Theorem caucvgprlemcanl 7958

Description: Lemma for cauappcvgprlemladdrl 7971. 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 7933	. . . 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 7861	. . . 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 7861	. . . . 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 7851	. . . 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 7861	. . . 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 7892	. . . 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 6223	. 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 7865	. . 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 7875	. . . . 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 7875	. . . . . 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 6065	. . . 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 4121	. . 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 7689	. . . . 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 7689	. . . . . . 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 7861	. . . . . 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 7851	. . . . 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 7865	. . . 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 7893	. . . . 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 1274	. . . 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 4120	. . 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 1005   = wceq 1398   e. wcel 2203   {cab 2218   <.cop 3691   class class class wbr 4108   ` cfv 5351  (class class class)co 6049   1stc1st 6331   Q.cnq 7594   +Q cplq 7596   <Q cltq 7599   P.cnp 7605   +P. cpp 7607   <P cltp 7609

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-eprel 4409  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-1o 6646  df-2o 6647  df-oadd 6650  df-omul 6651  df-er 6766  df-ec 6768  df-qs 6772  df-ni 7618  df-pli 7619  df-mi 7620  df-lti 7621  df-plpq 7658  df-mpq 7659  df-enq 7661  df-nqqs 7662  df-plqqs 7663  df-mqqs 7664  df-1nqqs 7665  df-rq 7666  df-ltnqqs 7667  df-enq0 7738  df-nq0 7739  df-0nq0 7740  df-plq0 7741  df-mq0 7742  df-inp 7780  df-iplp 7782  df-iltp 7784

This theorem is referenced by:  cauappcvgprlemladdrl  7971  caucvgprlemladdrl  7992