Theorem addnqprlemru 7925

Description: Lemma for addnqpr 7928. 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 7914	. . . . . 6 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
2		nqprlu 7914	. . . . . 6 |- ( B e. Q. -> <. { l | l <Q B } , { u | B <Q u } >. e. P. )
3		df-iplp 7835	. . . . . . 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 7742	. . . . . . 7 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
5	3, 4	genpelvu 7880	. . . . . 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 2824	. . . . . . . . . . . . 13 |- s e. _V
9		breq2 4134	. . . . . . . . . . . . 13 |- ( u = s -> ( A <Q u <-> A <Q s ) )
10		ltnqex 7916	. . . . . . . . . . . . . 14 |- { l | l <Q A } e. _V
11		gtnqex 7917	. . . . . . . . . . . . . 14 |- { u | A <Q u } e. _V
12	10, 11	op2nd 6381	. . . . . . . . . . . . 13 |- ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) = { u | A <Q u }
13	8, 9, 12	elab2 2974	. . . . . . . . . . . 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 494	. . . . . . . . . 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 2824	. . . . . . . . . . . . 13 |- t e. _V
18		breq2 4134	. . . . . . . . . . . . 13 |- ( u = t -> ( B <Q u <-> B <Q t ) )
19		ltnqex 7916	. . . . . . . . . . . . . 14 |- { l | l <Q B } e. _V
20		gtnqex 7917	. . . . . . . . . . . . . 14 |- { u | B <Q u } e. _V
21	19, 20	op2nd 6381	. . . . . . . . . . . . 13 |- ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) = { u | B <Q u }
22	17, 18, 21	elab2 2974	. . . . . . . . . . . 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 495	. . . . . . . . . 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 7732	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
27	26	brel 4827	. . . . . . . . . . 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 4827	. . . . . . . . . . 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 7771	. . . . . . . . . 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 415	. . . . . . . . 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 437	. . . . . . . 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 4134	. . . . . . . . 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 2824	. . . . . . . 8 |- r e. _V
38		breq2 4134	. . . . . . . 8 |- ( u = r -> ( ( A +Q B ) <Q u <-> ( A +Q B ) <Q r ) )
39		ltnqex 7916	. . . . . . . . 9 |- { l | l <Q ( A +Q B ) } e. _V
40		gtnqex 7917	. . . . . . . . 9 |- { u | ( A +Q B ) <Q u } e. _V
41	39, 40	op2nd 6381	. . . . . . . 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 2974	. . . . . . 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 2676	. . . 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 3254	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 } >. ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {cab 2224   E.wrex 2529   C_ wss 3220   <.cop 3712   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   2ndc2nd 6373   Q.cnq 7647   +Q cplq 7649   <Q cltq 7652   P.cnp 7658   +P. cpp 7660

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

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 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-inp 7833  df-iplp 7835

This theorem is used by:  addnqprlemfl  7926  addnqpr  7928