Theorem caucvgprprlemmu 7636

Description: Lemma for caucvgprpr 7653. The upper cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.)

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 } >. } >.

Assertion

Ref	Expression
caucvgprprlemmu	|- ( ph -> E. t e. Q. t e. ( 2nd ` L ) )

Distinct variable groups:   A, m   m, F   A, r, m   F, r, u   t, L   q, p, r, u

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

Proof of Theorem caucvgprprlemmu

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

Step	Hyp	Ref	Expression
1		caucvgprpr.f	. . . 4 |- ( ph -> F : N. --> P. )
2		1pi 7256	. . . . 5 |- 1o e. N.
3	2	a1i 9	. . . 4 |- ( ph -> 1o e. N. )
4	1, 3	ffvelrnd 5621	. . 3 |- ( ph -> ( F ` 1o ) e. P. )
5		prop 7416	. . 3 |- ( ( F ` 1o ) e. P. -> <. ( 1st ` ( F ` 1o ) ) , ( 2nd ` ( F ` 1o ) ) >. e. P. )
6		prmu 7419	. . 3 |- ( <. ( 1st ` ( F ` 1o ) ) , ( 2nd ` ( F ` 1o ) ) >. e. P. -> E. x e. Q. x e. ( 2nd ` ( F ` 1o ) ) )
7	4, 5, 6	3syl 17	. 2 |- ( ph -> E. x e. Q. x e. ( 2nd ` ( F ` 1o ) ) )
8		simprl 521	. . . 4 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> x e. Q. )
9		1nq 7307	. . . 4 |- 1Q e. Q.
10		addclnq 7316	. . . 4 |- ( ( x e. Q. /\ 1Q e. Q. ) -> ( x +Q 1Q ) e. Q. )
11	8, 9, 10	sylancl 410	. . 3 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> ( x +Q 1Q ) e. Q. )
12	2	a1i 9	. . . . 5 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> 1o e. N. )
13		simprr 522	. . . . . . . 8 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> x e. ( 2nd ` ( F ` 1o ) ) )
14	4	adantr 274	. . . . . . . . 9 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> ( F ` 1o ) e. P. )
15		nqpru 7493	. . . . . . . . 9 |- ( ( x e. Q. /\ ( F ` 1o ) e. P. ) -> ( x e. ( 2nd ` ( F ` 1o ) ) <-> ( F ` 1o ) <P <. { p | p <Q x } , { q | x <Q q } >. ) )
16	8, 14, 15	syl2anc 409	. . . . . . . 8 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> ( x e. ( 2nd ` ( F ` 1o ) ) <-> ( F ` 1o ) <P <. { p | p <Q x } , { q | x <Q q } >. ) )
17	13, 16	mpbid 146	. . . . . . 7 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> ( F ` 1o ) <P <. { p | p <Q x } , { q | x <Q q } >. )
18		ltaprg 7560	. . . . . . . . 9 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
19	18	adantl 275	. . . . . . . 8 |- ( ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) /\ ( f e. P. /\ g e. P. /\ h e. P. ) ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
20		nqprlu 7488	. . . . . . . . 9 |- ( x e. Q. -> <. { p | p <Q x } , { q | x <Q q } >. e. P. )
21	8, 20	syl 14	. . . . . . . 8 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> <. { p | p <Q x } , { q | x <Q q } >. e. P. )
22		nqprlu 7488	. . . . . . . . 9 |- ( 1Q e. Q. -> <. { p | p <Q 1Q } , { q | 1Q <Q q } >. e. P. )
23	9, 22	mp1i 10	. . . . . . . 8 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> <. { p | p <Q 1Q } , { q | 1Q <Q q } >. e. P. )
24		addcomprg 7519	. . . . . . . . 9 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
25	24	adantl 275	. . . . . . . 8 |- ( ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) = ( g +P. f ) )
26	19, 14, 21, 23, 25	caovord2d 6011	. . . . . . 7 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> ( ( F ` 1o ) <P <. { p | p <Q x } , { q | x <Q q } >. <-> ( ( F ` 1o ) +P. <. { p | p <Q 1Q } , { q | 1Q <Q q } >. ) <P ( <. { p | p <Q x } , { q | x <Q q } >. +P. <. { p | p <Q 1Q } , { q | 1Q <Q q } >. ) ) )
27	17, 26	mpbid 146	. . . . . 6 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> ( ( F ` 1o ) +P. <. { p | p <Q 1Q } , { q | 1Q <Q q } >. ) <P ( <. { p | p <Q x } , { q | x <Q q } >. +P. <. { p | p <Q 1Q } , { q | 1Q <Q q } >. ) )
28		df-1nqqs 7292	. . . . . . . . . . . . 13 |- 1Q = [ <. 1o , 1o >. ] ~Q
29	28	fveq2i 5489	. . . . . . . . . . . 12 |- ( *Q ` 1Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q )
30		rec1nq 7336	. . . . . . . . . . . 12 |- ( *Q ` 1Q ) = 1Q
31	29, 30	eqtr3i 2188	. . . . . . . . . . 11 |- ( *Q ` [ <. 1o , 1o >. ] ~Q ) = 1Q
32	31	breq2i 3990	. . . . . . . . . 10 |- ( p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) <-> p <Q 1Q )
33	32	abbii 2282	. . . . . . . . 9 |- { p | p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } = { p | p <Q 1Q }
34	31	breq1i 3989	. . . . . . . . . 10 |- ( ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q <-> 1Q <Q q )
35	34	abbii 2282	. . . . . . . . 9 |- { q | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q } = { q | 1Q <Q q }
36	33, 35	opeq12i 3763	. . . . . . . 8 |- <. { p | p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q } >. = <. { p | p <Q 1Q } , { q | 1Q <Q q } >.
37	36	oveq2i 5853	. . . . . . 7 |- ( ( F ` 1o ) +P. <. { p | p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q } >. ) = ( ( F ` 1o ) +P. <. { p | p <Q 1Q } , { q | 1Q <Q q } >. )
38	37	a1i 9	. . . . . 6 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> ( ( F ` 1o ) +P. <. { p | p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q } >. ) = ( ( F ` 1o ) +P. <. { p | p <Q 1Q } , { q | 1Q <Q q } >. ) )
39		addnqpr 7502	. . . . . . 7 |- ( ( x e. Q. /\ 1Q e. Q. ) -> <. { p | p <Q ( x +Q 1Q ) } , { q | ( x +Q 1Q ) <Q q } >. = ( <. { p | p <Q x } , { q | x <Q q } >. +P. <. { p | p <Q 1Q } , { q | 1Q <Q q } >. ) )
40	8, 9, 39	sylancl 410	. . . . . 6 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> <. { p | p <Q ( x +Q 1Q ) } , { q | ( x +Q 1Q ) <Q q } >. = ( <. { p | p <Q x } , { q | x <Q q } >. +P. <. { p | p <Q 1Q } , { q | 1Q <Q q } >. ) )
41	27, 38, 40	3brtr4d 4014	. . . . 5 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> ( ( F ` 1o ) +P. <. { p | p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q ( x +Q 1Q ) } , { q | ( x +Q 1Q ) <Q q } >. )
42		fveq2 5486	. . . . . . . 8 |- ( r = 1o -> ( F ` r ) = ( F ` 1o ) )
43		opeq1 3758	. . . . . . . . . . . . 13 |- ( r = 1o -> <. r , 1o >. = <. 1o , 1o >. )
44	43	eceq1d 6537	. . . . . . . . . . . 12 |- ( r = 1o -> [ <. r , 1o >. ] ~Q = [ <. 1o , 1o >. ] ~Q )
45	44	fveq2d 5490	. . . . . . . . . . 11 |- ( r = 1o -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q ) )
46	45	breq2d 3994	. . . . . . . . . 10 |- ( r = 1o -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) ) )
47	46	abbidv 2284	. . . . . . . . 9 |- ( r = 1o -> { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } )
48	45	breq1d 3992	. . . . . . . . . 10 |- ( r = 1o -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q ) )
49	48	abbidv 2284	. . . . . . . . 9 |- ( r = 1o -> { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q } )
50	47, 49	opeq12d 3766	. . . . . . . 8 |- ( r = 1o -> <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. = <. { p | p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q } >. )
51	42, 50	oveq12d 5860	. . . . . . 7 |- ( r = 1o -> ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) = ( ( F ` 1o ) +P. <. { p | p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q } >. ) )
52	51	breq1d 3992	. . . . . 6 |- ( r = 1o -> ( ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q ( x +Q 1Q ) } , { q | ( x +Q 1Q ) <Q q } >. <-> ( ( F ` 1o ) +P. <. { p | p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q ( x +Q 1Q ) } , { q | ( x +Q 1Q ) <Q q } >. ) )
53	52	rspcev 2830	. . . . 5 |- ( ( 1o e. N. /\ ( ( F ` 1o ) +P. <. { p | p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q ( x +Q 1Q ) } , { q | ( x +Q 1Q ) <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 ( x +Q 1Q ) } , { q | ( x +Q 1Q ) <Q q } >. )
54	12, 41, 53	syl2anc 409	. . . 4 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> 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 ( x +Q 1Q ) } , { q | ( x +Q 1Q ) <Q q } >. )
55		breq2 3986	. . . . . . . . 9 |- ( u = ( x +Q 1Q ) -> ( p <Q u <-> p <Q ( x +Q 1Q ) ) )
56	55	abbidv 2284	. . . . . . . 8 |- ( u = ( x +Q 1Q ) -> { p | p <Q u } = { p | p <Q ( x +Q 1Q ) } )
57		breq1 3985	. . . . . . . . 9 |- ( u = ( x +Q 1Q ) -> ( u <Q q <-> ( x +Q 1Q ) <Q q ) )
58	57	abbidv 2284	. . . . . . . 8 |- ( u = ( x +Q 1Q ) -> { q | u <Q q } = { q | ( x +Q 1Q ) <Q q } )
59	56, 58	opeq12d 3766	. . . . . . 7 |- ( u = ( x +Q 1Q ) -> <. { p | p <Q u } , { q | u <Q q } >. = <. { p | p <Q ( x +Q 1Q ) } , { q | ( x +Q 1Q ) <Q q } >. )
60	59	breq2d 3994	. . . . . 6 |- ( u = ( x +Q 1Q ) -> ( ( ( 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 ( x +Q 1Q ) } , { q | ( x +Q 1Q ) <Q q } >. ) )
61	60	rexbidv 2467	. . . . 5 |- ( u = ( x +Q 1Q ) -> ( 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 ( x +Q 1Q ) } , { q | ( x +Q 1Q ) <Q q } >. ) )
62		caucvgprpr.lim	. . . . . . 7 |- 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 } >. } >.
63	62	fveq2i 5489	. . . . . 6 |- ( 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 } >. } >. )
64		nqex 7304	. . . . . . . 8 |- Q. e. _V
65	64	rabex 4126	. . . . . . 7 |- { 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
66	64	rabex 4126	. . . . . . 7 |- { 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
67	65, 66	op2nd 6115	. . . . . 6 |- ( 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 } >. }
68	63, 67	eqtri 2186	. . . . 5 |- ( 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 } >. }
69	61, 68	elrab2 2885	. . . 4 |- ( ( x +Q 1Q ) e. ( 2nd ` L ) <-> ( ( x +Q 1Q ) 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 ( x +Q 1Q ) } , { q | ( x +Q 1Q ) <Q q } >. ) )
70	11, 54, 69	sylanbrc 414	. . 3 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> ( x +Q 1Q ) e. ( 2nd ` L ) )
71		eleq1 2229	. . . 4 |- ( t = ( x +Q 1Q ) -> ( t e. ( 2nd ` L ) <-> ( x +Q 1Q ) e. ( 2nd ` L ) ) )
72	71	rspcev 2830	. . 3 |- ( ( ( x +Q 1Q ) e. Q. /\ ( x +Q 1Q ) e. ( 2nd ` L ) ) -> E. t e. Q. t e. ( 2nd ` L ) )
73	11, 70, 72	syl2anc 409	. 2 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> E. t e. Q. t e. ( 2nd ` L ) )
74	7, 73	rexlimddv 2588	1 |- ( ph -> E. t e. Q. t e. ( 2nd ` L ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   {cab 2151   A.wral 2444   E.wrex 2445   {crab 2448   <.cop 3579   class class class wbr 3982   -->wf 5184   ` cfv 5188  (class class class)co 5842   1stc1st 6106   2ndc2nd 6107   1oc1o 6377   [cec 6499   N.cnpi 7213   <N clti 7216   ~Q ceq 7220   Q.cnq 7221   1Qc1q 7222   +Q cplq 7223   *Qcrq 7225   <Q cltq 7226   P.cnp 7232   +P. cpp 7234   <P cltp 7236

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-iplp 7409  df-iltp 7411

This theorem is referenced by:  caucvgprprlemm  7637