Theorem caucvgprlemm 7851

Description: Lemma for caucvgpr 7865. The putative limit is inhabited. (Contributed by Jim Kingdon, 27-Sep-2020.)

Hypotheses

Ref	Expression
caucvgpr.f	|- ( ph -> F : N. --> Q. )
caucvgpr.cau	|- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )
caucvgpr.bnd	|- ( ph -> A. j e. N. A <Q ( F ` j ) )
caucvgpr.lim	|- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.

Assertion

Ref	Expression
caucvgprlemm	|- ( ph -> ( E. s e. Q. s e. ( 1st ` L ) /\ E. r e. Q. r e. ( 2nd ` L ) ) )

Distinct variable groups:   A, j, s   j, F, l   F, r   u, F, j   L, r   ph, j, s   s, l

Allowed substitution hints:   ph( u, k, n, r, l)   A( u, k, n, r, l)   F( k, n, s)   L( u, j, k, n, s, l)

Proof of Theorem caucvgprlemm

Step	Hyp	Ref	Expression
1		fveq2 5626	. . . . . 6 |- ( j = 1o -> ( F ` j ) = ( F ` 1o ) )
2	1	breq2d 4094	. . . . 5 |- ( j = 1o -> ( A <Q ( F ` j ) <-> A <Q ( F ` 1o ) ) )
3		caucvgpr.bnd	. . . . 5 |- ( ph -> A. j e. N. A <Q ( F ` j ) )
4		1pi 7498	. . . . . 6 |- 1o e. N.
5	4	a1i 9	. . . . 5 |- ( ph -> 1o e. N. )
6	2, 3, 5	rspcdva 2912	. . . 4 |- ( ph -> A <Q ( F ` 1o ) )
7		ltrelnq 7548	. . . . . 6 |- <Q C_ ( Q. X. Q. )
8	7	brel 4770	. . . . 5 |- ( A <Q ( F ` 1o ) -> ( A e. Q. /\ ( F ` 1o ) e. Q. ) )
9	8	simpld 112	. . . 4 |- ( A <Q ( F ` 1o ) -> A e. Q. )
10		halfnqq 7593	. . . 4 |- ( A e. Q. -> E. s e. Q. ( s +Q s ) = A )
11	6, 9, 10	3syl 17	. . 3 |- ( ph -> E. s e. Q. ( s +Q s ) = A )
12		simplr 528	. . . . . 6 |- ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) -> s e. Q. )
13		archrecnq 7846	. . . . . . . 8 |- ( s e. Q. -> E. j e. N. ( *Q ` [ <. j , 1o >. ] ~Q ) <Q s )
14	12, 13	syl 14	. . . . . . 7 |- ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) -> E. j e. N. ( *Q ` [ <. j , 1o >. ] ~Q ) <Q s )
15		simpr 110	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q s ) -> ( *Q ` [ <. j , 1o >. ] ~Q ) <Q s )
16		simplr 528	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q s ) -> j e. N. )
17		nnnq 7605	. . . . . . . . . . . . . 14 |- ( j e. N. -> [ <. j , 1o >. ] ~Q e. Q. )
18		recclnq 7575	. . . . . . . . . . . . . 14 |- ( [ <. j , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. j , 1o >. ] ~Q ) e. Q. )
19	16, 17, 18	3syl 17	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q s ) -> ( *Q ` [ <. j , 1o >. ] ~Q ) e. Q. )
20	12	ad2antrr 488	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q s ) -> s e. Q. )
21		ltanqg 7583	. . . . . . . . . . . . 13 |- ( ( ( *Q ` [ <. j , 1o >. ] ~Q ) e. Q. /\ s e. Q. /\ s e. Q. ) -> ( ( *Q ` [ <. j , 1o >. ] ~Q ) <Q s <-> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( s +Q s ) ) )
22	19, 20, 20, 21	syl3anc 1271	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q s ) -> ( ( *Q ` [ <. j , 1o >. ] ~Q ) <Q s <-> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( s +Q s ) ) )
23	15, 22	mpbid 147	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q s ) -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( s +Q s ) )
24		simpllr 534	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q s ) -> ( s +Q s ) = A )
25	23, 24	breqtrd 4108	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q s ) -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q A )
26		rsp 2577	. . . . . . . . . . . . 13 |- ( A. j e. N. A <Q ( F ` j ) -> ( j e. N. -> A <Q ( F ` j ) ) )
27	3, 26	syl 14	. . . . . . . . . . . 12 |- ( ph -> ( j e. N. -> A <Q ( F ` j ) ) )
28	27	ad4antr 494	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q s ) -> ( j e. N. -> A <Q ( F ` j ) ) )
29	16, 28	mpd 13	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q s ) -> A <Q ( F ` j ) )
30		ltsonq 7581	. . . . . . . . . . 11 |- <Q Or Q.
31	30, 7	sotri 5123	. . . . . . . . . 10 |- ( ( ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q A /\ A <Q ( F ` j ) ) -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
32	25, 29, 31	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q s ) -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
33	32	ex 115	. . . . . . . 8 |- ( ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) /\ j e. N. ) -> ( ( *Q ` [ <. j , 1o >. ] ~Q ) <Q s -> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
34	33	reximdva 2632	. . . . . . 7 |- ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) -> ( E. j e. N. ( *Q ` [ <. j , 1o >. ] ~Q ) <Q s -> E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
35	14, 34	mpd 13	. . . . . 6 |- ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) -> E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) )
36		oveq1 6007	. . . . . . . . 9 |- ( l = s -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
37	36	breq1d 4092	. . . . . . . 8 |- ( l = s -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
38	37	rexbidv 2531	. . . . . . 7 |- ( l = s -> ( E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
39		caucvgpr.lim	. . . . . . . . 9 |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.
40	39	fveq2i 5629	. . . . . . . 8 |- ( 1st ` L ) = ( 1st ` <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. )
41		nqex 7546	. . . . . . . . . 10 |- Q. e. _V
42	41	rabex 4227	. . . . . . . . 9 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
43	41	rabex 4227	. . . . . . . . 9 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
44	42, 43	op1st 6290	. . . . . . . 8 |- ( 1st ` <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. ) = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
45	40, 44	eqtri 2250	. . . . . . 7 |- ( 1st ` L ) = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
46	38, 45	elrab2 2962	. . . . . 6 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. j e. N. ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
47	12, 35, 46	sylanbrc 417	. . . . 5 |- ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) -> s e. ( 1st ` L ) )
48	47	ex 115	. . . 4 |- ( ( ph /\ s e. Q. ) -> ( ( s +Q s ) = A -> s e. ( 1st ` L ) ) )
49	48	reximdva 2632	. . 3 |- ( ph -> ( E. s e. Q. ( s +Q s ) = A -> E. s e. Q. s e. ( 1st ` L ) ) )
50	11, 49	mpd 13	. 2 |- ( ph -> E. s e. Q. s e. ( 1st ` L ) )
51		caucvgpr.f	. . . . . 6 |- ( ph -> F : N. --> Q. )
52	51, 5	ffvelcdmd 5770	. . . . 5 |- ( ph -> ( F ` 1o ) e. Q. )
53		1nq 7549	. . . . 5 |- 1Q e. Q.
54		addclnq 7558	. . . . 5 |- ( ( ( F ` 1o ) e. Q. /\ 1Q e. Q. ) -> ( ( F ` 1o ) +Q 1Q ) e. Q. )
55	52, 53, 54	sylancl 413	. . . 4 |- ( ph -> ( ( F ` 1o ) +Q 1Q ) e. Q. )
56		addclnq 7558	. . . 4 |- ( ( ( ( F ` 1o ) +Q 1Q ) e. Q. /\ 1Q e. Q. ) -> ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) e. Q. )
57	55, 53, 56	sylancl 413	. . 3 |- ( ph -> ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) e. Q. )
58		df-1nqqs 7534	. . . . . . . . 9 |- 1Q = [ <. 1o , 1o >. ] ~Q
59	58	fveq2i 5629	. . . . . . . 8 |- ( *Q ` 1Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q )
60		rec1nq 7578	. . . . . . . 8 |- ( *Q ` 1Q ) = 1Q
61	59, 60	eqtr3i 2252	. . . . . . 7 |- ( *Q ` [ <. 1o , 1o >. ] ~Q ) = 1Q
62	61	oveq2i 6011	. . . . . 6 |- ( ( F ` 1o ) +Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) ) = ( ( F ` 1o ) +Q 1Q )
63		ltaddnq 7590	. . . . . . 7 |- ( ( ( ( F ` 1o ) +Q 1Q ) e. Q. /\ 1Q e. Q. ) -> ( ( F ` 1o ) +Q 1Q ) <Q ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) )
64	55, 53, 63	sylancl 413	. . . . . 6 |- ( ph -> ( ( F ` 1o ) +Q 1Q ) <Q ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) )
65	62, 64	eqbrtrid 4117	. . . . 5 |- ( ph -> ( ( F ` 1o ) +Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) ) <Q ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) )
66		opeq1 3856	. . . . . . . . . 10 |- ( j = 1o -> <. j , 1o >. = <. 1o , 1o >. )
67	66	eceq1d 6714	. . . . . . . . 9 |- ( j = 1o -> [ <. j , 1o >. ] ~Q = [ <. 1o , 1o >. ] ~Q )
68	67	fveq2d 5630	. . . . . . . 8 |- ( j = 1o -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q ) )
69	1, 68	oveq12d 6018	. . . . . . 7 |- ( j = 1o -> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` 1o ) +Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) ) )
70	69	breq1d 4092	. . . . . 6 |- ( j = 1o -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) <-> ( ( F ` 1o ) +Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) ) <Q ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) ) )
71	70	rspcev 2907	. . . . 5 |- ( ( 1o e. N. /\ ( ( F ` 1o ) +Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) ) <Q ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) ) -> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) )
72	5, 65, 71	syl2anc 411	. . . 4 |- ( ph -> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) )
73		breq2 4086	. . . . . 6 |- ( u = ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) ) )
74	73	rexbidv 2531	. . . . 5 |- ( u = ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) -> ( E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) ) )
75	39	fveq2i 5629	. . . . . 6 |- ( 2nd ` L ) = ( 2nd ` <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. )
76	42, 43	op2nd 6291	. . . . . 6 |- ( 2nd ` <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
77	75, 76	eqtri 2250	. . . . 5 |- ( 2nd ` L ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
78	74, 77	elrab2 2962	. . . 4 |- ( ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) e. ( 2nd ` L ) <-> ( ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) ) )
79	57, 72, 78	sylanbrc 417	. . 3 |- ( ph -> ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) e. ( 2nd ` L ) )
80		eleq1 2292	. . . 4 |- ( r = ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) -> ( r e. ( 2nd ` L ) <-> ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) e. ( 2nd ` L ) ) )
81	80	rspcev 2907	. . 3 |- ( ( ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) e. Q. /\ ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) e. ( 2nd ` L ) ) -> E. r e. Q. r e. ( 2nd ` L ) )
82	57, 79, 81	syl2anc 411	. 2 |- ( ph -> E. r e. Q. r e. ( 2nd ` L ) )
83	50, 82	jca 306	1 |- ( ph -> ( E. s e. Q. s e. ( 1st ` L ) /\ E. r e. Q. r e. ( 2nd ` L ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   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

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-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

This theorem is referenced by:  caucvgprlemcl  7859