Theorem addnqprlemrl 7624

Description: Lemma for addnqpr 7628. 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 7614	. . . . . 6 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
2		nqprlu 7614	. . . . . 6 |- ( B e. Q. -> <. { l | l <Q B } , { u | B <Q u } >. e. P. )
3		df-iplp 7535	. . . . . . 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 7442	. . . . . . 7 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
5	3, 4	genpelvl 7579	. . . . . 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 2766	. . . . . . . . . . . . 13 |- s e. _V
9		breq1 4036	. . . . . . . . . . . . 13 |- ( l = s -> ( l <Q A <-> s <Q A ) )
10		ltnqex 7616	. . . . . . . . . . . . . 14 |- { l | l <Q A } e. _V
11		gtnqex 7617	. . . . . . . . . . . . . 14 |- { u | A <Q u } e. _V
12	10, 11	op1st 6204	. . . . . . . . . . . . 13 |- ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) = { l | l <Q A }
13	8, 9, 12	elab2 2912	. . . . . . . . . . . 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 2766	. . . . . . . . . . . . 13 |- t e. _V
18		breq1 4036	. . . . . . . . . . . . 13 |- ( l = t -> ( l <Q B <-> t <Q B ) )
19		ltnqex 7616	. . . . . . . . . . . . . 14 |- { l | l <Q B } e. _V
20		gtnqex 7617	. . . . . . . . . . . . . 14 |- { u | B <Q u } e. _V
21	19, 20	op1st 6204	. . . . . . . . . . . . 13 |- ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) = { l | l <Q B }
22	17, 18, 21	elab2 2912	. . . . . . . . . . . 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 7432	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
27	26	brel 4715	. . . . . . . . . . 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 4715	. . . . . . . . . . 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 7471	. . . . . . . . . 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 4036	. . . . . . . . 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 2766	. . . . . . . 8 |- r e. _V
38		breq1 4036	. . . . . . . 8 |- ( l = r -> ( l <Q ( A +Q B ) <-> r <Q ( A +Q B ) ) )
39		ltnqex 7616	. . . . . . . . 9 |- { l | l <Q ( A +Q B ) } e. _V
40		gtnqex 7617	. . . . . . . . 9 |- { u | ( A +Q B ) <Q u } e. _V
41	39, 40	op1st 6204	. . . . . . . 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 2912	. . . . . . 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 2622	. . . 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 3189	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 1364   e. wcel 2167   {cab 2182   E.wrex 2476   C_ wss 3157   <.cop 3625   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   1stc1st 6196   Q.cnq 7347   +Q cplq 7349   <Q cltq 7352   P.cnp 7358   +P. cpp 7360

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420  df-inp 7533  df-iplp 7535

This theorem is referenced by:  addnqprlemfu  7627  addnqpr  7628