Theorem prplnqu 7680

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 7607	. . . . . . . 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 7657	. . . . . . 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 411	. . . . . 6 |- ( ph -> <. { l | l <Q Q } , { u | Q <Q u } >. <P ( <. { l | l <Q Q } , { u | Q <Q u } >. +P. X ) )
7		addcomprg 7638	. . . . . . 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 411	. . . . . 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 4055	. . . . 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 7597	. . . . . . . . 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 411	. . . . . . . 8 |- ( ph -> ( X +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) e. P. )
13		prop 7535	. . . . . . . . 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 7542	. . . . . . . . 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 411	. . . . . . 7 |- ( ph -> A e. Q. )
17		nqpru 7612	. . . . . . 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 411	. . . . . 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 7656	. . . . . 6 |- <P Or P.
21		ltrelpr 7565	. . . . . 6 |- <P C_ ( P. X. P. )
22	20, 21	sotri 5061	. . . . 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 411	. . . 4 |- ( ph -> <. { l | l <Q Q } , { u | Q <Q u } >. <P <. { l | l <Q A } , { u | A <Q u } >. )
24		ltnqpr 7653	. . . . 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 411	. . . 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 7469	. . 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 529	. . . . . . . . . 10 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> z e. Q. )
32		addcomnqg 7441	. . . . . . . . . 10 |- ( ( Q e. Q. /\ z e. Q. ) -> ( Q +Q z ) = ( z +Q Q ) )
33	30, 31, 32	syl2anc 411	. . . . . . . . 9 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> ( Q +Q z ) = ( z +Q Q ) )
34		simprr 531	. . . . . . . . 9 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> ( Q +Q z ) = A )
35	33, 34	eqtr3d 2228	. . . . . . . 8 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> ( z +Q Q ) = A )
36		breq2 4033	. . . . . . . . . 10 |- ( ( z +Q Q ) = A -> ( l <Q ( z +Q Q ) <-> l <Q A ) )
37	36	abbidv 2311	. . . . . . . . 9 |- ( ( z +Q Q ) = A -> { l | l <Q ( z +Q Q ) } = { l | l <Q A } )
38		breq1 4032	. . . . . . . . . 10 |- ( ( z +Q Q ) = A -> ( ( z +Q Q ) <Q u <-> A <Q u ) )
39	38	abbidv 2311	. . . . . . . . 9 |- ( ( z +Q Q ) = A -> { u | ( z +Q Q ) <Q u } = { u | A <Q u } )
40	37, 39	opeq12d 3812	. . . . . . . 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 7621	. . . . . . . 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 411	. . . . . . 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 2228	. . . . . 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 4055	. . . . 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 7679	. . . . . . 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 7607	. . . . . . 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 7638	. . . . . . 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 6088	. . . . 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 7612	. . . . 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 411	. . . 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 5925	. . . . 5 |- ( y = z -> ( y +Q Q ) = ( z +Q Q ) )
60	59	eqeq1d 2202	. . . 4 |- ( y = z -> ( ( y +Q Q ) = A <-> ( z +Q Q ) = A ) )
61	60	rspcev 2864	. . 3 |- ( ( z e. ( 2nd ` X ) /\ ( z +Q Q ) = A ) -> E. y e. ( 2nd ` X ) ( y +Q Q ) = A )
62	58, 35, 61	syl2anc 411	. 2 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> E. y e. ( 2nd ` X ) ( y +Q Q ) = A )
63	28, 62	rexlimddv 2616	1 |- ( ph -> E. y e. ( 2nd ` X ) ( y +Q Q ) = A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   {cab 2179   E.wrex 2473   <.cop 3621   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   1stc1st 6191   2ndc2nd 6192   Q.cnq 7340   +Q cplq 7342   <Q cltq 7345   P.cnp 7351   +P. cpp 7353   <P cltp 7355

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526  df-iplp 7528  df-iltp 7530

This theorem is referenced by:  caucvgprprlemexbt  7766