Theorem caucvgprlemcanl 7728

Description: Lemma for cauappcvgprlemladdrl 7741. 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 7703	. . . 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 7631	. . . 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 7631	. . . . 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 7621	. . . 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 7631	. . . 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 7662	. . . 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 6097	. 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 7635	. . 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 7645	. . . . 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 7645	. . . . . 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 5941	. . . 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 4047	. . 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 7459	. . . . 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 7459	. . . . . . 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 7631	. . . . . 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 7621	. . . . 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 7635	. . . 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 7663	. . . . 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 1249	. . . 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 4046	. . 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 980   = wceq 1364   e. wcel 2167   {cab 2182   <.cop 3626   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   1stc1st 6205   Q.cnq 7364   +Q cplq 7366   <Q cltq 7369   P.cnp 7375   +P. cpp 7377   <P cltp 7379

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-iplp 7552  df-iltp 7554

This theorem is referenced by:  cauappcvgprlemladdrl  7741  caucvgprlemladdrl  7762