Theorem prplnqu 7733

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 7660	. . . . . . . 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 7710	. . . . . . 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 7691	. . . . . . 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 4070	. . . . 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 7650	. . . . . . . . 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 7588	. . . . . . . . 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 7595	. . . . . . . . 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 7665	. . . . . . 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 7709	. . . . . 6 |- <P Or P.
21		ltrelpr 7618	. . . . . 6 |- <P C_ ( P. X. P. )
22	20, 21	sotri 5078	. . . . 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 7706	. . . . 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 7522	. . 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 7494	. . . . . . . . . 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 2240	. . . . . . . 8 |- ( ( ph /\ ( z e. Q. /\ ( Q +Q z ) = A ) ) -> ( z +Q Q ) = A )
36		breq2 4048	. . . . . . . . . 10 |- ( ( z +Q Q ) = A -> ( l <Q ( z +Q Q ) <-> l <Q A ) )
37	36	abbidv 2323	. . . . . . . . 9 |- ( ( z +Q Q ) = A -> { l | l <Q ( z +Q Q ) } = { l | l <Q A } )
38		breq1 4047	. . . . . . . . . 10 |- ( ( z +Q Q ) = A -> ( ( z +Q Q ) <Q u <-> A <Q u ) )
39	38	abbidv 2323	. . . . . . . . 9 |- ( ( z +Q Q ) = A -> { u | ( z +Q Q ) <Q u } = { u | A <Q u } )
40	37, 39	opeq12d 3827	. . . . . . . 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 7674	. . . . . . . 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 2240	. . . . . 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 4070	. . . . 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 7732	. . . . . . 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 7660	. . . . . . 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 7691	. . . . . . 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 6116	. . . . 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 7665	. . . . 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 5951	. . . . 5 |- ( y = z -> ( y +Q Q ) = ( z +Q Q ) )
60	59	eqeq1d 2214	. . . 4 |- ( y = z -> ( ( y +Q Q ) = A <-> ( z +Q Q ) = A ) )
61	60	rspcev 2877	. . 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 2628	1 |- ( ph -> E. y e. ( 2nd ` X ) ( y +Q Q ) = A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   {cab 2191   E.wrex 2485   <.cop 3636   class class class wbr 4044   ` cfv 5271  (class class class)co 5944   1stc1st 6224   2ndc2nd 6225   Q.cnq 7393   +Q cplq 7395   <Q cltq 7398   P.cnp 7404   +P. cpp 7406   <P cltp 7408

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-eprel 4336  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-irdg 6456  df-1o 6502  df-2o 6503  df-oadd 6506  df-omul 6507  df-er 6620  df-ec 6622  df-qs 6626  df-ni 7417  df-pli 7418  df-mi 7419  df-lti 7420  df-plpq 7457  df-mpq 7458  df-enq 7460  df-nqqs 7461  df-plqqs 7462  df-mqqs 7463  df-1nqqs 7464  df-rq 7465  df-ltnqqs 7466  df-enq0 7537  df-nq0 7538  df-0nq0 7539  df-plq0 7540  df-mq0 7541  df-inp 7579  df-iplp 7581  df-iltp 7583

This theorem is referenced by:  caucvgprprlemexbt  7819