Theorem caucvgprprlemexbt 7514

Description: Lemma for caucvgprpr 7520. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 16-Jun-2021.)

Hypotheses

Ref	Expression
caucvgprpr.f	|- ( ph -> F : N. --> P. )
caucvgprpr.cau	|- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )
caucvgprpr.bnd	|- ( ph -> A. m e. N. A <P ( F ` m ) )
caucvgprpr.lim	|- L = <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.
caucvgprprlemexbt.q	|- ( ph -> Q e. Q. )
caucvgprprlemexbt.t	|- ( ph -> T e. P. )
caucvgprprlemexbt.lt	|- ( ph -> ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P T )

Assertion

Ref	Expression
caucvgprprlemexbt	|- ( ph -> E. b e. N. ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P T )

Distinct variable groups:   A, m   m, F   A, r, m   F, b   k, F, l, n, u   F, r   L, b   k, L   Q, b, p, q   T, b   ph, b   r, b, p, q   k, p, q, r, l, u

Allowed substitution hints:   ph( u, k, m, n, r, q, p, l)   A( u, k, n, q, p, b, l)   Q( u, k, m, n, r, l)   T( u, k, m, n, r, q, p, l)   F( q, p)   L( u, m, n, r, q, p, l)

Proof of Theorem caucvgprprlemexbt

Dummy variables f g h x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgprprlemexbt.lt	. . . . 5 |- ( ph -> ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P T )
2		caucvgprpr.f	. . . . . . . 8 |- ( ph -> F : N. --> P. )
3		caucvgprpr.cau	. . . . . . . 8 |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )
4		caucvgprpr.bnd	. . . . . . . 8 |- ( ph -> A. m e. N. A <P ( F ` m ) )
5		caucvgprpr.lim	. . . . . . . 8 |- L = <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.
6	2, 3, 4, 5	caucvgprprlemclphr 7513	. . . . . . 7 |- ( ph -> L e. P. )
7		caucvgprprlemexbt.q	. . . . . . . 8 |- ( ph -> Q e. Q. )
8		nqprlu 7355	. . . . . . . 8 |- ( Q e. Q. -> <. { p | p <Q Q } , { q | Q <Q q } >. e. P. )
9	7, 8	syl 14	. . . . . . 7 |- ( ph -> <. { p | p <Q Q } , { q | Q <Q q } >. e. P. )
10		addclpr 7345	. . . . . . 7 |- ( ( L e. P. /\ <. { p | p <Q Q } , { q | Q <Q q } >. e. P. ) -> ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) e. P. )
11	6, 9, 10	syl2anc 408	. . . . . 6 |- ( ph -> ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) e. P. )
12		caucvgprprlemexbt.t	. . . . . 6 |- ( ph -> T e. P. )
13		ltdfpr 7314	. . . . . 6 |- ( ( ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) e. P. /\ T e. P. ) -> ( ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P T <-> E. x e. Q. ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) )
14	11, 12, 13	syl2anc 408	. . . . 5 |- ( ph -> ( ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P T <-> E. x e. Q. ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) )
15	1, 14	mpbid 146	. . . 4 |- ( ph -> E. x e. Q. ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) )
16	6	adantr 274	. . . . . . . 8 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) -> L e. P. )
17	7	adantr 274	. . . . . . . 8 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) -> Q e. Q. )
18		simprrl 528	. . . . . . . 8 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) -> x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) )
19	16, 17, 18	prplnqu 7428	. . . . . . 7 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) -> E. y e. ( 2nd ` L ) ( y +Q Q ) = x )
20		simprl 520	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) -> y e. ( 2nd ` L ) )
21		breq2 3933	. . . . . . . . . . . . . . . . 17 |- ( u = y -> ( p <Q u <-> p <Q y ) )
22	21	abbidv 2257	. . . . . . . . . . . . . . . 16 |- ( u = y -> { p | p <Q u } = { p | p <Q y } )
23		breq1 3932	. . . . . . . . . . . . . . . . 17 |- ( u = y -> ( u <Q q <-> y <Q q ) )
24	23	abbidv 2257	. . . . . . . . . . . . . . . 16 |- ( u = y -> { q | u <Q q } = { q | y <Q q } )
25	22, 24	opeq12d 3713	. . . . . . . . . . . . . . 15 |- ( u = y -> <. { p | p <Q u } , { q | u <Q q } >. = <. { p | p <Q y } , { q | y <Q q } >. )
26	25	breq2d 3941	. . . . . . . . . . . . . 14 |- ( u = y -> ( ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. <-> ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) )
27	26	rexbidv 2438	. . . . . . . . . . . . 13 |- ( u = y -> ( E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. <-> E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) )
28	5	fveq2i 5424	. . . . . . . . . . . . . 14 |- ( 2nd ` L ) = ( 2nd ` <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >. )
29		nqex 7171	. . . . . . . . . . . . . . . 16 |- Q. e. _V
30	29	rabex 4072	. . . . . . . . . . . . . . 15 |- { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } e. _V
31	29	rabex 4072	. . . . . . . . . . . . . . 15 |- { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } e. _V
32	30, 31	op2nd 6045	. . . . . . . . . . . . . 14 |- ( 2nd ` <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >. ) = { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. }
33	28, 32	eqtri 2160	. . . . . . . . . . . . 13 |- ( 2nd ` L ) = { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. }
34	27, 33	elrab2 2843	. . . . . . . . . . . 12 |- ( y e. ( 2nd ` L ) <-> ( y e. Q. /\ E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) )
35	34	biimpi 119	. . . . . . . . . . 11 |- ( y e. ( 2nd ` L ) -> ( y e. Q. /\ E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) )
36	35	simprd 113	. . . . . . . . . 10 |- ( y e. ( 2nd ` L ) -> E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. )
37	20, 36	syl 14	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) -> E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. )
38		fveq2 5421	. . . . . . . . . . . 12 |- ( r = b -> ( F ` r ) = ( F ` b ) )
39		opeq1 3705	. . . . . . . . . . . . . . . . 17 |- ( r = b -> <. r , 1o >. = <. b , 1o >. )
40	39	eceq1d 6465	. . . . . . . . . . . . . . . 16 |- ( r = b -> [ <. r , 1o >. ] ~Q = [ <. b , 1o >. ] ~Q )
41	40	fveq2d 5425	. . . . . . . . . . . . . . 15 |- ( r = b -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. b , 1o >. ] ~Q ) )
42	41	breq2d 3941	. . . . . . . . . . . . . 14 |- ( r = b -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
43	42	abbidv 2257	. . . . . . . . . . . . 13 |- ( r = b -> { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } )
44	41	breq1d 3939	. . . . . . . . . . . . . 14 |- ( r = b -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q ) )
45	44	abbidv 2257	. . . . . . . . . . . . 13 |- ( r = b -> { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } )
46	43, 45	opeq12d 3713	. . . . . . . . . . . 12 |- ( r = b -> <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. = <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. )
47	38, 46	oveq12d 5792	. . . . . . . . . . 11 |- ( r = b -> ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) = ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) )
48	47	breq1d 3939	. . . . . . . . . 10 |- ( r = b -> ( ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. <-> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) )
49	48	cbvrexv 2655	. . . . . . . . 9 |- ( E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. <-> E. b e. N. ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. )
50	37, 49	sylib 121	. . . . . . . 8 |- ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) -> E. b e. N. ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. )
51		simpr 109	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. )
52		ltaprg 7427	. . . . . . . . . . . . . . . . 17 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
53	52	adantl 275	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) /\ ( f e. P. /\ g e. P. /\ h e. P. ) ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
54	2	ad4antr 485	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> F : N. --> P. )
55		simplr 519	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> b e. N. )
56	54, 55	ffvelrnd 5556	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( F ` b ) e. P. )
57		recnnpr 7356	. . . . . . . . . . . . . . . . . 18 |- ( b e. N. -> <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. e. P. )
58	55, 57	syl 14	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. e. P. )
59		addclpr 7345	. . . . . . . . . . . . . . . . 17 |- ( ( ( F ` b ) e. P. /\ <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. e. P. ) -> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) e. P. )
60	56, 58, 59	syl2anc 408	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) e. P. )
61	20	ad2antrr 479	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> y e. ( 2nd ` L ) )
62	35	simpld 111	. . . . . . . . . . . . . . . . . 18 |- ( y e. ( 2nd ` L ) -> y e. Q. )
63	61, 62	syl 14	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> y e. Q. )
64		nqprlu 7355	. . . . . . . . . . . . . . . . 17 |- ( y e. Q. -> <. { p | p <Q y } , { q | y <Q q } >. e. P. )
65	63, 64	syl 14	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> <. { p | p <Q y } , { q | y <Q q } >. e. P. )
66	9	ad4antr 485	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> <. { p | p <Q Q } , { q | Q <Q q } >. e. P. )
67		addcomprg 7386	. . . . . . . . . . . . . . . . 17 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
68	67	adantl 275	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) = ( g +P. f ) )
69	53, 60, 65, 66, 68	caovord2d 5940	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. <-> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P ( <. { p | p <Q y } , { q | y <Q q } >. +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) )
70	51, 69	mpbid 146	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P ( <. { p | p <Q y } , { q | y <Q q } >. +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) )
71	7	ad4antr 485	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> Q e. Q. )
72		addnqpr 7369	. . . . . . . . . . . . . . 15 |- ( ( y e. Q. /\ Q e. Q. ) -> <. { p | p <Q ( y +Q Q ) } , { q | ( y +Q Q ) <Q q } >. = ( <. { p | p <Q y } , { q | y <Q q } >. +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) )
73	63, 71, 72	syl2anc 408	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> <. { p | p <Q ( y +Q Q ) } , { q | ( y +Q Q ) <Q q } >. = ( <. { p | p <Q y } , { q | y <Q q } >. +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) )
74	70, 73	breqtrrd 3956	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P <. { p | p <Q ( y +Q Q ) } , { q | ( y +Q Q ) <Q q } >. )
75		simplrr 525	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) -> ( y +Q Q ) = x )
76	75	adantr 274	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( y +Q Q ) = x )
77		breq2 3933	. . . . . . . . . . . . . . . . 17 |- ( ( y +Q Q ) = x -> ( p <Q ( y +Q Q ) <-> p <Q x ) )
78	77	abbidv 2257	. . . . . . . . . . . . . . . 16 |- ( ( y +Q Q ) = x -> { p | p <Q ( y +Q Q ) } = { p | p <Q x } )
79		breq1 3932	. . . . . . . . . . . . . . . . 17 |- ( ( y +Q Q ) = x -> ( ( y +Q Q ) <Q q <-> x <Q q ) )
80	79	abbidv 2257	. . . . . . . . . . . . . . . 16 |- ( ( y +Q Q ) = x -> { q | ( y +Q Q ) <Q q } = { q | x <Q q } )
81	78, 80	opeq12d 3713	. . . . . . . . . . . . . . 15 |- ( ( y +Q Q ) = x -> <. { p | p <Q ( y +Q Q ) } , { q | ( y +Q Q ) <Q q } >. = <. { p | p <Q x } , { q | x <Q q } >. )
82	81	breq2d 3941	. . . . . . . . . . . . . 14 |- ( ( y +Q Q ) = x -> ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P <. { p | p <Q ( y +Q Q ) } , { q | ( y +Q Q ) <Q q } >. <-> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P <. { p | p <Q x } , { q | x <Q q } >. ) )
83	76, 82	syl 14	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P <. { p | p <Q ( y +Q Q ) } , { q | ( y +Q Q ) <Q q } >. <-> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P <. { p | p <Q x } , { q | x <Q q } >. ) )
84	74, 83	mpbid 146	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P <. { p | p <Q x } , { q | x <Q q } >. )
85		simplrl 524	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) -> x e. Q. )
86	85	ad2antrr 479	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> x e. Q. )
87		addclpr 7345	. . . . . . . . . . . . . 14 |- ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) e. P. /\ <. { p | p <Q Q } , { q | Q <Q q } >. e. P. ) -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) e. P. )
88	60, 66, 87	syl2anc 408	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) e. P. )
89		nqpru 7360	. . . . . . . . . . . . 13 |- ( ( x e. Q. /\ ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) e. P. ) -> ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) <-> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P <. { p | p <Q x } , { q | x <Q q } >. ) )
90	86, 88, 89	syl2anc 408	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) <-> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P <. { p | p <Q x } , { q | x <Q q } >. ) )
91	84, 90	mpbird 166	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) )
92		simprrr 529	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) -> x e. ( 1st ` T ) )
93	92	ad3antrrr 483	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> x e. ( 1st ` T ) )
94	91, 93	jca 304	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. ) -> ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) )
95	94	ex 114	. . . . . . . . 9 |- ( ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) /\ b e. N. ) -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. -> ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) )
96	95	reximdva 2534	. . . . . . . 8 |- ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) -> ( E. b e. N. ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q y } , { q | y <Q q } >. -> E. b e. N. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) )
97	50, 96	mpd 13	. . . . . . 7 |- ( ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) /\ ( y e. ( 2nd ` L ) /\ ( y +Q Q ) = x ) ) -> E. b e. N. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) )
98	19, 97	rexlimddv 2554	. . . . . 6 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) ) -> E. b e. N. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) )
99	98	expr 372	. . . . 5 |- ( ( ph /\ x e. Q. ) -> ( ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) -> E. b e. N. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) )
100	99	reximdva 2534	. . . 4 |- ( ph -> ( E. x e. Q. ( x e. ( 2nd ` ( L +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) -> E. x e. Q. E. b e. N. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) )
101	15, 100	mpd 13	. . 3 |- ( ph -> E. x e. Q. E. b e. N. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) )
102		rexcom 2595	. . 3 |- ( E. x e. Q. E. b e. N. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) <-> E. b e. N. E. x e. Q. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) )
103	101, 102	sylib 121	. 2 |- ( ph -> E. b e. N. E. x e. Q. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) )
104	2	ffvelrnda 5555	. . . . . 6 |- ( ( ph /\ b e. N. ) -> ( F ` b ) e. P. )
105	57	adantl 275	. . . . . 6 |- ( ( ph /\ b e. N. ) -> <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. e. P. )
106	104, 105, 59	syl2anc 408	. . . . 5 |- ( ( ph /\ b e. N. ) -> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) e. P. )
107	9	adantr 274	. . . . 5 |- ( ( ph /\ b e. N. ) -> <. { p | p <Q Q } , { q | Q <Q q } >. e. P. )
108	106, 107, 87	syl2anc 408	. . . 4 |- ( ( ph /\ b e. N. ) -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) e. P. )
109	12	adantr 274	. . . 4 |- ( ( ph /\ b e. N. ) -> T e. P. )
110		ltdfpr 7314	. . . 4 |- ( ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) e. P. /\ T e. P. ) -> ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P T <-> E. x e. Q. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) )
111	108, 109, 110	syl2anc 408	. . 3 |- ( ( ph /\ b e. N. ) -> ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P T <-> E. x e. Q. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) )
112	111	rexbidva 2434	. 2 |- ( ph -> ( E. b e. N. ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P T <-> E. b e. N. E. x e. Q. ( x e. ( 2nd ` ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) ) /\ x e. ( 1st ` T ) ) ) )
113	103, 112	mpbird 166	1 |- ( ph -> E. b e. N. ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q Q } , { q | Q <Q q } >. ) <P T )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   {cab 2125   A.wral 2416   E.wrex 2417   {crab 2420   <.cop 3530   class class class wbr 3929   -->wf 5119   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   1oc1o 6306   [cec 6427   N.cnpi 7080   <N clti 7083   ~Q ceq 7087   Q.cnq 7088   +Q cplq 7090   *Qcrq 7092   <Q cltq 7093   P.cnp 7099   +P. cpp 7101   <P cltp 7103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-iplp 7276  df-iltp 7278

This theorem is referenced by:  caucvgprprlemexb  7515