Theorem cauappcvgprlemopl 7587

Description: Lemma for cauappcvgpr 7603. 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 5849	. . . . . . 7 |- ( l = s -> ( l +Q q ) = ( s +Q q ) )
2	1	breq1d 3992	. . . . . 6 |- ( l = s -> ( ( l +Q q ) <Q ( F ` q ) <-> ( s +Q q ) <Q ( F ` q ) ) )
3	2	rexbidv 2467	. . . . 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 5489	. . . . . 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 7304	. . . . . . . 8 |- Q. e. _V
7	6	rabex 4126	. . . . . . 7 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
8	6	rabex 4126	. . . . . . 7 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
9	7, 8	op1st 6114	. . . . . 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 2186	. . . . 5 |- ( 1st ` L ) = { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) }
11	3, 10	elrab2 2885	. . . 4 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. q e. Q. ( s +Q q ) <Q ( F ` q ) ) )
12	11	simprbi 273	. . 3 |- ( s e. ( 1st ` L ) -> E. q e. Q. ( s +Q q ) <Q ( F ` q ) )
13	12	adantl 275	. 2 |- ( ( ph /\ s e. ( 1st ` L ) ) -> E. q e. Q. ( s +Q q ) <Q ( F ` q ) )
14		simprr 522	. . . 4 |- ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) -> ( s +Q q ) <Q ( F ` q ) )
15		ltbtwnnqq 7356	. . . 4 |- ( ( s +Q q ) <Q ( F ` q ) <-> E. t e. Q. ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) )
16	14, 15	sylib 121	. . 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 525	. . . . . . . 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 272	. . . . . . . . 9 |- ( s e. ( 1st ` L ) -> s e. Q. )
19	18	ad3antlr 485	. . . . . . . 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 7348	. . . . . . . 8 |- ( ( q e. Q. /\ s e. Q. ) -> q <Q ( q +Q s ) )
21	17, 19, 20	syl2anc 409	. . . . . . 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 7322	. . . . . . . 8 |- ( ( q e. Q. /\ s e. Q. ) -> ( q +Q s ) = ( s +Q q ) )
23	17, 19, 22	syl2anc 409	. . . . . . 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 4008	. . . . . 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 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 ) ) ) ) -> ( s +Q q ) <Q t )
26		ltsonq 7339	. . . . . . 7 |- <Q Or Q.
27		ltrelnq 7306	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
28	26, 27	sotri 4999	. . . . . 6 |- ( ( q <Q ( s +Q q ) /\ ( s +Q q ) <Q t ) -> q <Q t )
29	24, 25, 28	syl2anc 409	. . . . 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 521	. . . . . 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 7349	. . . . . 6 |- ( ( q e. Q. /\ t e. Q. ) -> ( q <Q t <-> E. r e. Q. ( q +Q r ) = t ) )
32	17, 30, 31	syl2anc 409	. . . . 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 146	. . . 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 480	. . . . . . . . . 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 480	. . . . . . . . . . . 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 480	. . . . . . . . . . . 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 7322	. . . . . . . . . . . 12 |- ( ( s e. Q. /\ q e. Q. ) -> ( s +Q q ) = ( q +Q s ) )
38	35, 36, 37	syl2anc 409	. . . . . . . . . . 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 3992	. . . . . . . . . 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 146	. . . . . . . . 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 109	. . . . . . . . 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 4010	. . . . . . . 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 520	. . . . . . . . 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 7341	. . . . . . . . 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 1228	. . . . . . . 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 166	. . . . . . 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 530	. . . . . . . . . . 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 480	. . . . . . . . . 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 7322	. . . . . . . . . . . . 13 |- ( ( q e. Q. /\ r e. Q. ) -> ( q +Q r ) = ( r +Q q ) )
50	36, 43, 49	syl2anc 409	. . . . . . . . . . . 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 2200	. . . . . . . . . . 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 3992	. . . . . . . . . 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 166	. . . . . . . . 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 2515	. . . . . . . . 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 409	. . . . . . . 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 5849	. . . . . . . . . . 11 |- ( l = r -> ( l +Q q ) = ( r +Q q ) )
57	56	breq1d 3992	. . . . . . . . . 10 |- ( l = r -> ( ( l +Q q ) <Q ( F ` q ) <-> ( r +Q q ) <Q ( F ` q ) ) )
58	57	rexbidv 2467	. . . . . . . . 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 2885	. . . . . . . 8 |- ( r e. ( 1st ` L ) <-> ( r e. Q. /\ E. q e. Q. ( r +Q q ) <Q ( F ` q ) ) )
60	43, 55, 59	sylanbrc 414	. . . . . . 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 304	. . . . . 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 114	. . . . 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 2568	. . . 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 2588	. 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 2588	1 |- ( ( ph /\ s e. ( 1st ` L ) ) -> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   {crab 2448   <.cop 3579   class class class wbr 3982   -->wf 5184   ` cfv 5188  (class class class)co 5842   1stc1st 6106   Q.cnq 7221   +Q cplq 7223   <Q cltq 7226

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294

This theorem is referenced by:  cauappcvgprlemrnd  7591