Theorem caucvgprprlemmu 8026

Description: Lemma for caucvgprpr 8043. 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 7646	. . . . 5 |- 1o e. N.
3	2	a1i 9	. . . 4 |- ( ph -> 1o e. N. )
4	1, 3	ffvelcdmd 5818	. . 3 |- ( ph -> ( F ` 1o ) e. P. )
5		prop 7806	. . 3 |- ( ( F ` 1o ) e. P. -> <. ( 1st ` ( F ` 1o ) ) , ( 2nd ` ( F ` 1o ) ) >. e. P. )
6		prmu 7809	. . 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 7697	. . . 4 |- 1Q e. Q.
10		addclnq 7706	. . . 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 7883	. . . . . . . . 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 7950	. . . . . . . . 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 7878	. . . . . . . . 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 7878	. . . . . . . . 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 7909	. . . . . . . . 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 6232	. . . . . . 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 7682	. . . . . . . . . . . . 13 |- 1Q = [ <. 1o , 1o >. ] ~Q
29	28	fveq2i 5678	. . . . . . . . . . . 12 |- ( *Q ` 1Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q )
30		rec1nq 7726	. . . . . . . . . . . 12 |- ( *Q ` 1Q ) = 1Q
31	29, 30	eqtr3i 2257	. . . . . . . . . . 11 |- ( *Q ` [ <. 1o , 1o >. ] ~Q ) = 1Q
32	31	breq2i 4122	. . . . . . . . . 10 |- ( p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) <-> p <Q 1Q )
33	32	abbii 2350	. . . . . . . . 9 |- { p | p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } = { p | p <Q 1Q }
34	31	breq1i 4121	. . . . . . . . . 10 |- ( ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q <-> 1Q <Q q )
35	34	abbii 2350	. . . . . . . . 9 |- { q | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q } = { q | 1Q <Q q }
36	33, 35	opeq12i 3893	. . . . . . . 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 6069	. . . . . . 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 7892	. . . . . . 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 4146	. . . . 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 5675	. . . . . . . 8 |- ( r = 1o -> ( F ` r ) = ( F ` 1o ) )
43		opeq1 3888	. . . . . . . . . . . . 13 |- ( r = 1o -> <. r , 1o >. = <. 1o , 1o >. )
44	43	eceq1d 6816	. . . . . . . . . . . 12 |- ( r = 1o -> [ <. r , 1o >. ] ~Q = [ <. 1o , 1o >. ] ~Q )
45	44	fveq2d 5679	. . . . . . . . . . 11 |- ( r = 1o -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q ) )
46	45	breq2d 4126	. . . . . . . . . 10 |- ( r = 1o -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) ) )
47	46	abbidv 2354	. . . . . . . . 9 |- ( r = 1o -> { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } )
48	45	breq1d 4124	. . . . . . . . . 10 |- ( r = 1o -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q ) )
49	48	abbidv 2354	. . . . . . . . 9 |- ( r = 1o -> { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q } )
50	47, 49	opeq12d 3896	. . . . . . . 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 6076	. . . . . . 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 4124	. . . . . 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 2923	. . . . 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 4118	. . . . . . . . 9 |- ( u = ( x +Q 1Q ) -> ( p <Q u <-> p <Q ( x +Q 1Q ) ) )
56	55	abbidv 2354	. . . . . . . 8 |- ( u = ( x +Q 1Q ) -> { p | p <Q u } = { p | p <Q ( x +Q 1Q ) } )
57		breq1 4117	. . . . . . . . 9 |- ( u = ( x +Q 1Q ) -> ( u <Q q <-> ( x +Q 1Q ) <Q q ) )
58	57	abbidv 2354	. . . . . . . 8 |- ( u = ( x +Q 1Q ) -> { q | u <Q q } = { q | ( x +Q 1Q ) <Q q } )
59	56, 58	opeq12d 3896	. . . . . . 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 4126	. . . . . 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 2545	. . . . 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 5678	. . . . . 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 7694	. . . . . . . 8 |- Q. e. _V
65	64	rabex 4261	. . . . . . 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 4261	. . . . . . 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 6354	. . . . . 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 2255	. . . . 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 2979	. . . 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 2297	. . . 4 |- ( t = ( x +Q 1Q ) -> ( t e. ( 2nd ` L ) <-> ( x +Q 1Q ) e. ( 2nd ` L ) ) )
72	71	rspcev 2923	. . 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 2667	1 |- ( ph -> E. t e. Q. t e. ( 2nd ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   {cab 2220   A.wral 2522   E.wrex 2523   {crab 2526   <.cop 3697   class class class wbr 4114   -->wf 5353   ` cfv 5357  (class class class)co 6058   1stc1st 6345   2ndc2nd 6346   1oc1o 6653   [cec 6778   N.cnpi 7603   <N clti 7606   ~Q ceq 7610   Q.cnq 7611   1Qc1q 7612   +Q cplq 7613   *Qcrq 7615   <Q cltq 7616   P.cnp 7622   +P. cpp 7624   <P cltp 7626

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-2o 6661  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684  df-enq0 7755  df-nq0 7756  df-0nq0 7757  df-plq0 7758  df-mq0 7759  df-inp 7797  df-iplp 7799  df-iltp 7801

This theorem is referenced by:  caucvgprprlemm  8027