Theorem caucvgprprlemmu 7958

Description: Lemma for caucvgprpr 7975. 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 7578	. . . . 5 |- 1o e. N.
3	2	a1i 9	. . . 4 |- ( ph -> 1o e. N. )
4	1, 3	ffvelcdmd 5791	. . 3 |- ( ph -> ( F ` 1o ) e. P. )
5		prop 7738	. . 3 |- ( ( F ` 1o ) e. P. -> <. ( 1st ` ( F ` 1o ) ) , ( 2nd ` ( F ` 1o ) ) >. e. P. )
6		prmu 7741	. . 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 531	. . . 4 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> x e. Q. )
9		1nq 7629	. . . 4 |- 1Q e. Q.
10		addclnq 7638	. . . 4 |- ( ( x e. Q. /\ 1Q e. Q. ) -> ( x +Q 1Q ) e. Q. )
11	8, 9, 10	sylancl 413	. . 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 533	. . . . . . . 8 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> x e. ( 2nd ` ( F ` 1o ) ) )
14	4	adantr 276	. . . . . . . . 9 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> ( F ` 1o ) e. P. )
15		nqpru 7815	. . . . . . . . 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 411	. . . . . . . 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 147	. . . . . . 7 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> ( F ` 1o ) <P <. { p | p <Q x } , { q | x <Q q } >. )
18		ltaprg 7882	. . . . . . . . 9 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
19	18	adantl 277	. . . . . . . 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 7810	. . . . . . . . 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 7810	. . . . . . . . 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 7841	. . . . . . . . 9 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
25	24	adantl 277	. . . . . . . 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 6202	. . . . . . 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 147	. . . . . 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 7614	. . . . . . . . . . . . 13 |- 1Q = [ <. 1o , 1o >. ] ~Q
29	28	fveq2i 5651	. . . . . . . . . . . 12 |- ( *Q ` 1Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q )
30		rec1nq 7658	. . . . . . . . . . . 12 |- ( *Q ` 1Q ) = 1Q
31	29, 30	eqtr3i 2254	. . . . . . . . . . 11 |- ( *Q ` [ <. 1o , 1o >. ] ~Q ) = 1Q
32	31	breq2i 4101	. . . . . . . . . 10 |- ( p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) <-> p <Q 1Q )
33	32	abbii 2347	. . . . . . . . 9 |- { p | p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } = { p | p <Q 1Q }
34	31	breq1i 4100	. . . . . . . . . 10 |- ( ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q <-> 1Q <Q q )
35	34	abbii 2347	. . . . . . . . 9 |- { q | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q } = { q | 1Q <Q q }
36	33, 35	opeq12i 3872	. . . . . . . 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 6039	. . . . . . 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 7824	. . . . . . 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 413	. . . . . 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 4125	. . . . 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 5648	. . . . . . . 8 |- ( r = 1o -> ( F ` r ) = ( F ` 1o ) )
43		opeq1 3867	. . . . . . . . . . . . 13 |- ( r = 1o -> <. r , 1o >. = <. 1o , 1o >. )
44	43	eceq1d 6781	. . . . . . . . . . . 12 |- ( r = 1o -> [ <. r , 1o >. ] ~Q = [ <. 1o , 1o >. ] ~Q )
45	44	fveq2d 5652	. . . . . . . . . . 11 |- ( r = 1o -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q ) )
46	45	breq2d 4105	. . . . . . . . . 10 |- ( r = 1o -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) ) )
47	46	abbidv 2350	. . . . . . . . 9 |- ( r = 1o -> { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } )
48	45	breq1d 4103	. . . . . . . . . 10 |- ( r = 1o -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q ) )
49	48	abbidv 2350	. . . . . . . . 9 |- ( r = 1o -> { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q } )
50	47, 49	opeq12d 3875	. . . . . . . 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 6046	. . . . . . 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 4103	. . . . . 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 2911	. . . . 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 411	. . . 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 4097	. . . . . . . . 9 |- ( u = ( x +Q 1Q ) -> ( p <Q u <-> p <Q ( x +Q 1Q ) ) )
56	55	abbidv 2350	. . . . . . . 8 |- ( u = ( x +Q 1Q ) -> { p | p <Q u } = { p | p <Q ( x +Q 1Q ) } )
57		breq1 4096	. . . . . . . . 9 |- ( u = ( x +Q 1Q ) -> ( u <Q q <-> ( x +Q 1Q ) <Q q ) )
58	57	abbidv 2350	. . . . . . . 8 |- ( u = ( x +Q 1Q ) -> { q | u <Q q } = { q | ( x +Q 1Q ) <Q q } )
59	56, 58	opeq12d 3875	. . . . . . 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 4105	. . . . . 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 2534	. . . . 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 5651	. . . . . 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 7626	. . . . . . . 8 |- Q. e. _V
65	64	rabex 4239	. . . . . . 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 4239	. . . . . . 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 6319	. . . . . 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 2252	. . . . 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 2966	. . . 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 417	. . 3 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> ( x +Q 1Q ) e. ( 2nd ` L ) )
71		eleq1 2294	. . . 4 |- ( t = ( x +Q 1Q ) -> ( t e. ( 2nd ` L ) <-> ( x +Q 1Q ) e. ( 2nd ` L ) ) )
72	71	rspcev 2911	. . 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 411	. 2 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> E. t e. Q. t e. ( 2nd ` L ) )
74	7, 73	rexlimddv 2656	1 |- ( ph -> E. t e. Q. t e. ( 2nd ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   {cab 2217   A.wral 2511   E.wrex 2512   {crab 2515   <.cop 3676   class class class wbr 4093   -->wf 5329   ` cfv 5333  (class class class)co 6028   1stc1st 6310   2ndc2nd 6311   1oc1o 6618   [cec 6743   N.cnpi 7535   <N clti 7538   ~Q ceq 7542   Q.cnq 7543   1Qc1q 7544   +Q cplq 7545   *Qcrq 7547   <Q cltq 7548   P.cnp 7554   +P. cpp 7556   <P cltp 7558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-2o 6626  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7567  df-pli 7568  df-mi 7569  df-lti 7570  df-plpq 7607  df-mpq 7608  df-enq 7610  df-nqqs 7611  df-plqqs 7612  df-mqqs 7613  df-1nqqs 7614  df-rq 7615  df-ltnqqs 7616  df-enq0 7687  df-nq0 7688  df-0nq0 7689  df-plq0 7690  df-mq0 7691  df-inp 7729  df-iplp 7731  df-iltp 7733

This theorem is referenced by:  caucvgprprlemm  7959