e. Prime N = ( p + q ) ) ) ) )
57	56	3imp 1256	. . . 4 |- ( ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N e. Even /\ ( N + 2 ) = ( ( P + Q ) + R ) ) -> ( P = 2 -> E. p e. Prime E. q e. Prime N = ( p + q ) ) )
58	57	com12 32	. . 3 |- ( P = 2 -> ( ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N e. Even /\ ( N + 2 ) = ( ( P + Q ) + R ) ) -> E. p e. Prime E. q e. Prime N = ( p + q ) ) )
59		oveq2 6658	. . . . . . . . . . 11 |- ( Q = 2 -> ( P + Q ) = ( P + 2 ) )
60	59	oveq1d 6665	. . . . . . . . . 10 |- ( Q = 2 -> ( ( P + Q ) + R ) = ( ( P + 2 ) + R ) )
61	60	eqeq2d 2632	. . . . . . . . 9 |- ( Q = 2 -> ( ( N + 2 ) = ( ( P + Q ) + R ) <-> ( N + 2 ) = ( ( P + 2 ) + R ) ) )
62		prmz 15389	. . . . . . . . . . . . . . . . . 18 |- ( P e. Prime -> P e. ZZ )
63	62	zcnd 11483	. . . . . . . . . . . . . . . . 17 |- ( P e. Prime -> P e. CC )
64	63	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ R e. Prime ) -> P e. CC )
65		2cnd 11093	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ R e. Prime ) -> 2 e. CC )
66	17	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ R e. Prime ) -> R e. CC )
67	64, 65, 66	3jca 1242	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ R e. Prime ) -> ( P e. CC /\ 2 e. CC /\ R e. CC ) )
68	67	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ R e. Prime ) /\ N e. Even ) -> ( P e. CC /\ 2 e. CC /\ R e. CC ) )
69		add32 10254	. . . . . . . . . . . . . 14 |- ( ( P e. CC /\ 2 e. CC /\ R e. CC ) -> ( ( P + 2 ) + R ) = ( ( P + R ) + 2 ) )
70	68, 69	syl 17	. . . . . . . . . . . . 13 |- ( ( ( P e. Prime /\ R e. Prime ) /\ N e. Even ) -> ( ( P + 2 ) + R ) = ( ( P + R ) + 2 ) )
71	70	eqeq2d 2632	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ R e. Prime ) /\ N e. Even ) -> ( ( N + 2 ) = ( ( P + 2 ) + R ) <-> ( N + 2 ) = ( ( P + R ) + 2 ) ) )
72	28	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( P e. Prime /\ R e. Prime ) /\ N e. Even ) -> N e. CC )
73		zaddcl 11417	. . . . . . . . . . . . . . . 16 |- ( ( P e. ZZ /\ R e. ZZ ) -> ( P + R ) e. ZZ )
74	62, 16, 73	syl2an 494	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ R e. Prime ) -> ( P + R ) e. ZZ )
75	74	zcnd 11483	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ R e. Prime ) -> ( P + R ) e. CC )
76	75	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( P e. Prime /\ R e. Prime ) /\ N e. Even ) -> ( P + R ) e. CC )
77		2cnd 11093	. . . . . . . . . . . . 13 |- ( ( ( P e. Prime /\ R e. Prime ) /\ N e. Even ) -> 2 e. CC )
78	72, 76, 77	addcan2d 10240	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ R e. Prime ) /\ N e. Even ) -> ( ( N + 2 ) = ( ( P + R ) + 2 ) <-> N = ( P + R ) ) )
79	71, 78	bitrd 268	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ R e. Prime ) /\ N e. Even ) -> ( ( N + 2 ) = ( ( P + 2 ) + R ) <-> N = ( P + R ) ) )
80		simpll 790	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ R e. Prime ) /\ N = ( P + R ) ) -> P e. Prime )
81		oveq1 6657	. . . . . . . . . . . . . . . . 17 |- ( p = P -> ( p + q ) = ( P + q ) )
82	81	eqeq2d 2632	. . . . . . . . . . . . . . . 16 |- ( p = P -> ( N = ( p + q ) <-> N = ( P + q ) ) )
83	82	rexbidv 3052	. . . . . . . . . . . . . . 15 |- ( p = P -> ( E. q e. Prime N = ( p + q ) <-> E. q e. Prime N = ( P + q ) ) )
84	83	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( ( P e. Prime /\ R e. Prime ) /\ N = ( P + R ) ) /\ p = P ) -> ( E. q e. Prime N = ( p + q ) <-> E. q e. Prime N = ( P + q ) ) )
85		simplr 792	. . . . . . . . . . . . . . 15 |- ( ( ( P e. Prime /\ R e. Prime ) /\ N = ( P + R ) ) -> R e. Prime )
86		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( ( P e. Prime /\ R e. Prime ) /\ N = ( P + R ) ) -> N = ( P + R ) )
87		oveq2 6658	. . . . . . . . . . . . . . . . 17 |- ( q = R -> ( P + q ) = ( P + R ) )
88	87	eqcomd 2628	. . . . . . . . . . . . . . . 16 |- ( q = R -> ( P + R ) = ( P + q ) )
89	86, 88	sylan9eq 2676	. . . . . . . . . . . . . . 15 |- ( ( ( ( P e. Prime /\ R e. Prime ) /\ N = ( P + R ) ) /\ q = R ) -> N = ( P + q ) )
90	85, 89	rspcedeq2vd 3319	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ R e. Prime ) /\ N = ( P + R ) ) -> E. q e. Prime N = ( P + q ) )
91	80, 84, 90	rspcedvd 3317	. . . . . . . . . . . . 13 |- ( ( ( P e. Prime /\ R e. Prime ) /\ N = ( P + R ) ) -> E. p e. Prime E. q e. Prime N = ( p + q ) )
92	91	ex 450	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ R e. Prime ) -> ( N = ( P + R ) -> E. p e. Prime E. q e. Prime N = ( p + q ) ) )
93	92	adantr 481	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ R e. Prime ) /\ N e. Even ) -> ( N = ( P + R ) -> E. p e. Prime E. q e. Prime N = ( p + q ) ) )
94	79, 93	sylbid 230	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ R e. Prime ) /\ N e. Even ) -> ( ( N + 2 ) = ( ( P + 2 ) + R ) -> E. p e. Prime E. q e. Prime N = ( p + q ) ) )
95	94	com12 32	. . . . . . . . 9 |- ( ( N + 2 ) = ( ( P + 2 ) + R ) -> ( ( ( P e. Prime /\ R e. Prime ) /\ N e. Even ) -> E. p e. Prime E. q e. Prime N = ( p + q ) ) )
96	61, 95	syl6bi 243	. . . . . . . 8 |- ( Q = 2 -> ( ( N + 2 ) = ( ( P + Q ) + R ) -> ( ( ( P e. Prime /\ R e. Prime ) /\ N e. Even ) -> E. p e. Prime E. q e. Prime N = ( p + q ) ) ) )
97	96	com13 88	. . . . . . 7 |- ( ( ( P e. Prime /\ R e. Prime ) /\ N e. Even ) -> ( ( N + 2 ) = ( ( P + Q ) + R ) -> ( Q = 2 -> E. p e. Prime E. q e. Prime N = ( p + q ) ) ) )
98	97	ex 450	. . . . . 6 |- ( ( P e. Prime /\ R e. Prime ) -> ( N e. Even -> ( ( N + 2 ) = ( ( P + Q ) + R ) -> ( Q = 2 -> E. p e. Prime E. q e. Prime N = ( p + q ) ) ) ) )
99	98	3adant2 1080	. . . . 5 |- ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) -> ( N e. Even -> ( ( N + 2 ) = ( ( P + Q ) + R ) -> ( Q = 2 -> E. p e. Prime E. q e. Prime N = ( p + q ) ) ) ) )
100	99	3imp 1256	. . . 4 |- ( ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N e. Even /\ ( N + 2 ) = ( ( P + Q ) + R ) ) -> ( Q = 2 -> E. p e. Prime E. q e. Prime N = ( p + q ) ) )
101	100	com12 32	. . 3 |- ( Q = 2 -> ( ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N e. Even /\ ( N + 2 ) = ( ( P + Q ) + R ) ) -> E. p e. Prime E. q e. Prime N = ( p + q ) ) )
102		oveq2 6658	. . . . . . . . . 10 |- ( R = 2 -> ( ( P + Q ) + R ) = ( ( P + Q ) + 2 ) )
103	102	eqeq2d 2632	. . . . . . . . 9 |- ( R = 2 -> ( ( N + 2 ) = ( ( P + Q ) + R ) <-> ( N + 2 ) = ( ( P + Q ) + 2 ) ) )
104	28	adantl 482	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ N e. Even ) -> N e. CC )
105		zaddcl 11417	. . . . . . . . . . . . . . 15 |- ( ( P e. ZZ /\ Q e. ZZ ) -> ( P + Q ) e. ZZ )
106	62, 13, 105	syl2an 494	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ Q e. Prime ) -> ( P + Q ) e. ZZ )
107	106	zcnd 11483	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ Q e. Prime ) -> ( P + Q ) e. CC )
108	107	adantr 481	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ N e. Even ) -> ( P + Q ) e. CC )
109		2cnd 11093	. . . . . . . . . . . 12 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ N e. Even ) -> 2 e. CC )
110	104, 108, 109	addcan2d 10240	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ N e. Even ) -> ( ( N + 2 ) = ( ( P + Q ) + 2 ) <-> N = ( P + Q ) ) )
111		simpll 790	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ N = ( P + Q ) ) -> P e. Prime )
112	83	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( ( P e. Prime /\ Q e. Prime ) /\ N = ( P + Q ) ) /\ p = P ) -> ( E. q e. Prime N = ( p + q ) <-> E. q e. Prime N = ( P + q ) ) )
113		simplr 792	. . . . . . . . . . . . . . 15 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ N = ( P + Q ) ) -> Q e. Prime )
114		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ N = ( P + Q ) ) -> N = ( P + Q ) )
115		oveq2 6658	. . . . . . . . . . . . . . . . 17 |- ( q = Q -> ( P + q ) = ( P + Q ) )
116	115	eqcomd 2628	. . . . . . . . . . . . . . . 16 |- ( q = Q -> ( P + Q ) = ( P + q ) )
117	114, 116	sylan9eq 2676	. . . . . . . . . . . . . . 15 |- ( ( ( ( P e. Prime /\ Q e. Prime ) /\ N = ( P + Q ) ) /\ q = Q ) -> N = ( P + q ) )
118	113, 117	rspcedeq2vd 3319	. . . . . . . . . . . . . 14 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ N = ( P + Q ) ) -> E. q e. Prime N = ( P + q ) )
119	111, 112, 118	rspcedvd 3317	. . . . . . . . . . . . 13 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ N = ( P + Q ) ) -> E. p e. Prime E. q e. Prime N = ( p + q ) )
120	119	ex 450	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ Q e. Prime ) -> ( N = ( P + Q ) -> E. p e. Prime E. q e. Prime N = ( p + q ) ) )
121	120	adantr 481	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ N e. Even ) -> ( N = ( P + Q ) -> E. p e. Prime E. q e. Prime N = ( p + q ) ) )
122	110, 121	sylbid 230	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ N e. Even ) -> ( ( N + 2 ) = ( ( P + Q ) + 2 ) -> E. p e. Prime E. q e. Prime N = ( p + q ) ) )
123	122	com12 32	. . . . . . . . 9 |- ( ( N + 2 ) = ( ( P + Q ) + 2 ) -> ( ( ( P e. Prime /\ Q e. Prime ) /\ N e. Even ) -> E. p e. Prime E. q e. Prime N = ( p + q ) ) )
124	103, 123	syl6bi 243	. . . . . . . 8 |- ( R = 2 -> ( ( N + 2 ) = ( ( P + Q ) + R ) -> ( ( ( P e. Prime /\ Q e. Prime ) /\ N e. Even ) -> E. p e. Prime E. q e. Prime N = ( p + q ) ) ) )
125	124	com13 88	. . . . . . 7 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ N e. Even ) -> ( ( N + 2 ) = ( ( P + Q ) + R ) -> ( R = 2 -> E. p e. Prime E. q e. Prime N = ( p + q ) ) ) )
126	125	ex 450	. . . . . 6 |- ( ( P e. Prime /\ Q e. Prime ) -> ( N e. Even -> ( ( N + 2 ) = ( ( P + Q ) + R ) -> ( R = 2 -> E. p e. Prime E. q e. Prime N = ( p + q ) ) ) ) )
127	126	3adant3 1081	. . . . 5 |- ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) -> ( N e. Even -> ( ( N + 2 ) = ( ( P + Q ) + R ) -> ( R = 2 -> E. p e. Prime E. q e. Prime N = ( p + q ) ) ) ) )
128	127	3imp 1256	. . . 4 |- ( ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N e. Even /\ ( N + 2 ) = ( ( P + Q ) + R ) ) -> ( R = 2 -> E. p e. Prime E. q e. Prime N = ( p + q ) ) )
129	128	com12 32	. . 3 |- ( R = 2 -> ( ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N e. Even /\ ( N + 2 ) = ( ( P + Q ) + R ) ) -> E. p e. Prime E. q e. Prime N = ( p + q ) ) )
130	58, 101, 129	3jaoi 1391	. 2 |- ( ( P = 2 \/ Q = 2 \/ R = 2 ) -> ( ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N e. Even /\ ( N + 2 ) = ( ( P + Q ) + R ) ) -> E. p e. Prime E. q e. Prime N = ( p + q ) ) )
131	8, 130	mpcom 38	1 |- ( ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N e. Even /\ ( N + 2 ) = ( ( P + Q ) + R ) ) -> E. p e. Prime E. q e. Prime N = ( p + q ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913  (class class class)co 6650   CCcc 9934   + caddc 9939   2c2 11070   ZZcz 11377   Primecprime 15385   Even ceven 41537

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-cnex 9992  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  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-prm 15386  df-even 41539  df-odd 41540

This theorem is referenced by:  mogoldbb  41673