Theorem caucvgprlemcanl 7645

Description: Lemma for cauappcvgprlemladdrl 7658. 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 7620	. . . 4 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
2	1	adantl 277	. . 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 7548	. . . 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 7548	. . . . 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 7538	. . . 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 411	. . 3 |- ( ph -> ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) e. P. )
12		caucvgprlemcanl.q	. . . 4 |- ( ph -> Q e. Q. )
13		nqprlu 7548	. . . 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 7579	. . . 4 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
16	15	adantl 277	. . 3 |- ( ( ph /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) = ( g +P. f ) )
17	2, 5, 11, 14, 16	caovord2d 6046	. 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 7552	. . 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 411	. 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 7562	. . . . 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 411	. . . 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 7562	. . . . . 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 411	. . . . 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 5893	. . . 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 4018	. . 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 7376	. . . . 5 |- ( ( R e. Q. /\ Q e. Q. ) -> ( R +Q Q ) e. Q. )
27	3, 12, 26	syl2anc 411	. . . 4 |- ( ph -> ( R +Q Q ) e. Q. )
28		addclnq 7376	. . . . . . 7 |- ( ( S e. Q. /\ Q e. Q. ) -> ( S +Q Q ) e. Q. )
29	7, 12, 28	syl2anc 411	. . . . . 6 |- ( ph -> ( S +Q Q ) e. Q. )
30		nqprlu 7548	. . . . . 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 7538	. . . . 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 411	. . . 4 |- ( ph -> ( L +P. <. { l | l <Q ( S +Q Q ) } , { u | ( S +Q Q ) <Q u } >. ) e. P. )
34		nqprl 7552	. . . 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 411	. . 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 7580	. . . . 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 1238	. . . 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 4017	. . 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 220	. 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 221	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 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   {cab 2163   <.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   <P cltp 7296

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-2o 6420  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-enq0 7425  df-nq0 7426  df-0nq0 7427  df-plq0 7428  df-mq0 7429  df-inp 7467  df-iplp 7469  df-iltp 7471

This theorem is referenced by:  cauappcvgprlemladdrl  7658  caucvgprlemladdrl  7679