Theorem addnqprlemru 7653

Description: Lemma for addnqpr 7656. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)

Assertion

Ref	Expression
addnqprlemru	|- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) C_ ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) )

Distinct variable groups:   A, l, u   B, l, u

Proof of Theorem addnqprlemru

Dummy variables f g h r s t x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nqprlu 7642	. . . . . 6 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
2		nqprlu 7642	. . . . . 6 |- ( B e. Q. -> <. { l | l <Q B } , { u | B <Q u } >. e. P. )
3		df-iplp 7563	. . . . . . 7 |- +P. = ( x e. P. , y e. P. |-> <. { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 1st ` x ) /\ h e. ( 1st ` y ) /\ f = ( g +Q h ) ) } , { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 2nd ` x ) /\ h e. ( 2nd ` y ) /\ f = ( g +Q h ) ) } >. )
4		addclnq 7470	. . . . . . 7 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
5	3, 4	genpelvu 7608	. . . . . 6 |- ( ( <. { l | l <Q A } , { u | A <Q u } >. e. P. /\ <. { l | l <Q B } , { u | B <Q u } >. e. P. ) -> ( r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) <-> E. s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) r = ( s +Q t ) ) )
6	1, 2, 5	syl2an 289	. . . . 5 |- ( ( A e. Q. /\ B e. Q. ) -> ( r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) <-> E. s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) r = ( s +Q t ) ) )
7	6	biimpa 296	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> E. s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) r = ( s +Q t ) )
8		vex 2774	. . . . . . . . . . . . 13 |- s e. _V
9		breq2 4047	. . . . . . . . . . . . 13 |- ( u = s -> ( A <Q u <-> A <Q s ) )
10		ltnqex 7644	. . . . . . . . . . . . . 14 |- { l | l <Q A } e. _V
11		gtnqex 7645	. . . . . . . . . . . . . 14 |- { u | A <Q u } e. _V
12	10, 11	op2nd 6223	. . . . . . . . . . . . 13 |- ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) = { u | A <Q u }
13	8, 9, 12	elab2 2920	. . . . . . . . . . . 12 |- ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) <-> A <Q s )
14	13	biimpi 120	. . . . . . . . . . 11 |- ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) -> A <Q s )
15	14	ad2antrl 490	. . . . . . . . . 10 |- ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> A <Q s )
16	15	adantr 276	. . . . . . . . 9 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> A <Q s )
17		vex 2774	. . . . . . . . . . . . 13 |- t e. _V
18		breq2 4047	. . . . . . . . . . . . 13 |- ( u = t -> ( B <Q u <-> B <Q t ) )
19		ltnqex 7644	. . . . . . . . . . . . . 14 |- { l | l <Q B } e. _V
20		gtnqex 7645	. . . . . . . . . . . . . 14 |- { u | B <Q u } e. _V
21	19, 20	op2nd 6223	. . . . . . . . . . . . 13 |- ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) = { u | B <Q u }
22	17, 18, 21	elab2 2920	. . . . . . . . . . . 12 |- ( t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) <-> B <Q t )
23	22	biimpi 120	. . . . . . . . . . 11 |- ( t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) -> B <Q t )
24	23	ad2antll 491	. . . . . . . . . 10 |- ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> B <Q t )
25	24	adantr 276	. . . . . . . . 9 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> B <Q t )
26		ltrelnq 7460	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
27	26	brel 4725	. . . . . . . . . . 11 |- ( A <Q s -> ( A e. Q. /\ s e. Q. ) )
28	16, 27	syl 14	. . . . . . . . . 10 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( A e. Q. /\ s e. Q. ) )
29	26	brel 4725	. . . . . . . . . . 11 |- ( B <Q t -> ( B e. Q. /\ t e. Q. ) )
30	25, 29	syl 14	. . . . . . . . . 10 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( B e. Q. /\ t e. Q. ) )
31		lt2addnq 7499	. . . . . . . . . 10 |- ( ( ( A e. Q. /\ s e. Q. ) /\ ( B e. Q. /\ t e. Q. ) ) -> ( ( A <Q s /\ B <Q t ) -> ( A +Q B ) <Q ( s +Q t ) ) )
32	28, 30, 31	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( ( A <Q s /\ B <Q t ) -> ( A +Q B ) <Q ( s +Q t ) ) )
33	16, 25, 32	mp2and 433	. . . . . . . 8 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( A +Q B ) <Q ( s +Q t ) )
34		breq2 4047	. . . . . . . . 9 |- ( r = ( s +Q t ) -> ( ( A +Q B ) <Q r <-> ( A +Q B ) <Q ( s +Q t ) ) )
35	34	adantl 277	. . . . . . . 8 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( ( A +Q B ) <Q r <-> ( A +Q B ) <Q ( s +Q t ) ) )
36	33, 35	mpbird 167	. . . . . . 7 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( A +Q B ) <Q r )
37		vex 2774	. . . . . . . 8 |- r e. _V
38		breq2 4047	. . . . . . . 8 |- ( u = r -> ( ( A +Q B ) <Q u <-> ( A +Q B ) <Q r ) )
39		ltnqex 7644	. . . . . . . . 9 |- { l | l <Q ( A +Q B ) } e. _V
40		gtnqex 7645	. . . . . . . . 9 |- { u | ( A +Q B ) <Q u } e. _V
41	39, 40	op2nd 6223	. . . . . . . 8 |- ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) = { u | ( A +Q B ) <Q u }
42	37, 38, 41	elab2 2920	. . . . . . 7 |- ( r e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) <-> ( A +Q B ) <Q r )
43	36, 42	sylibr 134	. . . . . 6 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> r e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) )
44	43	ex 115	. . . . 5 |- ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> ( r = ( s +Q t ) -> r e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) ) )
45	44	rexlimdvva 2630	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> ( E. s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) r = ( s +Q t ) -> r e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) ) )
46	7, 45	mpd 13	. . 3 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> r e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) )
47	46	ex 115	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) -> r e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) ) )
48	47	ssrdv 3198	1 |- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) C_ ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1372   e. wcel 2175   {cab 2190   E.wrex 2484   C_ wss 3165   <.cop 3635   class class class wbr 4043   ` cfv 5268  (class class class)co 5934   2ndc2nd 6215   Q.cnq 7375   +Q cplq 7377   <Q cltq 7380   P.cnp 7386   +P. cpp 7388

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-iinf 4634

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-eprel 4334  df-id 4338  df-po 4341  df-iso 4342  df-iord 4411  df-on 4413  df-suc 4416  df-iom 4637  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-ov 5937  df-oprab 5938  df-mpo 5939  df-1st 6216  df-2nd 6217  df-recs 6381  df-irdg 6446  df-1o 6492  df-oadd 6496  df-omul 6497  df-er 6610  df-ec 6612  df-qs 6616  df-ni 7399  df-pli 7400  df-mi 7401  df-lti 7402  df-plpq 7439  df-mpq 7440  df-enq 7442  df-nqqs 7443  df-plqqs 7444  df-mqqs 7445  df-1nqqs 7446  df-rq 7447  df-ltnqqs 7448  df-inp 7561  df-iplp 7563

This theorem is referenced by:  addnqprlemfl  7654  addnqpr  7656