Theorem caucvgprprlemmu 7878

Description: Lemma for caucvgprpr 7895. 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 7498	. . . . 5 |- 1o e. N.
3	2	a1i 9	. . . 4 |- ( ph -> 1o e. N. )
4	1, 3	ffvelcdmd 5770	. . 3 |- ( ph -> ( F ` 1o ) e. P. )
5		prop 7658	. . 3 |- ( ( F ` 1o ) e. P. -> <. ( 1st ` ( F ` 1o ) ) , ( 2nd ` ( F ` 1o ) ) >. e. P. )
6		prmu 7661	. . 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 529	. . . 4 |- ( ( ph /\ ( x e. Q. /\ x e. ( 2nd ` ( F ` 1o ) ) ) ) -> x e. Q. )
9		1nq 7549	. . . 4 |- 1Q e. Q.
10		addclnq 7558	. . . 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 531	. . . . . . . 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 7735	. . . . . . . . 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 7802	. . . . . . . . 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 7730	. . . . . . . . 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 7730	. . . . . . . . 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 7761	. . . . . . . . 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 6174	. . . . . . 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 7534	. . . . . . . . . . . . 13 |- 1Q = [ <. 1o , 1o >. ] ~Q
29	28	fveq2i 5629	. . . . . . . . . . . 12 |- ( *Q ` 1Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q )
30		rec1nq 7578	. . . . . . . . . . . 12 |- ( *Q ` 1Q ) = 1Q
31	29, 30	eqtr3i 2252	. . . . . . . . . . 11 |- ( *Q ` [ <. 1o , 1o >. ] ~Q ) = 1Q
32	31	breq2i 4090	. . . . . . . . . 10 |- ( p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) <-> p <Q 1Q )
33	32	abbii 2345	. . . . . . . . 9 |- { p | p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } = { p | p <Q 1Q }
34	31	breq1i 4089	. . . . . . . . . 10 |- ( ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q <-> 1Q <Q q )
35	34	abbii 2345	. . . . . . . . 9 |- { q | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q } = { q | 1Q <Q q }
36	33, 35	opeq12i 3861	. . . . . . . 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 6011	. . . . . . 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 7744	. . . . . . 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 4114	. . . . 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 5626	. . . . . . . 8 |- ( r = 1o -> ( F ` r ) = ( F ` 1o ) )
43		opeq1 3856	. . . . . . . . . . . . 13 |- ( r = 1o -> <. r , 1o >. = <. 1o , 1o >. )
44	43	eceq1d 6714	. . . . . . . . . . . 12 |- ( r = 1o -> [ <. r , 1o >. ] ~Q = [ <. 1o , 1o >. ] ~Q )
45	44	fveq2d 5630	. . . . . . . . . . 11 |- ( r = 1o -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q ) )
46	45	breq2d 4094	. . . . . . . . . 10 |- ( r = 1o -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) ) )
47	46	abbidv 2347	. . . . . . . . 9 |- ( r = 1o -> { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) } )
48	45	breq1d 4092	. . . . . . . . . 10 |- ( r = 1o -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q ) )
49	48	abbidv 2347	. . . . . . . . 9 |- ( r = 1o -> { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. 1o , 1o >. ] ~Q ) <Q q } )
50	47, 49	opeq12d 3864	. . . . . . . 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 6018	. . . . . . 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 4092	. . . . . 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 2907	. . . . 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 4086	. . . . . . . . 9 |- ( u = ( x +Q 1Q ) -> ( p <Q u <-> p <Q ( x +Q 1Q ) ) )
56	55	abbidv 2347	. . . . . . . 8 |- ( u = ( x +Q 1Q ) -> { p | p <Q u } = { p | p <Q ( x +Q 1Q ) } )
57		breq1 4085	. . . . . . . . 9 |- ( u = ( x +Q 1Q ) -> ( u <Q q <-> ( x +Q 1Q ) <Q q ) )
58	57	abbidv 2347	. . . . . . . 8 |- ( u = ( x +Q 1Q ) -> { q | u <Q q } = { q | ( x +Q 1Q ) <Q q } )
59	56, 58	opeq12d 3864	. . . . . . 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 4094	. . . . . 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 2531	. . . . 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 5629	. . . . . 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 7546	. . . . . . . 8 |- Q. e. _V
65	64	rabex 4227	. . . . . . 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 4227	. . . . . . 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 6291	. . . . . 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 2250	. . . . 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 2962	. . . 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 2292	. . . 4 |- ( t = ( x +Q 1Q ) -> ( t e. ( 2nd ` L ) <-> ( x +Q 1Q ) e. ( 2nd ` L ) ) )
72	71	rspcev 2907	. . 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 2653	1 |- ( ph -> E. t e. Q. t e. ( 2nd ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   {cab 2215   A.wral 2508   E.wrex 2509   {crab 2512   <.cop 3669   class class class wbr 4082   -->wf 5313   ` cfv 5317  (class class class)co 6000   1stc1st 6282   2ndc2nd 6283   1oc1o 6553   [cec 6676   N.cnpi 7455   <N clti 7458   ~Q ceq 7462   Q.cnq 7463   1Qc1q 7464   +Q cplq 7465   *Qcrq 7467   <Q cltq 7468   P.cnp 7474   +P. cpp 7476   <P cltp 7478

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-eprel 4379  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-1o 6560  df-2o 6561  df-oadd 6564  df-omul 6565  df-er 6678  df-ec 6680  df-qs 6684  df-ni 7487  df-pli 7488  df-mi 7489  df-lti 7490  df-plpq 7527  df-mpq 7528  df-enq 7530  df-nqqs 7531  df-plqqs 7532  df-mqqs 7533  df-1nqqs 7534  df-rq 7535  df-ltnqqs 7536  df-enq0 7607  df-nq0 7608  df-0nq0 7609  df-plq0 7610  df-mq0 7611  df-inp 7649  df-iplp 7651  df-iltp 7653

This theorem is referenced by:  caucvgprprlemm  7879