Theorem prplnqu 7561

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 7488	. . . . . . . 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 7538	. . . . . . 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 409	. . . . . 6 |- ( ph -> <. { l | l <Q Q } , { u | Q <Q u } >. <P ( <. { l | l <Q Q } , { u | Q <Q u } >. +P. X ) )
7		addcomprg 7519	. . . . . . 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 409	. . . . . 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 4008	. . . . 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 7478	. . . . . . . . 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 409	. . . . . . . 8 |- ( ph -> ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) e. P. )
13		prop 7416	. . . . . . . . 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 7423	. . . . . . . . 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 281	. . . . . . . 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 409	. . . . . . 7 |- ( ph -> A e. Q. )
17		nqpru 7493	. . . . . . 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 409	. . . . . 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 146	. . . . 5 |- ( ph -> ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) <P <. { l | l <Q A } , { u | A <Q u } >. )
20		ltsopr 7537	. . . . . 6 |- <P Or P.
21		ltrelpr 7446	. . . . . 6 |- <P C_ ( P. X. P. )
22	20, 21	sotri 4999	. . . . 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 409	. . . 4 |- ( ph -> <. { l | l <Q Q } , { u | Q <Q u } >. <P <. { l | l <Q A } , { u | A <Q u } >. )
24		ltnqpr 7534	. . . . 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 409	. . . 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 166	. . 3 |- ( ph -> Q <Q A )
27		ltexnqi 7350	. . 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 274	. . . . . 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 274	. . . . . . . . . 10 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> Q e. Q. )
31		simprl 521	. . . . . . . . . 10 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> z e. Q. )
32		addcomnqg 7322	. . . . . . . . . 10 |- ( ( Q e. Q. /\ z e. Q. ) -> ( Q +Q z ) = ( z +Q Q ) )
33	30, 31, 32	syl2anc 409	. . . . . . . . 9 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> ( Q +Q z ) = ( z +Q Q ) )
34		simprr 522	. . . . . . . . 9 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> ( Q +Q z ) = A )
35	33, 34	eqtr3d 2200	. . . . . . . 8 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> ( z +Q Q ) = A )
36		breq2 3986	. . . . . . . . . 10 |- ( ( z +Q Q ) = A -> ( l <Q ( z +Q Q ) <-> l <Q A ) )
37	36	abbidv 2284	. . . . . . . . 9 |- ( ( z +Q Q ) = A -> { l | l <Q ( z +Q Q ) } = { l | l <Q A } )
38		breq1 3985	. . . . . . . . . 10 |- ( ( z +Q Q ) = A -> ( ( z +Q Q ) <Q u <-> A <Q u ) )
39	38	abbidv 2284	. . . . . . . . 9 |- ( ( z +Q Q ) = A -> { u | ( z +Q Q ) <Q u } = { u | A <Q u } )
40	37, 39	opeq12d 3766	. . . . . . . 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 7502	. . . . . . . 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 409	. . . . . . 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 2200	. . . . . 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 4008	. . . . 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 7560	. . . . . . 7 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
47	46	adantl 275	. . . . . 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 274	. . . . . 6 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> X e. P. )
49		nqprlu 7488	. . . . . . 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 7519	. . . . . . 7 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
53	52	adantl 275	. . . . . 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 6011	. . . . 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 166	. . . 4 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> X <P <. { l | l <Q z } , { u | z <Q u } >. )
56		nqpru 7493	. . . . 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 409	. . . 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 166	. . 3 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> z e. ( 2nd ` X ) )
59		oveq1 5849	. . . . 5 |- ( y = z -> ( y +Q Q ) = ( z +Q Q ) )
60	59	eqeq1d 2174	. . . 4 |- ( y = z -> ( ( y +Q Q ) = A <-> ( z +Q Q ) = A ) )
61	60	rspcev 2830	. . 3 |- ( ( z e. ( 2nd ` X ) /\ ( z +Q Q ) = A ) -> E. y e. ( 2nd ` X ) ( y +Q Q ) = A )
62	58, 35, 61	syl2anc 409	. 2 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> E. y e. ( 2nd ` X ) ( y +Q Q ) = A )
63	28, 62	rexlimddv 2588	1 |- ( ph -> E. y e. ( 2nd ` X ) ( y +Q Q ) = A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   {cab 2151   E.wrex 2445   <.cop 3579   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   1stc1st 6106   2ndc2nd 6107   Q.cnq 7221   +Q cplq 7223   <Q cltq 7226   P.cnp 7232   +P. cpp 7234   <P cltp 7236

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-2o 6385  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  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-iplp 7409  df-iltp 7411

This theorem is referenced by:  caucvgprprlemexbt  7647