Theorem caucvgprlemm 7630

Description: Lemma for caucvgpr 7644. 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 5496	. . . . . 6 |- ( j = 1o -> ( F ` j ) = ( F ` 1o ) )
2	1	breq2d 4001	. . . . 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 7277	. . . . . 6 |- 1o e. N.
5	4	a1i 9	. . . . 5 |- ( ph -> 1o e. N. )
6	2, 3, 5	rspcdva 2839	. . . 4 |- ( ph -> A <Q ( F ` 1o ) )
7		ltrelnq 7327	. . . . . 6 |- <Q C_ ( Q. X. Q. )
8	7	brel 4663	. . . . 5 |- ( A <Q ( F ` 1o ) -> ( A e. Q. /\ ( F ` 1o ) e. Q. ) )
9	8	simpld 111	. . . 4 |- ( A <Q ( F ` 1o ) -> A e. Q. )
10		halfnqq 7372	. . . 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 525	. . . . . 6 |- ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) -> s e. Q. )
13		archrecnq 7625	. . . . . . . 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 109	. . . . . . . . . . . 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 525	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q s ) -> j e. N. )
17		nnnq 7384	. . . . . . . . . . . . . 14 |- ( j e. N. -> [ <. j , 1o >. ] ~Q e. Q. )
18		recclnq 7354	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) /\ j e. N. ) /\ ( *Q ` [ <. j , 1o >. ] ~Q ) <Q s ) -> s e. Q. )
21		ltanqg 7362	. . . . . . . . . . . . 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 1233	. . . . . . . . . . . 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 146	. . . . . . . . . . 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 529	. . . . . . . . . . 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 4015	. . . . . . . . . 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 2517	. . . . . . . . . . . . 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 491	. . . . . . . . . . 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 7360	. . . . . . . . . . 11 |- <Q Or Q.
31	30, 7	sotri 5006	. . . . . . . . . 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 409	. . . . . . . . 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 114	. . . . . . . 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 2572	. . . . . . 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 5860	. . . . . . . . 9 |- ( l = s -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) )
37	36	breq1d 3999	. . . . . . . 8 |- ( l = s -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( s +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) ) )
38	37	rexbidv 2471	. . . . . . 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 5499	. . . . . . . 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 7325	. . . . . . . . . 10 |- Q. e. _V
42	41	rabex 4133	. . . . . . . . 9 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
43	41	rabex 4133	. . . . . . . . 9 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
44	42, 43	op1st 6125	. . . . . . . 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 2191	. . . . . . 7 |- ( 1st ` L ) = { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) }
46	38, 45	elrab2 2889	. . . . . 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 415	. . . . 5 |- ( ( ( ph /\ s e. Q. ) /\ ( s +Q s ) = A ) -> s e. ( 1st ` L ) )
48	47	ex 114	. . . 4 |- ( ( ph /\ s e. Q. ) -> ( ( s +Q s ) = A -> s e. ( 1st ` L ) ) )
49	48	reximdva 2572	. . 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	ffvelrnd 5632	. . . . 5 |- ( ph -> ( F ` 1o ) e. Q. )
53		1nq 7328	. . . . 5 |- 1Q e. Q.
54		addclnq 7337	. . . . 5 |- ( ( ( F ` 1o ) e. Q. /\ 1Q e. Q. ) -> ( ( F ` 1o ) +Q 1Q ) e. Q. )
55	52, 53, 54	sylancl 411	. . . 4 |- ( ph -> ( ( F ` 1o ) +Q 1Q ) e. Q. )
56		addclnq 7337	. . . 4 |- ( ( ( ( F ` 1o ) +Q 1Q ) e. Q. /\ 1Q e. Q. ) -> ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) e. Q. )
57	55, 53, 56	sylancl 411	. . 3 |- ( ph -> ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) e. Q. )
58		df-1nqqs 7313	. . . . . . . . 9 |- 1Q = [ <. 1o , 1o >. ] ~Q
59	58	fveq2i 5499	. . . . . . . 8 |- ( *Q ` 1Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q )
60		rec1nq 7357	. . . . . . . 8 |- ( *Q ` 1Q ) = 1Q
61	59, 60	eqtr3i 2193	. . . . . . 7 |- ( *Q ` [ <. 1o , 1o >. ] ~Q ) = 1Q
62	61	oveq2i 5864	. . . . . 6 |- ( ( F ` 1o ) +Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) ) = ( ( F ` 1o ) +Q 1Q )
63		ltaddnq 7369	. . . . . . 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 411	. . . . . 6 |- ( ph -> ( ( F ` 1o ) +Q 1Q ) <Q ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) )
65	62, 64	eqbrtrid 4024	. . . . 5 |- ( ph -> ( ( F ` 1o ) +Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) ) <Q ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) )
66		opeq1 3765	. . . . . . . . . 10 |- ( j = 1o -> <. j , 1o >. = <. 1o , 1o >. )
67	66	eceq1d 6549	. . . . . . . . 9 |- ( j = 1o -> [ <. j , 1o >. ] ~Q = [ <. 1o , 1o >. ] ~Q )
68	67	fveq2d 5500	. . . . . . . 8 |- ( j = 1o -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. 1o , 1o >. ] ~Q ) )
69	1, 68	oveq12d 5871	. . . . . . 7 |- ( j = 1o -> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` 1o ) +Q ( *Q ` [ <. 1o , 1o >. ] ~Q ) ) )
70	69	breq1d 3999	. . . . . 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 2834	. . . . 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 409	. . . 4 |- ( ph -> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) )
73		breq2 3993	. . . . . 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 2471	. . . . 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 5499	. . . . . 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 6126	. . . . . 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 2191	. . . . 5 |- ( 2nd ` L ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
78	74, 77	elrab2 2889	. . . 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 415	. . 3 |- ( ph -> ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) e. ( 2nd ` L ) )
80		eleq1 2233	. . . 4 |- ( r = ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) -> ( r e. ( 2nd ` L ) <-> ( ( ( F ` 1o ) +Q 1Q ) +Q 1Q ) e. ( 2nd ` L ) ) )
81	80	rspcev 2834	. . 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 409	. 2 |- ( ph -> E. r e. Q. r e. ( 2nd ` L ) )
83	50, 82	jca 304	1 |- ( ph -> ( E. s e. Q. s e. ( 1st ` L ) /\ E. r e. Q. r e. ( 2nd ` L ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   {crab 2452   <.cop 3586   class class class wbr 3989   -->wf 5194   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   1oc1o 6388   [cec 6511   N.cnpi 7234   <N clti 7237   ~Q ceq 7241   Q.cnq 7242   1Qc1q 7243   +Q cplq 7244   *Qcrq 7246   <Q cltq 7247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315

This theorem is referenced by:  caucvgprlemcl  7638