Theorem caucvgprlemcanl 6896

Description: Lemma for cauappcvgprlemladdrl 6909. Cancelling a term from both sides. (Contributed by Jim Kingdon, 15-Aug-2020.)

Hypotheses

Ref	Expression
caucvgprlemcanl.l	|- ( ph -> L e. P. )
caucvgprlemcanl.s	|- ( ph -> S e. Q. )
caucvgprlemcanl.r	|- ( ph -> R e. Q. )
caucvgprlemcanl.q	|- ( ph -> Q e. Q. )

Assertion

Ref	Expression
caucvgprlemcanl	|- ( ph -> ( ( R +Q Q ) e. ( 1st ` ( L +P. <. { l | l <Q ( S +Q Q ) } , { u | ( S +Q Q ) <Q u } >. ) ) <-> R e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) )

Distinct variable groups:   Q, l, u   R, l, u   S, l, u

Allowed substitution hints:   ph( u, l)   L( u, l)

Proof of Theorem caucvgprlemcanl

Dummy variables f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltaprg 6871	. . . 4 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
2	1	adantl 271	. . 3 |- ( ( ph /\ ( f e. P. /\ g e. P. /\ h e. P. ) ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
3		caucvgprlemcanl.r	. . . 4 |- ( ph -> R e. Q. )
4		nqprlu 6799	. . . 4 |- ( R e. Q. -> <. { l | l <Q R } , { u | R <Q u } >. e. P. )
5	3, 4	syl 14	. . 3 |- ( ph -> <. { l | l <Q R } , { u | R <Q u } >. e. P. )
6		caucvgprlemcanl.l	. . . 4 |- ( ph -> L e. P. )
7		caucvgprlemcanl.s	. . . . 5 |- ( ph -> S e. Q. )
8		nqprlu 6799	. . . . 5 |- ( S e. Q. -> <. { l | l <Q S } , { u | S <Q u } >. e. P. )
9	7, 8	syl 14	. . . 4 |- ( ph -> <. { l | l <Q S } , { u | S <Q u } >. e. P. )
10		addclpr 6789	. . . 4 |- ( ( L e. P. /\ <. { l | l <Q S } , { u | S <Q u } >. e. P. ) -> ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) e. P. )
11	6, 9, 10	syl2anc 403	. . 3 |- ( ph -> ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) e. P. )
12		caucvgprlemcanl.q	. . . 4 |- ( ph -> Q e. Q. )
13		nqprlu 6799	. . . 4 |- ( Q e. Q. -> <. { l | l <Q Q } , { u | Q <Q u } >. e. P. )
14	12, 13	syl 14	. . 3 |- ( ph -> <. { l | l <Q Q } , { u | Q <Q u } >. e. P. )
15		addcomprg 6830	. . . 4 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
16	15	adantl 271	. . 3 |- ( ( ph /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) = ( g +P. f ) )
17	2, 5, 11, 14, 16	caovord2d 5701	. 2 |- ( ph -> ( <. { l | l <Q R } , { u | R <Q u } >. <P ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) <-> ( <. { l | l <Q R } , { u | R <Q u } >. +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) <P ( ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) )
18		nqprl 6803	. . 3 |- ( ( R e. Q. /\ ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) e. P. ) -> ( R e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) <-> <. { l | l <Q R } , { u | R <Q u } >. <P ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) )
19	3, 11, 18	syl2anc 403	. 2 |- ( ph -> ( R e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) <-> <. { l | l <Q R } , { u | R <Q u } >. <P ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) )
20		addnqpr 6813	. . . . 5 |- ( ( R e. Q. /\ Q e. Q. ) -> <. { l | l <Q ( R +Q Q ) } , { u | ( R +Q Q ) <Q u } >. = ( <. { l | l <Q R } , { u | R <Q u } >. +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) )
21	3, 12, 20	syl2anc 403	. . . 4 |- ( ph -> <. { l | l <Q ( R +Q Q ) } , { u | ( R +Q Q ) <Q u } >. = ( <. { l | l <Q R } , { u | R <Q u } >. +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) )
22		addnqpr 6813	. . . . . 6 |- ( ( S e. Q. /\ Q e. Q. ) -> <. { l | l <Q ( S +Q Q ) } , { u | ( S +Q Q ) <Q u } >. = ( <. { l | l <Q S } , { u | S <Q u } >. +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) )
23	7, 12, 22	syl2anc 403	. . . . 5 |- ( ph -> <. { l | l <Q ( S +Q Q ) } , { u | ( S +Q Q ) <Q u } >. = ( <. { l | l <Q S } , { u | S <Q u } >. +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) )
24	23	oveq2d 5559	. . . 4 |- ( ph -> ( L +P. <. { l | l <Q ( S +Q Q ) } , { u | ( S +Q Q ) <Q u } >. ) = ( L +P. ( <. { l | l <Q S } , { u | S <Q u } >. +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) )
25	21, 24	breq12d 3806	. . 3 |- ( ph -> ( <. { l | l <Q ( R +Q Q ) } , { u | ( R +Q Q ) <Q u } >. <P ( L +P. <. { l | l <Q ( S +Q Q ) } , { u | ( S +Q Q ) <Q u } >. ) <-> ( <. { l | l <Q R } , { u | R <Q u } >. +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) <P ( L +P. ( <. { l | l <Q S } , { u | S <Q u } >. +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) ) )
26		addclnq 6627	. . . . 5 |- ( ( R e. Q. /\ Q e. Q. ) -> ( R +Q Q ) e. Q. )
27	3, 12, 26	syl2anc 403	. . . 4 |- ( ph -> ( R +Q Q ) e. Q. )
28		addclnq 6627	. . . . . . 7 |- ( ( S e. Q. /\ Q e. Q. ) -> ( S +Q Q ) e. Q. )
29	7, 12, 28	syl2anc 403	. . . . . 6 |- ( ph -> ( S +Q Q ) e. Q. )
30		nqprlu 6799	. . . . . 6 |- ( ( S +Q Q ) e. Q. -> <. { l | l <Q ( S +Q Q ) } , { u | ( S +Q Q ) <Q u } >. e. P. )
31	29, 30	syl 14	. . . . 5 |- ( ph -> <. { l | l <Q ( S +Q Q ) } , { u | ( S +Q Q ) <Q u } >. e. P. )
32		addclpr 6789	. . . . 5 |- ( ( L e. P. /\ <. { l | l <Q ( S +Q Q ) } , { u | ( S +Q Q ) <Q u } >. e. P. ) -> ( L +P. <. { l | l <Q ( S +Q Q ) } , { u | ( S +Q Q ) <Q u } >. ) e. P. )
33	6, 31, 32	syl2anc 403	. . . 4 |- ( ph -> ( L +P. <. { l | l <Q ( S +Q Q ) } , { u | ( S +Q Q ) <Q u } >. ) e. P. )
34		nqprl 6803	. . . 4 |- ( ( ( R +Q Q ) e. Q. /\ ( L +P. <. { l | l <Q ( S +Q Q ) } , { u | ( S +Q Q ) <Q u } >. ) e. P. ) -> ( ( R +Q Q ) e. ( 1st ` ( L +P. <. { l | l <Q ( S +Q Q ) } , { u | ( S +Q Q ) <Q u } >. ) ) <-> <. { l | l <Q ( R +Q Q ) } , { u | ( R +Q Q ) <Q u } >. <P ( L +P. <. { l | l <Q ( S +Q Q ) } , { u | ( S +Q Q ) <Q u } >. ) ) )
35	27, 33, 34	syl2anc 403	. . 3 |- ( ph -> ( ( R +Q Q ) e. ( 1st ` ( L +P. <. { l | l <Q ( S +Q Q ) } , { u | ( S +Q Q ) <Q u } >. ) ) <-> <. { l | l <Q ( R +Q Q ) } , { u | ( R +Q Q ) <Q u } >. <P ( L +P. <. { l | l <Q ( S +Q Q ) } , { u | ( S +Q Q ) <Q u } >. ) ) )
36		addassprg 6831	. . . . 5 |- ( ( L e. P. /\ <. { l | l <Q S } , { u | S <Q u } >. e. P. /\ <. { l | l <Q Q } , { u | Q <Q u } >. e. P. ) -> ( ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) = ( L +P. ( <. { l | l <Q S } , { u | S <Q u } >. +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) )
37	6, 9, 14, 36	syl3anc 1170	. . . 4 |- ( ph -> ( ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) = ( L +P. ( <. { l | l <Q S } , { u | S <Q u } >. +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) )
38	37	breq2d 3805	. . 3 |- ( ph -> ( ( <. { l | l <Q R } , { u | R <Q u } >. +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) <P ( ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) <-> ( <. { l | l <Q R } , { u | R <Q u } >. +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) <P ( L +P. ( <. { l | l <Q S } , { u | S <Q u } >. +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) ) )
39	25, 35, 38	3bitr4d 218	. 2 |- ( ph -> ( ( R +Q Q ) e. ( 1st ` ( L +P. <. { l | l <Q ( S +Q Q ) } , { u | ( S +Q Q ) <Q u } >. ) ) <-> ( <. { l | l <Q R } , { u | R <Q u } >. +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) <P ( ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) +P. <. { l | l <Q Q } , { u | Q <Q u } >. ) ) )
40	17, 19, 39	3bitr4rd 219	1 |- ( ph -> ( ( R +Q Q ) e. ( 1st ` ( L +P. <. { l | l <Q ( S +Q Q ) } , { u | ( S +Q Q ) <Q u } >. ) ) <-> R e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 920   = wceq 1285   e. wcel 1434   {cab 2068   <.cop 3409   class class class wbr 3793   ` cfv 4932  (class class class)co 5543   1stc1st 5796   Q.cnq 6532   +Q cplq 6534   <Q cltq 6537   P.cnp 6543   +P. cpp 6545   <P cltp 6547

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-iplp 6720  df-iltp 6722

This theorem is referenced by:  cauappcvgprlemladdrl  6909  caucvgprlemladdrl  6930