Theorem prplnqu 6872

Description: Membership in the upper cut of a sum of a positive real and a fraction. (Contributed by Jim Kingdon, 16-Jun-2021.)

Hypotheses

Ref	Expression
prplnqu.x	|- ( ph -> X e. P. )
prplnqu.q	|- ( ph -> Q e. Q. )
prplnqu.sum	|- ( ph -> A e. ( 2nd ` ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) )

Assertion

Ref	Expression
prplnqu	|- ( ph -> E. y e. ( 2nd ` X ) ( y +Q Q ) = A )

Distinct variable groups:   A, l, u   y, A   Q, l, u   y, Q   y, X

Allowed substitution hints:   ph( y, u, l)   X( u, l)

Proof of Theorem prplnqu

Dummy variables f g h z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prplnqu.q	. . . . . . . 8 |- ( ph -> Q e. Q. )
2		nqprlu 6799	. . . . . . . 8 |- ( Q e. Q. -> <. { l | l <Q Q } , { u | Q <Q u } >. e. P. )
3	1, 2	syl 14	. . . . . . 7 |- ( ph -> <. { l | l <Q Q } , { u | Q <Q u } >. e. P. )
4		prplnqu.x	. . . . . . 7 |- ( ph -> X e. P. )
5		ltaddpr 6849	. . . . . . 7 |- ( ( <. { l | l <Q Q } , { u | Q <Q u } >. e. P. /\ X e. P. ) -> <. { l | l <Q Q } , { u | Q <Q u } >. <P ( <. { l | l <Q Q } , { u | Q <Q u } >. +P. X ) )
6	3, 4, 5	syl2anc 403	. . . . . 6 |- ( ph -> <. { l | l <Q Q } , { u | Q <Q u } >. <P ( <. { l | l <Q Q } , { u | Q <Q u } >. +P. X ) )
7		addcomprg 6830	. . . . . . 7 |- ( ( <. { l | l <Q Q } , { u | Q <Q u } >. e. P. /\ X e. P. ) -> ( <. { l | l <Q Q } , { u | Q <Q u } >. +P. X ) = ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) )
8	3, 4, 7	syl2anc 403	. . . . . 6 |- ( ph -> ( <. { l | l <Q Q } , { u | Q <Q u } >. +P. X ) = ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) )
9	6, 8	breqtrd 3817	. . . . 5 |- ( ph -> <. { l | l <Q Q } , { u | Q <Q u } >. <P ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) )
10		prplnqu.sum	. . . . . 6 |- ( ph -> A e. ( 2nd ` ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) )
11		addclpr 6789	. . . . . . . . 9 |- ( ( X e. P. /\ <. { l | l <Q Q } , { u | Q <Q u } >. e. P. ) -> ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) e. P. )
12	4, 3, 11	syl2anc 403	. . . . . . . 8 |- ( ph -> ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) e. P. )
13		prop 6727	. . . . . . . . 9 |- ( ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) e. P. -> <. ( 1st ` ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) , ( 2nd ` ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) >. e. P. )
14		elprnqu 6734	. . . . . . . . 9 |- ( ( <. ( 1st ` ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) , ( 2nd ` ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) >. e. P. /\ A e. ( 2nd ` ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) ) -> A e. Q. )
15	13, 14	sylan 277	. . . . . . . 8 |- ( ( ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) e. P. /\ A e. ( 2nd ` ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) ) -> A e. Q. )
16	12, 10, 15	syl2anc 403	. . . . . . 7 |- ( ph -> A e. Q. )
17		nqpru 6804	. . . . . . 7 |- ( ( A e. Q. /\ ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) e. P. ) -> ( A e. ( 2nd ` ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) <-> ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) <P <. { l | l <Q A } , { u | A <Q u } >. ) )
18	16, 12, 17	syl2anc 403	. . . . . 6 |- ( ph -> ( A e. ( 2nd ` ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) <-> ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) <P <. { l | l <Q A } , { u | A <Q u } >. ) )
19	10, 18	mpbid 145	. . . . 5 |- ( ph -> ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) <P <. { l | l <Q A } , { u | A <Q u } >. )
20		ltsopr 6848	. . . . . 6 |- <P Or P.
21		ltrelpr 6757	. . . . . 6 |- <P C_ ( P. X. P. )
22	20, 21	sotri 4750	. . . . 5 |- ( ( <. { l | l <Q Q } , { u | Q <Q u } >. <P ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) /\ ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) <P <. { l | l <Q A } , { u | A <Q u } >. ) -> <. { l | l <Q Q } , { u | Q <Q u } >. <P <. { l | l <Q A } , { u | A <Q u } >. )
23	9, 19, 22	syl2anc 403	. . . 4 |- ( ph -> <. { l | l <Q Q } , { u | Q <Q u } >. <P <. { l | l <Q A } , { u | A <Q u } >. )
24		ltnqpr 6845	. . . . 5 |- ( ( Q e. Q. /\ A e. Q. ) -> ( Q <Q A <-> <. { l | l <Q Q } , { u | Q <Q u } >. <P <. { l | l <Q A } , { u | A <Q u } >. ) )
25	1, 16, 24	syl2anc 403	. . . 4 |- ( ph -> ( Q <Q A <-> <. { l | l <Q Q } , { u | Q <Q u } >. <P <. { l | l <Q A } , { u | A <Q u } >. ) )
26	23, 25	mpbird 165	. . 3 |- ( ph -> Q <Q A )
27		ltexnqi 6661	. . 3 |- ( Q <Q A -> E. z e. Q. ( Q +Q z ) = A )
28	26, 27	syl 14	. 2 |- ( ph -> E. z e. Q. ( Q +Q z ) = A )
29	19	adantr 270	. . . . . 6 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) <P <. { l | l <Q A } , { u | A <Q u } >. )
30	1	adantr 270	. . . . . . . . . 10 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> Q e. Q. )
31		simprl 498	. . . . . . . . . 10 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> z e. Q. )
32		addcomnqg 6633	. . . . . . . . . 10 |- ( ( Q e. Q. /\ z e. Q. ) -> ( Q +Q z ) = ( z +Q Q ) )
33	30, 31, 32	syl2anc 403	. . . . . . . . 9 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> ( Q +Q z ) = ( z +Q Q ) )
34		simprr 499	. . . . . . . . 9 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> ( Q +Q z ) = A )
35	33, 34	eqtr3d 2116	. . . . . . . 8 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> ( z +Q Q ) = A )
36		breq2 3797	. . . . . . . . . 10 |- ( ( z +Q Q ) = A -> ( l <Q ( z +Q Q ) <-> l <Q A ) )
37	36	abbidv 2197	. . . . . . . . 9 |- ( ( z +Q Q ) = A -> { l | l <Q ( z +Q Q ) } = { l | l <Q A } )
38		breq1 3796	. . . . . . . . . 10 |- ( ( z +Q Q ) = A -> ( ( z +Q Q ) <Q u <-> A <Q u ) )
39	38	abbidv 2197	. . . . . . . . 9 |- ( ( z +Q Q ) = A -> { u | ( z +Q Q ) <Q u } = { u | A <Q u } )
40	37, 39	opeq12d 3586	. . . . . . . 8 |- ( ( z +Q Q ) = A -> <. { l | l <Q ( z +Q Q ) } , { u | ( z +Q Q ) <Q u } >. = <. { l | l <Q A } , { u | A <Q u } >. )
41	35, 40	syl 14	. . . . . . 7 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> <. { l | l <Q ( z +Q Q ) } , { u | ( z +Q Q ) <Q u } >. = <. { l | l <Q A } , { u | A <Q u } >. )
42		addnqpr 6813	. . . . . . . 8 |- ( ( z e. Q. /\ Q e. Q. ) -> <. { l | l <Q ( z +Q Q ) } , { u | ( z +Q Q ) <Q u } >. = ( <. { l | l <Q z } , { u | z <Q u } >. +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) )
43	31, 30, 42	syl2anc 403	. . . . . . 7 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> <. { l | l <Q ( z +Q Q ) } , { u | ( z +Q Q ) <Q u } >. = ( <. { l | l <Q z } , { u | z <Q u } >. +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) )
44	41, 43	eqtr3d 2116	. . . . . 6 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> <. { l | l <Q A } , { u | A <Q u } >. = ( <. { l | l <Q z } , { u | z <Q u } >. +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) )
45	29, 44	breqtrd 3817	. . . . 5 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) <P ( <. { l | l <Q z } , { u | z <Q u } >. +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) )
46		ltaprg 6871	. . . . . . 7 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
47	46	adantl 271	. . . . . 6 |- ( ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) /\ ( f e. P. /\ g e. P. /\ h e. P. ) ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
48	4	adantr 270	. . . . . 6 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> X e. P. )
49		nqprlu 6799	. . . . . . 7 |- ( z e. Q. -> <. { l | l <Q z } , { u | z <Q u } >. e. P. )
50	31, 49	syl 14	. . . . . 6 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> <. { l | l <Q z } , { u | z <Q u } >. e. P. )
51	30, 2	syl 14	. . . . . 6 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> <. { l | l <Q Q } , { u | Q <Q u } >. e. P. )
52		addcomprg 6830	. . . . . . 7 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
53	52	adantl 271	. . . . . 6 |- ( ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) = ( g +P. f ) )
54	47, 48, 50, 51, 53	caovord2d 5701	. . . . 5 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> ( X <P <. { l | l <Q z } , { u | z <Q u } >. <-> ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) <P ( <. { l | l <Q z } , { u | z <Q u } >. +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) )
55	45, 54	mpbird 165	. . . 4 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> X <P <. { l | l <Q z } , { u | z <Q u } >. )
56		nqpru 6804	. . . . 5 |- ( ( z e. Q. /\ X e. P. ) -> ( z e. ( 2nd ` X ) <-> X <P <. { l | l <Q z } , { u | z <Q u } >. ) )
57	31, 48, 56	syl2anc 403	. . . 4 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> ( z e. ( 2nd ` X ) <-> X <P <. { l | l <Q z } , { u | z <Q u } >. ) )
58	55, 57	mpbird 165	. . 3 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> z e. ( 2nd ` X ) )
59		oveq1 5550	. . . . 5 |- ( y = z -> ( y +Q Q ) = ( z +Q Q ) )
60	59	eqeq1d 2090	. . . 4 |- ( y = z -> ( ( y +Q Q ) = A <-> ( z +Q Q ) = A ) )
61	60	rspcev 2702	. . 3 |- ( ( z e. ( 2nd ` X ) /\ ( z +Q Q ) = A ) -> E. y e. ( 2nd ` X ) ( y +Q Q ) = A )
62	58, 35, 61	syl2anc 403	. 2 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> E. y e. ( 2nd ` X ) ( y +Q Q ) = A )
63	28, 62	rexlimddv 2482	1 |- ( ph -> E. y e. ( 2nd ` X ) ( y +Q Q ) = A )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 920   = wceq 1285   e. wcel 1434   {cab 2068   E.wrex 2350   <.cop 3409   class class class wbr 3793   ` cfv 4932  (class class class)co 5543   1stc1st 5796   2ndc2nd 5797   Q.cnq 6532   +Q cplq 6534   <Q cltq 6537   P.cnp 6543   +P. cpp 6545   <P cltp 6547

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 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337

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 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-iplp 6720  df-iltp 6722

This theorem is referenced by:  caucvgprprlemexbt  6958