Theorem addnqprlemfu 7780

Description: Lemma for addnqpr 7781. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)

Assertion

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

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

Proof of Theorem addnqprlemfu

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		addnqprlemrl 7777	. . . . . 6 |- ( ( 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 } >. ) )
2		ltsonq 7618	. . . . . . . . 9 |- <Q Or Q.
3		addclnq 7595	. . . . . . . . 9 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) e. Q. )
4		sonr 4414	. . . . . . . . 9 |- ( ( <Q Or Q. /\ ( A +Q B ) e. Q. ) -> -. ( A +Q B ) <Q ( A +Q B ) )
5	2, 3, 4	sylancr 414	. . . . . . . 8 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A +Q B ) <Q ( A +Q B ) )
6		ltrelnq 7585	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
7	6	brel 4778	. . . . . . . . . . 11 |- ( ( A +Q B ) <Q ( A +Q B ) -> ( ( A +Q B ) e. Q. /\ ( A +Q B ) e. Q. ) )
8	7	simpld 112	. . . . . . . . . 10 |- ( ( A +Q B ) <Q ( A +Q B ) -> ( A +Q B ) e. Q. )
9		elex 2814	. . . . . . . . . 10 |- ( ( A +Q B ) e. Q. -> ( A +Q B ) e. _V )
10	8, 9	syl 14	. . . . . . . . 9 |- ( ( A +Q B ) <Q ( A +Q B ) -> ( A +Q B ) e. _V )
11		breq1 4091	. . . . . . . . 9 |- ( l = ( A +Q B ) -> ( l <Q ( A +Q B ) <-> ( A +Q B ) <Q ( A +Q B ) ) )
12	10, 11	elab3 2958	. . . . . . . 8 |- ( ( A +Q B ) e. { l | l <Q ( A +Q B ) } <-> ( A +Q B ) <Q ( A +Q B ) )
13	5, 12	sylnibr 683	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A +Q B ) e. { l | l <Q ( A +Q B ) } )
14		ltnqex 7769	. . . . . . . . 9 |- { l | l <Q ( A +Q B ) } e. _V
15		gtnqex 7770	. . . . . . . . 9 |- { u | ( A +Q B ) <Q u } e. _V
16	14, 15	op1st 6309	. . . . . . . 8 |- ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) = { l | l <Q ( A +Q B ) }
17	16	eleq2i 2298	. . . . . . 7 |- ( ( A +Q B ) e. ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) <-> ( A +Q B ) e. { l | l <Q ( A +Q B ) } )
18	13, 17	sylnibr 683	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A +Q B ) e. ( 1st ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) )
19	1, 18	ssneldd 3230	. . . . 5 |- ( ( A e. Q. /\ B e. Q. ) -> -. ( A +Q B ) e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )
20	19	adantr 276	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) ) -> -. ( A +Q B ) e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )
21		nqprlu 7767	. . . . . . . 8 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
22		nqprlu 7767	. . . . . . . 8 |- ( B e. Q. -> <. { l | l <Q B } , { u | B <Q u } >. e. P. )
23		addclpr 7757	. . . . . . . 8 |- ( ( <. { l | l <Q A } , { u | A <Q u } >. e. P. /\ <. { l | l <Q B } , { u | B <Q u } >. e. P. ) -> ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) e. P. )
24	21, 22, 23	syl2an 289	. . . . . . 7 |- ( ( A e. Q. /\ B e. Q. ) -> ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) e. P. )
25		prop 7695	. . . . . . 7 |- ( ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) e. P. -> <. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) , ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) >. e. P. )
26	24, 25	syl 14	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. ) -> <. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) , ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) >. e. P. )
27		vex 2805	. . . . . . . 8 |- r e. _V
28		breq2 4092	. . . . . . . 8 |- ( u = r -> ( ( A +Q B ) <Q u <-> ( A +Q B ) <Q r ) )
29	14, 15	op2nd 6310	. . . . . . . 8 |- ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) = { u | ( A +Q B ) <Q u }
30	27, 28, 29	elab2 2954	. . . . . . 7 |- ( r e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) <-> ( A +Q B ) <Q r )
31	30	biimpi 120	. . . . . 6 |- ( r e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) -> ( A +Q B ) <Q r )
32		prloc 7711	. . . . . 6 |- ( ( <. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) , ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) >. e. P. /\ ( A +Q B ) <Q r ) -> ( ( A +Q B ) e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) \/ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) )
33	26, 31, 32	syl2an 289	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) ) -> ( ( A +Q B ) e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) \/ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) )
34	33	orcomd 736	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) ) -> ( r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) \/ ( A +Q B ) e. ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) )
35	20, 34	ecased 1385	. . 3 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) ) -> r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )
36	35	ex 115	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( r e. ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) -> r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) )
37	36	ssrdv 3233	1 |- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` <. { l | l <Q ( A +Q B ) } , { u | ( A +Q B ) <Q u } >. ) C_ ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. +P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 715   e. wcel 2202   {cab 2217   _Vcvv 2802   C_ wss 3200   <.cop 3672   class class class wbr 4088   Or wor 4392   ` cfv 5326  (class class class)co 6018   1stc1st 6301   2ndc2nd 6302   Q.cnq 7500   +Q cplq 7502   <Q cltq 7505   P.cnp 7511   +P. cpp 7513

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-1o 6582  df-2o 6583  df-oadd 6586  df-omul 6587  df-er 6702  df-ec 6704  df-qs 6708  df-ni 7524  df-pli 7525  df-mi 7526  df-lti 7527  df-plpq 7564  df-mpq 7565  df-enq 7567  df-nqqs 7568  df-plqqs 7569  df-mqqs 7570  df-1nqqs 7571  df-rq 7572  df-ltnqqs 7573  df-enq0 7644  df-nq0 7645  df-0nq0 7646  df-plq0 7647  df-mq0 7648  df-inp 7686  df-iplp 7688

This theorem is referenced by:  addnqpr  7781