Theorem jm2.27b 37573

Description: Lemma for jm2.27 37575. Expand existential quantifiers for reverse direction. (Contributed by Stefan O'Rear, 4-Oct-2014.)

Hypotheses

Ref	Expression
jm2.27a1	|- ( ph -> A e. ( ZZ>= ` 2 ) )
jm2.27a2	|- ( ph -> B e. NN )
jm2.27a3	|- ( ph -> C e. NN )
jm2.27a4	|- ( ph -> D e. NN0 )
jm2.27a5	|- ( ph -> E e. NN0 )
jm2.27a6	|- ( ph -> F e. NN0 )
jm2.27a7	|- ( ph -> G e. NN0 )
jm2.27a8	|- ( ph -> H e. NN0 )
jm2.27a9	|- ( ph -> I e. NN0 )
jm2.27a10	|- ( ph -> J e. NN0 )
jm2.27a11	|- ( ph -> ( ( D ^ 2 ) - ( ( ( A ^ 2 ) - 1 ) x. ( C ^ 2 ) ) ) = 1 )
jm2.27a12	|- ( ph -> ( ( F ^ 2 ) - ( ( ( A ^ 2 ) - 1 ) x. ( E ^ 2 ) ) ) = 1 )
jm2.27a13	|- ( ph -> G e. ( ZZ>= ` 2 ) )
jm2.27a14	|- ( ph -> ( ( I ^ 2 ) - ( ( ( G ^ 2 ) - 1 ) x. ( H ^ 2 ) ) ) = 1 )
jm2.27a15	|- ( ph -> E = ( ( J + 1 ) x. ( 2 x. ( C ^ 2 ) ) ) )
jm2.27a16	|- ( ph -> F || ( G - A ) )
jm2.27a17	|- ( ph -> ( 2 x. C ) || ( G - 1 ) )
jm2.27a18	|- ( ph -> F || ( H - C ) )
jm2.27a19	|- ( ph -> ( 2 x. C ) || ( H - B ) )
jm2.27a20	|- ( ph -> B <_ C )

Assertion

Ref	Expression
jm2.27b	|- ( ph -> C = ( A Yrm B ) )

Proof of Theorem jm2.27b

Dummy variables p q r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		jm2.27a11	. . 3 |- ( ph -> ( ( D ^ 2 ) - ( ( ( A ^ 2 ) - 1 ) x. ( C ^ 2 ) ) ) = 1 )
2		jm2.27a1	. . . 4 |- ( ph -> A e. ( ZZ>= ` 2 ) )
3		jm2.27a4	. . . 4 |- ( ph -> D e. NN0 )
4		jm2.27a3	. . . . 5 |- ( ph -> C e. NN )
5	4	nnzd 11481	. . . 4 |- ( ph -> C e. ZZ )
6		rmxycomplete 37482	. . . 4 |- ( ( A e. ( ZZ>= ` 2 ) /\ D e. NN0 /\ C e. ZZ ) -> ( ( ( D ^ 2 ) - ( ( ( A ^ 2 ) - 1 ) x. ( C ^ 2 ) ) ) = 1 <-> E. p e. ZZ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) )
7	2, 3, 5, 6	syl3anc 1326	. . 3 |- ( ph -> ( ( ( D ^ 2 ) - ( ( ( A ^ 2 ) - 1 ) x. ( C ^ 2 ) ) ) = 1 <-> E. p e. ZZ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) )
8	1, 7	mpbid 222	. 2 |- ( ph -> E. p e. ZZ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) )
9		jm2.27a12	. . . . 5 |- ( ph -> ( ( F ^ 2 ) - ( ( ( A ^ 2 ) - 1 ) x. ( E ^ 2 ) ) ) = 1 )
10	9	adantr 481	. . . 4 |- ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) -> ( ( F ^ 2 ) - ( ( ( A ^ 2 ) - 1 ) x. ( E ^ 2 ) ) ) = 1 )
11	2	adantr 481	. . . . 5 |- ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) -> A e. ( ZZ>= ` 2 ) )
12		jm2.27a6	. . . . . 6 |- ( ph -> F e. NN0 )
13	12	adantr 481	. . . . 5 |- ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) -> F e. NN0 )
14		jm2.27a5	. . . . . . 7 |- ( ph -> E e. NN0 )
15	14	nn0zd 11480	. . . . . 6 |- ( ph -> E e. ZZ )
16	15	adantr 481	. . . . 5 |- ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) -> E e. ZZ )
17		rmxycomplete 37482	. . . . 5 |- ( ( A e. ( ZZ>= ` 2 ) /\ F e. NN0 /\ E e. ZZ ) -> ( ( ( F ^ 2 ) - ( ( ( A ^ 2 ) - 1 ) x. ( E ^ 2 ) ) ) = 1 <-> E. q e. ZZ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) )
18	11, 13, 16, 17	syl3anc 1326	. . . 4 |- ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) -> ( ( ( F ^ 2 ) - ( ( ( A ^ 2 ) - 1 ) x. ( E ^ 2 ) ) ) = 1 <-> E. q e. ZZ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) )
19	10, 18	mpbid 222	. . 3 |- ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) -> E. q e. ZZ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) )
20		jm2.27a14	. . . . . 6 |- ( ph -> ( ( I ^ 2 ) - ( ( ( G ^ 2 ) - 1 ) x. ( H ^ 2 ) ) ) = 1 )
21	20	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) -> ( ( I ^ 2 ) - ( ( ( G ^ 2 ) - 1 ) x. ( H ^ 2 ) ) ) = 1 )
22		jm2.27a13	. . . . . . 7 |- ( ph -> G e. ( ZZ>= ` 2 ) )
23	22	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) -> G e. ( ZZ>= ` 2 ) )
24		jm2.27a9	. . . . . . 7 |- ( ph -> I e. NN0 )
25	24	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) -> I e. NN0 )
26		jm2.27a8	. . . . . . . 8 |- ( ph -> H e. NN0 )
27	26	nn0zd 11480	. . . . . . 7 |- ( ph -> H e. ZZ )
28	27	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) -> H e. ZZ )
29		rmxycomplete 37482	. . . . . 6 |- ( ( G e. ( ZZ>= ` 2 ) /\ I e. NN0 /\ H e. ZZ ) -> ( ( ( I ^ 2 ) - ( ( ( G ^ 2 ) - 1 ) x. ( H ^ 2 ) ) ) = 1 <-> E. r e. ZZ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) )
30	23, 25, 28, 29	syl3anc 1326	. . . . 5 |- ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) -> ( ( ( I ^ 2 ) - ( ( ( G ^ 2 ) - 1 ) x. ( H ^ 2 ) ) ) = 1 <-> E. r e. ZZ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) )
31	21, 30	mpbid 222	. . . 4 |- ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) -> E. r e. ZZ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) )
32	2	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> A e. ( ZZ>= ` 2 ) )
33		jm2.27a2	. . . . . 6 |- ( ph -> B e. NN )
34	33	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> B e. NN )
35	4	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> C e. NN )
36	3	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> D e. NN0 )
37	14	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> E e. NN0 )
38	12	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> F e. NN0 )
39		jm2.27a7	. . . . . 6 |- ( ph -> G e. NN0 )
40	39	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> G e. NN0 )
41	26	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> H e. NN0 )
42	24	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> I e. NN0 )
43		jm2.27a10	. . . . . 6 |- ( ph -> J e. NN0 )
44	43	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> J e. NN0 )
45	1	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> ( ( D ^ 2 ) - ( ( ( A ^ 2 ) - 1 ) x. ( C ^ 2 ) ) ) = 1 )
46	9	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> ( ( F ^ 2 ) - ( ( ( A ^ 2 ) - 1 ) x. ( E ^ 2 ) ) ) = 1 )
47	22	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> G e. ( ZZ>= ` 2 ) )
48	20	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> ( ( I ^ 2 ) - ( ( ( G ^ 2 ) - 1 ) x. ( H ^ 2 ) ) ) = 1 )
49		jm2.27a15	. . . . . 6 |- ( ph -> E = ( ( J + 1 ) x. ( 2 x. ( C ^ 2 ) ) ) )
50	49	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> E = ( ( J + 1 ) x. ( 2 x. ( C ^ 2 ) ) ) )
51		jm2.27a16	. . . . . 6 |- ( ph -> F || ( G - A ) )
52	51	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> F || ( G - A ) )
53		jm2.27a17	. . . . . 6 |- ( ph -> ( 2 x. C ) || ( G - 1 ) )
54	53	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> ( 2 x. C ) || ( G - 1 ) )
55		jm2.27a18	. . . . . 6 |- ( ph -> F || ( H - C ) )
56	55	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> F || ( H - C ) )
57		jm2.27a19	. . . . . 6 |- ( ph -> ( 2 x. C ) || ( H - B ) )
58	57	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> ( 2 x. C ) || ( H - B ) )
59		jm2.27a20	. . . . . 6 |- ( ph -> B <_ C )
60	59	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> B <_ C )
61		simprl 794	. . . . . 6 |- ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) -> p e. ZZ )
62	61	ad2antrr 762	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> p e. ZZ )
63		simprrl 804	. . . . . 6 |- ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) -> D = ( A Xrm p ) )
64	63	ad2antrr 762	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> D = ( A Xrm p ) )
65		simprrr 805	. . . . . 6 |- ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) -> C = ( A Yrm p ) )
66	65	ad2antrr 762	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> C = ( A Yrm p ) )
67		simplrl 800	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> q e. ZZ )
68		simprl 794	. . . . . 6 |- ( ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) -> F = ( A Xrm q ) )
69	68	ad2antlr 763	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> F = ( A Xrm q ) )
70		simprr 796	. . . . . 6 |- ( ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) -> E = ( A Yrm q ) )
71	70	ad2antlr 763	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> E = ( A Yrm q ) )
72		simprl 794	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> r e. ZZ )
73		simprrl 804	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> I = ( G Xrm r ) )
74		simprrr 805	. . . . 5 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> H = ( G Yrm r ) )
75	32, 34, 35, 36, 37, 38, 40, 41, 42, 44, 45, 46, 47, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 67, 69, 71, 72, 73, 74	jm2.27a 37572	. . . 4 |- ( ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) /\ ( r e. ZZ /\ ( I = ( G Xrm r ) /\ H = ( G Yrm r ) ) ) ) -> C = ( A Yrm B ) )
76	31, 75	rexlimddv 3035	. . 3 |- ( ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) /\ ( q e. ZZ /\ ( F = ( A Xrm q ) /\ E = ( A Yrm q ) ) ) ) -> C = ( A Yrm B ) )
77	19, 76	rexlimddv 3035	. 2 |- ( ( ph /\ ( p e. ZZ /\ ( D = ( A Xrm p ) /\ C = ( A Yrm p ) ) ) ) -> C = ( A Yrm B ) )
78	8, 77	rexlimddv 3035	1 |- ( ph -> C = ( A Yrm B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   1c1 9937   + caddc 9939   x. cmul 9941   <_ cle 10075   - cmin 10266   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ^cexp 12860   || cdvds 14983   X_rm crmx 37464   Y_rm crmy 37465

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  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  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-int 4476  df-iun 4522  df-iin 4523  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-dvds 14984  df-gcd 15217  df-prm 15386  df-numer 15443  df-denom 15444  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-squarenn 37405  df-pell1qr 37406  df-pell14qr 37407  df-pell1234qr 37408  df-pellfund 37409  df-rmx 37466  df-rmy 37467

This theorem is referenced by:  jm2.27  37575