Theorem cauappcvgprlemopl 7647

Description: Lemma for cauappcvgpr 7663. 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 5884	. . . . . . 7 |- ( l = s -> ( l +Q q ) = ( s +Q q ) )
2	1	breq1d 4015	. . . . . 6 |- ( l = s -> ( ( l +Q q ) <Q ( F ` q ) <-> ( s +Q q ) <Q ( F ` q ) ) )
3	2	rexbidv 2478	. . . . 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 5520	. . . . . 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 7364	. . . . . . . 8 |- Q. e. _V
7	6	rabex 4149	. . . . . . 7 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
8	6	rabex 4149	. . . . . . 7 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
9	7, 8	op1st 6149	. . . . . 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 2198	. . . . 5 |- ( 1st ` L ) = { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) }
11	3, 10	elrab2 2898	. . . 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 7416	. . . 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 7408	. . . . . . . 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 7382	. . . . . . . 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 4031	. . . . . 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 7399	. . . . . . 7 |- <Q Or Q.
27		ltrelnq 7366	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
28	26, 27	sotri 5026	. . . . . 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 7409	. . . . . 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 7382	. . . . . . . . . . . 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 4015	. . . . . . . . . 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 4033	. . . . . . . 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 7401	. . . . . . . . 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 1238	. . . . . . . 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 7382	. . . . . . . . . . . . 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 2212	. . . . . . . . . . 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 4015	. . . . . . . . . 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 2526	. . . . . . . . 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 5884	. . . . . . . . . . 11 |- ( l = r -> ( l +Q q ) = ( r +Q q ) )
57	56	breq1d 4015	. . . . . . . . . 10 |- ( l = r -> ( ( l +Q q ) <Q ( F ` q ) <-> ( r +Q q ) <Q ( F ` q ) ) )
58	57	rexbidv 2478	. . . . . . . . 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 2898	. . . . . . . 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 2579	. . . 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 2599	. 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 2599	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 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459   <.cop 3597   class class class wbr 4005   -->wf 5214   ` cfv 5218  (class class class)co 5877   1stc1st 6141   Q.cnq 7281   +Q cplq 7283   <Q cltq 7286

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354

This theorem is referenced by:  cauappcvgprlemrnd  7651