Theorem cauappcvgprlemopl 6887

Description: Lemma for cauappcvgpr 6903. 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 5544	. . . . . . 7 |- ( l = s -> ( l +Q q ) = ( s +Q q ) )
2	1	breq1d 3797	. . . . . 6 |- ( l = s -> ( ( l +Q q ) <Q ( F ` q ) <-> ( s +Q q ) <Q ( F ` q ) ) )
3	2	rexbidv 2370	. . . . 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 5206	. . . . . 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 6604	. . . . . . . 8 |- Q. e. _V
7	6	rabex 3924	. . . . . . 7 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
8	6	rabex 3924	. . . . . . 7 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
9	7, 8	op1st 5798	. . . . . 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 2102	. . . . 5 |- ( 1st ` L ) = { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) }
11	3, 10	elrab2 2752	. . . 4 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. q e. Q. ( s +Q q ) <Q ( F ` q ) ) )
12	11	simprbi 269	. . 3 |- ( s e. ( 1st ` L ) -> E. q e. Q. ( s +Q q ) <Q ( F ` q ) )
13	12	adantl 271	. 2 |- ( ( ph /\ s e. ( 1st ` L ) ) -> E. q e. Q. ( s +Q q ) <Q ( F ` q ) )
14		simprr 499	. . . 4 |- ( ( ( ph /\ s e. ( 1st ` L ) ) /\ ( q e. Q. /\ ( s +Q q ) <Q ( F ` q ) ) ) -> ( s +Q q ) <Q ( F ` q ) )
15		ltbtwnnqq 6656	. . . 4 |- ( ( s +Q q ) <Q ( F ` q ) <-> E. t e. Q. ( ( s +Q q ) <Q t /\ t <Q ( F ` q ) ) )
16	14, 15	sylib 120	. . 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 502	. . . . . . . 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 268	. . . . . . . . 9 |- ( s e. ( 1st ` L ) -> s e. Q. )
19	18	ad3antlr 477	. . . . . . . 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 6648	. . . . . . . 8 |- ( ( q e. Q. /\ s e. Q. ) -> q <Q ( q +Q s ) )
21	17, 19, 20	syl2anc 403	. . . . . . 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 6622	. . . . . . . 8 |- ( ( q e. Q. /\ s e. Q. ) -> ( q +Q s ) = ( s +Q q ) )
23	17, 19, 22	syl2anc 403	. . . . . . 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 3811	. . . . . 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 506	. . . . . 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 6639	. . . . . . 7 |- <Q Or Q.
27		ltrelnq 6606	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
28	26, 27	sotri 4744	. . . . . 6 |- ( ( q <Q ( s +Q q ) /\ ( s +Q q ) <Q t ) -> q <Q t )
29	24, 25, 28	syl2anc 403	. . . . 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 498	. . . . . 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 6649	. . . . . 6 |- ( ( q e. Q. /\ t e. Q. ) -> ( q <Q t <-> E. r e. Q. ( q +Q r ) = t ) )
32	17, 30, 31	syl2anc 403	. . . . 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 145	. . . 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 472	. . . . . . . . . 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 472	. . . . . . . . . . . 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 472	. . . . . . . . . . . 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 6622	. . . . . . . . . . . 12 |- ( ( s e. Q. /\ q e. Q. ) -> ( s +Q q ) = ( q +Q s ) )
38	35, 36, 37	syl2anc 403	. . . . . . . . . . 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 3797	. . . . . . . . . 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 145	. . . . . . . . 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 108	. . . . . . . . 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 3813	. . . . . . . 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 497	. . . . . . . . 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 6641	. . . . . . . . 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 1170	. . . . . . . 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 165	. . . . . . 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 507	. . . . . . . . . . 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 472	. . . . . . . . . 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 6622	. . . . . . . . . . . . 13 |- ( ( q e. Q. /\ r e. Q. ) -> ( q +Q r ) = ( r +Q q ) )
50	36, 43, 49	syl2anc 403	. . . . . . . . . . . 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 2116	. . . . . . . . . . 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 3797	. . . . . . . . . 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 165	. . . . . . . . 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 2413	. . . . . . . . 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 403	. . . . . . . 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 5544	. . . . . . . . . . 11 |- ( l = r -> ( l +Q q ) = ( r +Q q ) )
57	56	breq1d 3797	. . . . . . . . . 10 |- ( l = r -> ( ( l +Q q ) <Q ( F ` q ) <-> ( r +Q q ) <Q ( F ` q ) ) )
58	57	rexbidv 2370	. . . . . . . . 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 2752	. . . . . . . 8 |- ( r e. ( 1st ` L ) <-> ( r e. Q. /\ E. q e. Q. ( r +Q q ) <Q ( F ` q ) ) )
60	43, 55, 59	sylanbrc 408	. . . . . . 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 300	. . . . . 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 113	. . . . 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 2464	. . . 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 2482	. 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 2482	1 |- ( ( ph /\ s e. ( 1st ` L ) ) -> E. r e. Q. ( s <Q r /\ r e. ( 1st ` L ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   e. wcel 1434   A.wral 2349   E.wrex 2350   {crab 2353   <.cop 3403   class class class wbr 3787   -->wf 4922   ` cfv 4926  (class class class)co 5537   1stc1st 5790   Q.cnq 6521   +Q cplq 6523   <Q cltq 6526

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-eprel 4046  df-id 4050  df-po 4053  df-iso 4054  df-iord 4123  df-on 4125  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 6013  df-1o 6059  df-oadd 6063  df-omul 6064  df-er 6165  df-ec 6167  df-qs 6171  df-ni 6545  df-pli 6546  df-mi 6547  df-lti 6548  df-plpq 6585  df-mpq 6586  df-enq 6588  df-nqqs 6589  df-plqqs 6590  df-mqqs 6591  df-1nqqs 6592  df-rq 6593  df-ltnqqs 6594

This theorem is referenced by:  cauappcvgprlemrnd  6891