Theorem cauappcvgprlemopl 7676

Description: Lemma for cauappcvgpr 7692. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.)

Hypotheses

Ref	Expression
cauappcvgpr.f	|- ( ph -> F : Q. --> Q. )
cauappcvgpr.app	|- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )
cauappcvgpr.bnd	|- ( ph -> A. p e. Q. A <Q ( F ` p ) )
cauappcvgpr.lim	|- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.

Assertion

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

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

Allowed substitution hints:   ph( u, l)   A( u, r, q, l)   L( u, l)

Proof of Theorem cauappcvgprlemopl

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 5904	. . . . . . 7 |- ( l = s -> ( l +Q q ) = ( s +Q q ) )
2	1	breq1d 4028	. . . . . 6 |- ( l = s -> ( ( l +Q q ) <Q ( F ` q ) <-> ( s +Q q ) <Q ( F ` q ) ) )
3	2	rexbidv 2491	. . . . 5 |- ( l = s -> ( E. q e. Q. ( l +Q q ) <Q ( F ` q ) <-> E. q e. Q. ( s +Q q ) <Q ( F ` q ) ) )
4		cauappcvgpr.lim	. . . . . . 7 |- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.
5	4	fveq2i 5537	. . . . . 6 |- ( 1st ` L ) = ( 1st ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >. )
6		nqex 7393	. . . . . . . 8 |- Q. e. _V
7	6	rabex 4162	. . . . . . 7 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
8	6	rabex 4162	. . . . . . 7 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
9	7, 8	op1st 6172	. . . . . 6 |- ( 1st ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >. ) = { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) }
10	5, 9	eqtri 2210	. . . . 5 |- ( 1st ` L ) = { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) }
11	3, 10	elrab2 2911	. . . 4 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. q e. Q. ( s +Q q ) <Q ( F ` q ) ) )
12	11	simprbi 275	. . 3 |- ( s e. ( 1st ` L ) -> E. q e. Q. ( s +Q q ) <Q ( F ` q ) )
13	12	adantl 277	. 2 |- ( ( ph /\ s e. ( 1st ` L ) ) -> E. q e. Q. ( s +Q q ) <Q ( F ` q ) )
14		simprr 531	. . . 4 |- ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) -> ( s +Q q ) <Q ( F ` q ) )
15		ltbtwnnqq 7445	. . . 4 |- ( ( s +Q q ) <Q ( F ` q ) <-> E. t e. Q. ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) )
16	14, 15	sylib 122	. . 3 |- ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) -> E. t e. Q. ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) )
17		simplrl 535	. . . . . . . 8 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) -> q e. Q. )
18	11	simplbi 274	. . . . . . . . 9 |- ( s e. ( 1st ` L ) -> s e. Q. )
19	18	ad3antlr 493	. . . . . . . 8 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) -> s e. Q. )
20		ltaddnq 7437	. . . . . . . 8 |- ( ( q e. Q. /\ s e. Q. ) -> q <Q ( q +Q s ) )
21	17, 19, 20	syl2anc 411	. . . . . . 7 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) -> q <Q ( q +Q s ) )
22		addcomnqg 7411	. . . . . . . 8 |- ( ( q e. Q. /\ s e. Q. ) -> ( q +Q s ) = ( s +Q q ) )
23	17, 19, 22	syl2anc 411	. . . . . . 7 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) -> ( q +Q s ) = ( s +Q q ) )
24	21, 23	breqtrd 4044	. . . . . 6 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) -> q <Q ( s +Q q ) )
25		simprrl 539	. . . . . 6 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) -> ( s +Q q ) <Q t )
26		ltsonq 7428	. . . . . . 7 |- <Q Or Q.
27		ltrelnq 7395	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
28	26, 27	sotri 5042	. . . . . 6 |- ( ( q <Q ( s +Q q ) /\ ( s +Q q ) <Q t ) -> q <Q t )
29	24, 25, 28	syl2anc 411	. . . . 5 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) -> q <Q t )
30		simprl 529	. . . . . 6 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) -> t e. Q. )
31		ltexnqq 7438	. . . . . 6 |- ( ( q e. Q. /\ t e. Q. ) -> ( q <Q t <-> E. r e. Q. ( q +Q r ) = t ) )
32	17, 30, 31	syl2anc 411	. . . . 5 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) -> ( q <Q t <-> E. r e. Q. ( q +Q r ) = t ) )
33	29, 32	mpbid 147	. . . 4 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) -> E. r e. Q. ( q +Q r ) = t )
34	25	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) /\ r e. Q. ) /\ ( q +Q r ) = t ) -> ( s +Q q ) <Q t )
35	19	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) /\ r e. Q. ) /\ ( q +Q r ) = t ) -> s e. Q. )
36	17	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) /\ r e. Q. ) /\ ( q +Q r ) = t ) -> q e. Q. )
37		addcomnqg 7411	. . . . . . . . . . . 12 |- ( ( s e. Q. /\ q e. Q. ) -> ( s +Q q ) = ( q +Q s ) )
38	35, 36, 37	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) /\ r e. Q. ) /\ ( q +Q r ) = t ) -> ( s +Q q ) = ( q +Q s ) )
39	38	breq1d 4028	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) /\ r e. Q. ) /\ ( q +Q r ) = t ) -> ( ( s +Q q ) <Q t <-> ( q +Q s ) <Q t ) )
40	34, 39	mpbid 147	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) /\ r e. Q. ) /\ ( q +Q r ) = t ) -> ( q +Q s ) <Q t )
41		simpr 110	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) /\ r e. Q. ) /\ ( q +Q r ) = t ) -> ( q +Q r ) = t )
42	40, 41	breqtrrd 4046	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) /\ r e. Q. ) /\ ( q +Q r ) = t ) -> ( q +Q s ) <Q ( q +Q r ) )
43		simplr 528	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) /\ r e. Q. ) /\ ( q +Q r ) = t ) -> r e. Q. )
44		ltanqg 7430	. . . . . . . . 9 |- ( ( s e. Q. /\ r e. Q. /\ q e. Q. ) -> ( s <Q r <-> ( q +Q s ) <Q ( q +Q r ) ) )
45	35, 43, 36, 44	syl3anc 1249	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) /\ r e. Q. ) /\ ( q +Q r ) = t ) -> ( s <Q r <-> ( q +Q s ) <Q ( q +Q r ) ) )
46	42, 45	mpbird 167	. . . . . . 7 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) /\ r e. Q. ) /\ ( q +Q r ) = t ) -> s <Q r )
47		simprrr 540	. . . . . . . . . . 11 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) -> t <Q ( F ` q ) )
48	47	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) /\ r e. Q. ) /\ ( q +Q r ) = t ) -> t <Q ( F ` q ) )
49		addcomnqg 7411	. . . . . . . . . . . . 13 |- ( ( q e. Q. /\ r e. Q. ) -> ( q +Q r ) = ( r +Q q ) )
50	36, 43, 49	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) /\ r e. Q. ) /\ ( q +Q r ) = t ) -> ( q +Q r ) = ( r +Q q ) )
51	50, 41	eqtr3d 2224	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) /\ r e. Q. ) /\ ( q +Q r ) = t ) -> ( r +Q q ) = t )
52	51	breq1d 4028	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) /\ r e. Q. ) /\ ( q +Q r ) = t ) -> ( ( r +Q q ) <Q ( F ` q ) <-> t <Q ( F ` q ) ) )
53	48, 52	mpbird 167	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) /\ r e. Q. ) /\ ( q +Q r ) = t ) -> ( r +Q q ) <Q ( F ` q ) )
54		rspe 2539	. . . . . . . . 9 |- ( ( q e. Q. /\ ( r +Q q ) <Q ( F ` q ) ) -> E. q e. Q. ( r +Q q ) <Q ( F ` q ) )
55	36, 53, 54	syl2anc 411	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) /\ r e. Q. ) /\ ( q +Q r ) = t ) -> E. q e. Q. ( r +Q q ) <Q ( F ` q ) )
56		oveq1 5904	. . . . . . . . . . 11 |- ( l = r -> ( l +Q q ) = ( r +Q q ) )
57	56	breq1d 4028	. . . . . . . . . 10 |- ( l = r -> ( ( l +Q q ) <Q ( F ` q ) <-> ( r +Q q ) <Q ( F ` q ) ) )
58	57	rexbidv 2491	. . . . . . . . 9 |- ( l = r -> ( E. q e. Q. ( l +Q q ) <Q ( F ` q ) <-> E. q e. Q. ( r +Q q ) <Q ( F ` q ) ) )
59	58, 10	elrab2 2911	. . . . . . . 8 |- ( r e. ( 1st ` L ) <-> ( r e. Q. /\ E. q e. Q. ( r +Q q ) <Q ( F ` q ) ) )
60	43, 55, 59	sylanbrc 417	. . . . . . 7 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) /\ r e. Q. ) /\ ( q +Q r ) = t ) -> r e. ( 1st ` L ) )
61	46, 60	jca 306	. . . . . 6 |- ( ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) /\ r e. Q. ) /\ ( q +Q r ) = t ) -> ( s <Q r /\ r e. ( 1st ` L ) ) )
62	61	ex 115	. . . . 5 |- ( ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) /\ r e. Q. ) -> ( ( q +Q r ) = t -> ( s <Q r /\ r e. ( 1st ` L ) ) ) )
63	62	reximdva 2592	. . . 4 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) -> ( E. r e. Q. ( q +Q r ) = t -> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) ) )
64	33, 63	mpd 13	. . 3 |- ( ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) /\ ( t e. Q. /\ ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) ) ) -> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) )
65	16, 64	rexlimddv 2612	. 2 |- ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) -> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) )
66	13, 65	rexlimddv 2612	1 |- ( ( ph /\ s e. ( 1st ` L ) ) -> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2160   A.wral 2468   E.wrex 2469   {crab 2472   <.cop 3610   class class class wbr 4018   -->wf 5231   ` cfv 5235  (class class class)co 5897   1stc1st 6164   Q.cnq 7310   +Q cplq 7312   <Q cltq 7315

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-1o 6442  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-pli 7335  df-mi 7336  df-lti 7337  df-plpq 7374  df-mpq 7375  df-enq 7377  df-nqqs 7378  df-plqqs 7379  df-mqqs 7380  df-1nqqs 7381  df-rq 7382  df-ltnqqs 7383

This theorem is referenced by:  cauappcvgprlemrnd  7680