Theorem caucvgprlemm 8041

Description: Lemma for caucvgpr 8055. 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 5698	. . . . . 6 |- ( j = 1o -> ( F ` j ) = ( F ` 1o ) )
2	1	breq2d 4143	. . . . 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 7688	. . . . . 6 |- 1o e. N.
5	4	a1i 9	. . . . 5 |- ( ph -> 1o e. N. )
6	2, 3, 5	rspcdva 2934	. . . 4 |- ( ph -> A <Q ( F ` 1o ) )
7		ltrelnq 7738	. . . . . 6 |- <Q C_ ( Q. X. Q. )
8	7	brel 4829	. . . . 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 7783	. . . 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 533	. . . . . 6 |- ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) -> s e. Q. )
13		archrecnq 8036	. . . . . . . 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 533	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q s ) -> j e. N. )
17		nnnq 7795	. . . . . . . . . . . . . 14 |- ( j e. N. -> [ <. j , 1o >. ] ~Q e. Q. )
18		recclnq 7765	. . . . . . . . . . . . . 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 492	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q s ) -> s e. Q. )
21		ltanqg 7773	. . . . . . . . . . . . 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 1278	. . . . . . . . . . . 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 540	. . . . . . . . . . 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 4157	. . . . . . . . . 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 2597	. . . . . . . . . . . . 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 498	. . . . . . . . . . 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 7771	. . . . . . . . . . 11 |- <Q Or Q.
31	30, 7	sotri 5186	. . . . . . . . . 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 415	. . . . . . . . 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 2652	. . . . . . 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 6095	. . . . . . . . 9 |- ( l = s -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
37	36	breq1d 4141	. . . . . . . 8 |- ( l = s -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
38	37	rexbidv 2551	. . . . . . 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 5701	. . . . . . . 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 7736	. . . . . . . . . 10 |- Q. e. _V
42	41	rabex 4282	. . . . . . . . 9 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
43	41	rabex 4282	. . . . . . . . 9 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
44	42, 43	op1st 6384	. . . . . . . 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 2259	. . . . . . 7 |- ( 1st ` L ) = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
46	38, 45	elrab2 2985	. . . . . 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 421	. . . . 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 2652	. . 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 5847	. . . . 5 |- ( ph -> ( F ` 1o ) e. Q. )
53		1nq 7739	. . . . 5 |- 1Q e. Q.
54		addclnq 7748	. . . . 5 |- ( ( ( F ` 1o ) e. Q. /\ 1Q e. Q. ) -> ( ( F ` 1o ) +Q 1Q ) e. Q. )
55	52, 53, 54	sylancl 417	. . . 4 |- ( ph -> ( ( F ` 1o ) +Q 1Q ) e. Q. )
56		addclnq 7748	. . . 4 |- ( ( ( ( F ` 1o ) +Q 1Q ) e. Q. /\ 1Q e. Q. ) -> ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) e. Q. )
57	55, 53, 56	sylancl 417	. . 3 |- ( ph -> ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) e. Q. )
58		df-1nqqs 7724	. . . . . . . . 9 |- 1Q = [ <. 1o , 1o >. ] ~Q
59	58	fveq2i 5701	. . . . . . . 8 |- ( *Q ` 1Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q )
60		rec1nq 7768	. . . . . . . 8 |- ( *Q ` 1Q ) = 1Q
61	59, 60	eqtr3i 2261	. . . . . . 7 |- ( *Q ` [ <. 1o , 1o >. ] ~Q ) = 1Q
62	61	oveq2i 6099	. . . . . 6 |- ( ( F ` 1o ) +Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) ) = ( ( F ` 1o ) +Q 1Q )
63		ltaddnq 7780	. . . . . . 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 417	. . . . . 6 |- ( ph -> ( ( F ` 1o ) +Q 1Q ) <Q ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) )
65	62, 64	eqbrtrid 4166	. . . . 5 |- ( ph -> ( ( F ` 1o ) +Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) ) <Q ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) )
66		opeq1 3904	. . . . . . . . . 10 |- ( j = 1o -> <. j , 1o >. = <. 1o , 1o >. )
67	66	eceq1d 6847	. . . . . . . . 9 |- ( j = 1o -> [ <. j , 1o >. ] ~Q = [ <. 1o , 1o >. ] ~Q )
68	67	fveq2d 5702	. . . . . . . 8 |- ( j = 1o -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q ) )
69	1, 68	oveq12d 6106	. . . . . . 7 |- ( j = 1o -> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` 1o ) +Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) ) )
70	69	breq1d 4141	. . . . . 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 2929	. . . . 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 415	. . . 4 |- ( ph -> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) )
73		breq2 4135	. . . . . 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 2551	. . . . 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 5701	. . . . . 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 6385	. . . . . 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 2259	. . . . 5 |- ( 2nd ` L ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
78	74, 77	elrab2 2985	. . . 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 421	. . 3 |- ( ph -> ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) e. ( 2nd ` L ) )
80		eleq1 2301	. . . 4 |- ( r = ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) -> ( r e. ( 2nd ` L ) <-> ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) e. ( 2nd ` L ) ) )
81	80	rspcev 2929	. . 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 415	. 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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   {crab 2532   <.cop 3712   class class class wbr 4131   -->wf 5376   ` cfv 5380  (class class class)co 6088   1stc1st 6376   2ndc2nd 6377   1oc1o 6684   [cec 6809   N.cnpi 7645   <N clti 7648   ~Q ceq 7652   Q.cnq 7653   1Qc1q 7654   +Q cplq 7655   *Qcrq 7657   <Q cltq 7658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7677  df-pli 7678  df-mi 7679  df-lti 7680  df-plpq 7717  df-mpq 7718  df-enq 7720  df-nqqs 7721  df-plqqs 7722  df-mqqs 7723  df-1nqqs 7724  df-rq 7725  df-ltnqqs 7726

This theorem is used by:  caucvgprlemcl  8049