Theorem addnqprlemrl 7519

Description: Lemma for addnqpr 7523. 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 7509	. . . . . 6 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
2		nqprlu 7509	. . . . . 6 |- ( B e. Q. -> <. { l | l <Q B } , { u | B <Q u } >. e. P. )
3		df-iplp 7430	. . . . . . 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 7337	. . . . . . 7 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
5	3, 4	genpelvl 7474	. . . . . 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 287	. . . . 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 294	. . . 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 2733	. . . . . . . . . . . . 13 |- s e. _V
9		breq1 3992	. . . . . . . . . . . . 13 |- ( l = s -> ( l <Q A <-> s <Q A ) )
10		ltnqex 7511	. . . . . . . . . . . . . 14 |- { l | l <Q A } e. _V
11		gtnqex 7512	. . . . . . . . . . . . . 14 |- { u | A <Q u } e. _V
12	10, 11	op1st 6125	. . . . . . . . . . . . 13 |- ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) = { l | l <Q A }
13	8, 9, 12	elab2 2878	. . . . . . . . . . . 12 |- ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) <-> s <Q A )
14	13	biimpi 119	. . . . . . . . . . 11 |- ( s e. ( 1st ` <. { l | l <Q A } , { u | A <Q u } >. ) -> s <Q A )
15	14	ad2antrl 487	. . . . . . . . . 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 274	. . . . . . . . 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 2733	. . . . . . . . . . . . 13 |- t e. _V
18		breq1 3992	. . . . . . . . . . . . 13 |- ( l = t -> ( l <Q B <-> t <Q B ) )
19		ltnqex 7511	. . . . . . . . . . . . . 14 |- { l | l <Q B } e. _V
20		gtnqex 7512	. . . . . . . . . . . . . 14 |- { u | B <Q u } e. _V
21	19, 20	op1st 6125	. . . . . . . . . . . . 13 |- ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) = { l | l <Q B }
22	17, 18, 21	elab2 2878	. . . . . . . . . . . 12 |- ( t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) <-> t <Q B )
23	22	biimpi 119	. . . . . . . . . . 11 |- ( t e. ( 1st ` <. { l | l <Q B } , { u | B <Q u } >. ) -> t <Q B )
24	23	ad2antll 488	. . . . . . . . . 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 274	. . . . . . . . 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 7327	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
27	26	brel 4663	. . . . . . . . . . 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 4663	. . . . . . . . . . 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 7366	. . . . . . . . . 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 409	. . . . . . . . 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 431	. . . . . . . 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 3992	. . . . . . . . 9 |- ( r = ( s +Q t ) -> ( r <Q ( A +Q B ) <-> ( s +Q t ) <Q ( A +Q B ) ) )
35	34	adantl 275	. . . . . . . 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 166	. . . . . . 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 2733	. . . . . . . 8 |- r e. _V
38		breq1 3992	. . . . . . . 8 |- ( l = r -> ( l <Q ( A +Q B ) <-> r <Q ( A +Q B ) ) )
39		ltnqex 7511	. . . . . . . . 9 |- { l | l <Q ( A +Q B ) } e. _V
40		gtnqex 7512	. . . . . . . . 9 |- { u | ( A +Q B ) <Q u } e. _V
41	39, 40	op1st 6125	. . . . . . . 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 2878	. . . . . . 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 133	. . . . . 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 114	. . . . 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 2595	. . . 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 114	. 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 3153	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 103   <-> wb 104   = wceq 1348   e. wcel 2141   {cab 2156   E.wrex 2449   C_ wss 3121   <.cop 3586   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   1stc1st 6117   Q.cnq 7242   +Q cplq 7244   <Q cltq 7247   P.cnp 7253   +P. cpp 7255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-inp 7428  df-iplp 7430

This theorem is referenced by:  addnqprlemfu  7522  addnqpr  7523