Theorem addnqprlemrl 7558

Description: Lemma for addnqpr 7562. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)

Assertion

Ref	Expression
addnqprlemrl	|- ( ( A e. Q. /\ B e. Q. ) -> ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) C_ ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) )

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

Proof of Theorem addnqprlemrl

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 7548	. . . . . 6 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
2		nqprlu 7548	. . . . . 6 |- ( B e. Q. -> <. { l | l <Q B } , { u | B <Q u } >. e. P. )
3		df-iplp 7469	. . . . . . 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 7376	. . . . . . 7 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
5	3, 4	genpelvl 7513	. . . . . 6 |- ( ( <. { l | l <Q A } , { u | A <Q u } >. e. P. /\ <. { l | l <Q B } , { u | B <Q u } >. e. P. ) -> ( r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) <-> E. s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 1st ` <. { 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. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) <-> E. s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 1st ` <. { 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. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> E. s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) r = ( s +Q t ) )
8		vex 2742	. . . . . . . . . . . . 13 |- s e. _V
9		breq1 4008	. . . . . . . . . . . . 13 |- ( l = s -> ( l <Q A <-> s <Q A ) )
10		ltnqex 7550	. . . . . . . . . . . . . 14 |- { l | l <Q A } e. _V
11		gtnqex 7551	. . . . . . . . . . . . . 14 |- { u | A <Q u } e. _V
12	10, 11	op1st 6149	. . . . . . . . . . . . 13 |- ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) = { l | l <Q A }
13	8, 9, 12	elab2 2887	. . . . . . . . . . . 12 |- ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) <-> s <Q A )
14	13	biimpi 120	. . . . . . . . . . 11 |- ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) -> s <Q A )
15	14	ad2antrl 490	. . . . . . . . . 10 |- ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> s <Q A )
16	15	adantr 276	. . . . . . . . 9 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> s <Q A )
17		vex 2742	. . . . . . . . . . . . 13 |- t e. _V
18		breq1 4008	. . . . . . . . . . . . 13 |- ( l = t -> ( l <Q B <-> t <Q B ) )
19		ltnqex 7550	. . . . . . . . . . . . . 14 |- { l | l <Q B } e. _V
20		gtnqex 7551	. . . . . . . . . . . . . 14 |- { u | B <Q u } e. _V
21	19, 20	op1st 6149	. . . . . . . . . . . . 13 |- ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) = { l | l <Q B }
22	17, 18, 21	elab2 2887	. . . . . . . . . . . 12 |- ( t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) <-> t <Q B )
23	22	biimpi 120	. . . . . . . . . . 11 |- ( t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) -> t <Q B )
24	23	ad2antll 491	. . . . . . . . . 10 |- ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> t <Q B )
25	24	adantr 276	. . . . . . . . 9 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> t <Q B )
26		ltrelnq 7366	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
27	26	brel 4680	. . . . . . . . . . 11 |- ( s <Q A -> ( s e. Q. /\ A e. Q. ) )
28	16, 27	syl 14	. . . . . . . . . 10 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( s e. Q. /\ A e. Q. ) )
29	26	brel 4680	. . . . . . . . . . 11 |- ( t <Q B -> ( t e. Q. /\ B e. Q. ) )
30	25, 29	syl 14	. . . . . . . . . 10 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( t e. Q. /\ B e. Q. ) )
31		lt2addnq 7405	. . . . . . . . . 10 |- ( ( ( s e. Q. /\ A e. Q. ) /\ ( t e. Q. /\ B e. Q. ) ) -> ( ( s <Q A /\ t <Q B ) -> ( s +Q t ) <Q ( A +Q B ) ) )
32	28, 30, 31	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( ( s <Q A /\ t <Q B ) -> ( s +Q t ) <Q ( A +Q B ) ) )
33	16, 25, 32	mp2and 433	. . . . . . . 8 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( s +Q t ) <Q ( A +Q B ) )
34		breq1 4008	. . . . . . . . 9 |- ( r = ( s +Q t ) -> ( r <Q ( A +Q B ) <-> ( s +Q t ) <Q ( A +Q B ) ) )
35	34	adantl 277	. . . . . . . 8 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( r <Q ( A +Q B ) <-> ( s +Q t ) <Q ( A +Q B ) ) )
36	33, 35	mpbird 167	. . . . . . 7 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> r <Q ( A +Q B ) )
37		vex 2742	. . . . . . . 8 |- r e. _V
38		breq1 4008	. . . . . . . 8 |- ( l = r -> ( l <Q ( A +Q B ) <-> r <Q ( A +Q B ) ) )
39		ltnqex 7550	. . . . . . . . 9 |- { l | l <Q ( A +Q B ) } e. _V
40		gtnqex 7551	. . . . . . . . 9 |- { u | ( A +Q B ) <Q u } e. _V
41	39, 40	op1st 6149	. . . . . . . 8 |- ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) = { l | l <Q ( A +Q B ) }
42	37, 38, 41	elab2 2887	. . . . . . 7 |- ( r e. ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) <-> r <Q ( A +Q B ) )
43	36, 42	sylibr 134	. . . . . 6 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> r e. ( 1st ` <. { 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. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> ( r = ( s +Q t ) -> r e. ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) ) )
45	44	rexlimdvva 2602	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> ( E. s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) r = ( s +Q t ) -> r e. ( 1st ` <. { 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. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> r e. ( 1st ` <. { 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. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) -> r e. ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) ) )
48	47	ssrdv 3163	1 |- ( ( A e. Q. /\ B e. Q. ) -> ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) C_ ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   {cab 2163   E.wrex 2456   C_ wss 3131   <.cop 3597   class class class wbr 4005   ` cfv 5218  (class class class)co 5877   1stc1st 6141   Q.cnq 7281   +Q cplq 7283   <Q cltq 7286   P.cnp 7292   +P. cpp 7294

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-inp 7467  df-iplp 7469

This theorem is referenced by:  addnqprlemfu  7561  addnqpr  7562