Theorem prplnqu 7993

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 7920	. . . . . . . 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 7970	. . . . . . 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 415	. . . . . 6 |- ( ph -> <. { l | l <Q Q } , { u | Q <Q u } >. <P ( <. { l | l <Q Q } , { u | Q <Q u } >. +P. X ) )
7		addcomprg 7951	. . . . . . 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 415	. . . . . 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 4157	. . . . 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 7910	. . . . . . . . 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 415	. . . . . . . 8 |- ( ph -> ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) e. P. )
13		prop 7848	. . . . . . . . 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 7855	. . . . . . . . 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 283	. . . . . . . 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 415	. . . . . . 7 |- ( ph -> A e. Q. )
17		nqpru 7925	. . . . . . 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 415	. . . . . 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 147	. . . . 5 |- ( ph -> ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) <P <. { l | l <Q A } , { u | A <Q u } >. )
20		ltsopr 7969	. . . . . 6 |- <P Or P.
21		ltrelpr 7878	. . . . . 6 |- <P C_ ( P. X. P. )
22	20, 21	sotri 5186	. . . . 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 415	. . . 4 |- ( ph -> <. { l | l <Q Q } , { u | Q <Q u } >. <P <. { l | l <Q A } , { u | A <Q u } >. )
24		ltnqpr 7966	. . . . 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 415	. . . 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 167	. . 3 |- ( ph -> Q <Q A )
27		ltexnqi 7782	. . 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 276	. . . . . 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 276	. . . . . . . . . 10 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> Q e. Q. )
31		simprl 535	. . . . . . . . . 10 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> z e. Q. )
32		addcomnqg 7754	. . . . . . . . . 10 |- ( ( Q e. Q. /\ z e. Q. ) -> ( Q +Q z ) = ( z +Q Q ) )
33	30, 31, 32	syl2anc 415	. . . . . . . . 9 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> ( Q +Q z ) = ( z +Q Q ) )
34		simprr 537	. . . . . . . . 9 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> ( Q +Q z ) = A )
35	33, 34	eqtr3d 2273	. . . . . . . 8 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> ( z +Q Q ) = A )
36		breq2 4135	. . . . . . . . . 10 |- ( ( z +Q Q ) = A -> ( l <Q ( z +Q Q ) <-> l <Q A ) )
37	36	abbidv 2358	. . . . . . . . 9 |- ( ( z +Q Q ) = A -> { l | l <Q ( z +Q Q ) } = { l | l <Q A } )
38		breq1 4134	. . . . . . . . . 10 |- ( ( z +Q Q ) = A -> ( ( z +Q Q ) <Q u <-> A <Q u ) )
39	38	abbidv 2358	. . . . . . . . 9 |- ( ( z +Q Q ) = A -> { u | ( z +Q Q ) <Q u } = { u | A <Q u } )
40	37, 39	opeq12d 3912	. . . . . . . 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 7934	. . . . . . . 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 415	. . . . . . 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 2273	. . . . . 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 4157	. . . . 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 7992	. . . . . . 7 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
47	46	adantl 277	. . . . . 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 276	. . . . . 6 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> X e. P. )
49		nqprlu 7920	. . . . . . 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 7951	. . . . . . 7 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
53	52	adantl 277	. . . . . 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 6262	. . . . 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 167	. . . 4 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> X <P <. { l | l <Q z } , { u | z <Q u } >. )
56		nqpru 7925	. . . . 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 415	. . . 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 167	. . 3 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> z e. ( 2nd ` X ) )
59		oveq1 6095	. . . . 5 |- ( y = z -> ( y +Q Q ) = ( z +Q Q ) )
60	59	eqeq1d 2247	. . . 4 |- ( y = z -> ( ( y +Q Q ) = A <-> ( z +Q Q ) = A ) )
61	60	rspcev 2929	. . 3 |- ( ( z e. ( 2nd ` X ) /\ ( z +Q Q ) = A ) -> E. y e. ( 2nd ` X ) ( y +Q Q ) = A )
62	58, 35, 61	syl2anc 415	. 2 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> E. y e. ( 2nd ` X ) ( y +Q Q ) = A )
63	28, 62	rexlimddv 2673	1 |- ( ph -> E. y e. ( 2nd ` X ) ( y +Q Q ) = A )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   {cab 2224   E.wrex 2529   <.cop 3712   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   1stc1st 6376   2ndc2nd 6377   Q.cnq 7653   +Q cplq 7655   <Q cltq 7658   P.cnp 7664   +P. cpp 7666   <P cltp 7668

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-2o 6692  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7677  df-pli 7678  df-mi 7679  df-lti 7680  df-plpq 7717  df-mpq 7718  df-enq 7720  df-nqqs 7721  df-plqqs 7722  df-mqqs 7723  df-1nqqs 7724  df-rq 7725  df-ltnqqs 7726  df-enq0 7797  df-nq0 7798  df-0nq0 7799  df-plq0 7800  df-mq0 7801  df-inp 7839  df-iplp 7841  df-iltp 7843

This theorem is used by:  caucvgprprlemexbt  8079