Theorem caucvgprlemcanl 7854

Description: Lemma for cauappcvgprlemladdrl 7867. 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 7829	. . . 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 7757	. . . 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 7757	. . . . 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 7747	. . . 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 7757	. . . 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 7788	. . . 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 6187	. 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 7761	. . 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 7771	. . . . 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 7771	. . . . . 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 6029	. . . 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 4099	. . 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 7585	. . . . 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 7585	. . . . . . 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 7757	. . . . . 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 7747	. . . . 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 7761	. . . 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 7789	. . . . 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 1271	. . . 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 4098	. . 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 1002   = wceq 1395   e. wcel 2200   {cab 2215   <.cop 3670   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   1stc1st 6296   Q.cnq 7490   +Q cplq 7492   <Q cltq 7495   P.cnp 7501   +P. cpp 7503   <P cltp 7505

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-2o 6578  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-pli 7515  df-mi 7516  df-lti 7517  df-plpq 7554  df-mpq 7555  df-enq 7557  df-nqqs 7558  df-plqqs 7559  df-mqqs 7560  df-1nqqs 7561  df-rq 7562  df-ltnqqs 7563  df-enq0 7634  df-nq0 7635  df-0nq0 7636  df-plq0 7637  df-mq0 7638  df-inp 7676  df-iplp 7678  df-iltp 7680

This theorem is referenced by:  cauappcvgprlemladdrl  7867  caucvgprlemladdrl  7888