Theorem distrlem5prl 6684

Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)

Assertion

Ref	Expression
distrlem5prl	|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) C_ ( 1st ` ( A .P. ( B +P. C ) ) ) )

Proof of Theorem distrlem5prl

Dummy variables x y z w v u f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulclpr 6670	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )
2	1	3adant3 924	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. B ) e. P. )
3		mulclpr 6670	. . . . 5 |- ( ( A e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
4	3	3adant2 923	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
5		df-iplp 6566	. . . . 5 |- +P. = ( x e. P. , y e. P. |-> <. { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 1st ` x ) /\ h e. ( 1st ` y ) /\ f = ( g +Q h ) ) } , { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 2nd ` x ) /\ h e. ( 2nd ` y ) /\ f = ( g +Q h ) ) } >. )
6		addclnq 6473	. . . . 5 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
7	5, 6	genpelvl 6610	. . . 4 |- ( ( ( A .P. B ) e. P. /\ ( A .P. C ) e. P. ) -> ( w e. ( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) <-> E. v e. ( 1st ` ( A .P. B ) ) E. u e. ( 1st ` ( A .P. C ) ) w = ( v +Q u ) ) )
8	2, 4, 7	syl2anc 391	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w e. ( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) <-> E. v e. ( 1st ` ( A .P. B ) ) E. u e. ( 1st ` ( A .P. C ) ) w = ( v +Q u ) ) )
9		df-imp 6567	. . . . . . . 8 |- .P. = ( w e. P. , v e. P. |-> <. { x e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 1st ` w ) /\ h e. ( 1st ` v ) /\ x = ( g .Q h ) ) } , { x e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 2nd ` w ) /\ h e. ( 2nd ` v ) /\ x = ( g .Q h ) ) } >. )
10		mulclnq 6474	. . . . . . . 8 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
11	9, 10	genpelvl 6610	. . . . . . 7 |- ( ( A e. P. /\ C e. P. ) -> ( u e. ( 1st ` ( A .P. C ) ) <-> E. f e. ( 1st ` A ) E. z e. ( 1st ` C ) u = ( f .Q z ) ) )
12	11	3adant2 923	. . . . . 6 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( u e. ( 1st ` ( A .P. C ) ) <-> E. f e. ( 1st ` A ) E. z e. ( 1st ` C ) u = ( f .Q z ) ) )
13	12	anbi2d 437	. . . . 5 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( v e. ( 1st ` ( A .P. B ) ) /\ u e. ( 1st ` ( A .P. C ) ) ) <-> ( v e. ( 1st ` ( A .P. B ) ) /\ E. f e. ( 1st ` A ) E. z e. ( 1st ` C ) u = ( f .Q z ) ) ) )
14		df-imp 6567	. . . . . . . . 9 |- .P. = ( w e. P. , v e. P. |-> <. { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 1st ` w ) /\ h e. ( 1st ` v ) /\ f = ( g .Q h ) ) } , { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 2nd ` w ) /\ h e. ( 2nd ` v ) /\ f = ( g .Q h ) ) } >. )
15	14, 10	genpelvl 6610	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( v e. ( 1st ` ( A .P. B ) ) <-> E. x e. ( 1st ` A ) E. y e. ( 1st ` B ) v = ( x .Q y ) ) )
16	15	3adant3 924	. . . . . . 7 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( v e. ( 1st ` ( A .P. B ) ) <-> E. x e. ( 1st ` A ) E. y e. ( 1st ` B ) v = ( x .Q y ) ) )
17		distrlem4prl 6682	. . . . . . . . . . . . . . 15 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) )
18		oveq12 5521	. . . . . . . . . . . . . . . . . 18 |- ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( v +Q u ) = ( ( x .Q y ) +Q ( f .Q z ) ) )
19	18	eqeq2d 2051	. . . . . . . . . . . . . . . . 17 |- ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( w = ( v +Q u ) <-> w = ( ( x .Q y ) +Q ( f .Q z ) ) ) )
20		eleq1 2100	. . . . . . . . . . . . . . . . 17 |- ( w = ( ( x .Q y ) +Q ( f .Q z ) ) -> ( w e. ( 1st ` ( A .P. ( B +P. C ) ) ) <-> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) )
21	19, 20	syl6bi 152	. . . . . . . . . . . . . . . 16 |- ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( w = ( v +Q u ) -> ( w e. ( 1st ` ( A .P. ( B +P. C ) ) ) <-> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) ) )
22	21	imp 115	. . . . . . . . . . . . . . 15 |- ( ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) /\ w = ( v +Q u ) ) -> ( w e. ( 1st ` ( A .P. ( B +P. C ) ) ) <-> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) )
23	17, 22	syl5ibrcom 146	. . . . . . . . . . . . . 14 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) ) -> ( ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) /\ w = ( v +Q u ) ) -> w e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) )
24	23	exp4b 349	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) -> ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( w = ( v +Q u ) -> w e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) ) ) )
25	24	com3l 75	. . . . . . . . . . . 12 |- ( ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) /\ ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) ) -> ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w = ( v +Q u ) -> w e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) ) ) )
26	25	exp4b 349	. . . . . . . . . . 11 |- ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) -> ( ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) -> ( v = ( x .Q y ) -> ( u = ( f .Q z ) -> ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w = ( v +Q u ) -> w e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) ) ) ) ) )
27	26	com23 72	. . . . . . . . . 10 |- ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) -> ( v = ( x .Q y ) -> ( ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) -> ( u = ( f .Q z ) -> ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w = ( v +Q u ) -> w e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) ) ) ) ) )
28	27	rexlimivv 2438	. . . . . . . . 9 |- ( E. x e. ( 1st ` A ) E. y e. ( 1st ` B ) v = ( x .Q y ) -> ( ( f e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) -> ( u = ( f .Q z ) -> ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w = ( v +Q u ) -> w e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) ) ) ) )
29	28	rexlimdvv 2439	. . . . . . . 8 |- ( E. x e. ( 1st ` A ) E. y e. ( 1st ` B ) v = ( x .Q y ) -> ( E. f e. ( 1st ` A ) E. z e. ( 1st ` C ) u = ( f .Q z ) -> ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w = ( v +Q u ) -> w e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) ) ) )
30	29	com3r 73	. . . . . . 7 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( E. x e. ( 1st ` A ) E. y e. ( 1st ` B ) v = ( x .Q y ) -> ( E. f e. ( 1st ` A ) E. z e. ( 1st ` C ) u = ( f .Q z ) -> ( w = ( v +Q u ) -> w e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) ) ) )
31	16, 30	sylbid 139	. . . . . 6 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( v e. ( 1st ` ( A .P. B ) ) -> ( E. f e. ( 1st ` A ) E. z e. ( 1st ` C ) u = ( f .Q z ) -> ( w = ( v +Q u ) -> w e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) ) ) )
32	31	impd 242	. . . . 5 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( v e. ( 1st ` ( A .P. B ) ) /\ E. f e. ( 1st ` A ) E. z e. ( 1st ` C ) u = ( f .Q z ) ) -> ( w = ( v +Q u ) -> w e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) ) )
33	13, 32	sylbid 139	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( v e. ( 1st ` ( A .P. B ) ) /\ u e. ( 1st ` ( A .P. C ) ) ) -> ( w = ( v +Q u ) -> w e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) ) )
34	33	rexlimdvv 2439	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( E. v e. ( 1st ` ( A .P. B ) ) E. u e. ( 1st ` ( A .P. C ) ) w = ( v +Q u ) -> w e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) )
35	8, 34	sylbid 139	. 2 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w e. ( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) -> w e. ( 1st ` ( A .P. ( B +P. C ) ) ) ) )
36	35	ssrdv 2951	1 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) C_ ( 1st ` ( A .P. ( B +P. C ) ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 885   = wceq 1243   e. wcel 1393   E.wrex 2307   C_ wss 2917   ` cfv 4902  (class class class)co 5512   1stc1st 5765   +Q cplq 6380   .Q cmq 6381   P.cnp 6389   +P. cpp 6391   .P. cmp 6392

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-iplp 6566  df-imp 6567

This theorem is referenced by:  distrprg  6686