Theorem distrlem5pru 7904

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

Assertion

Ref	Expression
distrlem5pru	|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) C_ ( 2nd ` ( A .P. ( B +P. C ) ) ) )

Proof of Theorem distrlem5pru

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 7889	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )
2	1	3adant3 1044	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. B ) e. P. )
3		mulclpr 7889	. . . . 5 |- ( ( A e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
4	3	3adant2 1043	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
5		df-iplp 7785	. . . . 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 7692	. . . . 5 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
7	5, 6	genpelvu 7830	. . . 4 |- ( ( ( A .P. B ) e. P. /\ ( A .P. C ) e. P. ) -> ( w e. ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) <-> E. v e. ( 2nd ` ( A .P. B ) ) E. u e. ( 2nd ` ( A .P. C ) ) w = ( v +Q u ) ) )
8	2, 4, 7	syl2anc 411	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w e. ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) <-> E. v e. ( 2nd ` ( A .P. B ) ) E. u e. ( 2nd ` ( A .P. C ) ) w = ( v +Q u ) ) )
9		df-imp 7786	. . . . . . . 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 7693	. . . . . . . 8 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
11	9, 10	genpelvu 7830	. . . . . . 7 |- ( ( A e. P. /\ C e. P. ) -> ( u e. ( 2nd ` ( A .P. C ) ) <-> E. f e. ( 2nd ` A ) E. z e. ( 2nd ` C ) u = ( f .Q z ) ) )
12	11	3adant2 1043	. . . . . 6 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( u e. ( 2nd ` ( A .P. C ) ) <-> E. f e. ( 2nd ` A ) E. z e. ( 2nd ` C ) u = ( f .Q z ) ) )
13	12	anbi2d 464	. . . . 5 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( v e. ( 2nd ` ( A .P. B ) ) /\ u e. ( 2nd ` ( A .P. C ) ) ) <-> ( v e. ( 2nd ` ( A .P. B ) ) /\ E. f e. ( 2nd ` A ) E. z e. ( 2nd ` C ) u = ( f .Q z ) ) ) )
14		df-imp 7786	. . . . . . . . 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	genpelvu 7830	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( v e. ( 2nd ` ( A .P. B ) ) <-> E. x e. ( 2nd ` A ) E. y e. ( 2nd ` B ) v = ( x .Q y ) ) )
16	15	3adant3 1044	. . . . . . 7 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( v e. ( 2nd ` ( A .P. B ) ) <-> E. x e. ( 2nd ` A ) E. y e. ( 2nd ` B ) v = ( x .Q y ) ) )
17		distrlem4pru 7902	. . . . . . . . . . . . . . 15 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) )
18		oveq12 6061	. . . . . . . . . . . . . . . . . 18 |- ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( v +Q u ) = ( ( x .Q y ) +Q ( f .Q z ) ) )
19	18	eqeq2d 2246	. . . . . . . . . . . . . . . . 17 |- ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( w = ( v +Q u ) <-> w = ( ( x .Q y ) +Q ( f .Q z ) ) ) )
20		eleq1 2297	. . . . . . . . . . . . . . . . 17 |- ( w = ( ( x .Q y ) +Q ( f .Q z ) ) -> ( w e. ( 2nd ` ( A .P. ( B +P. C ) ) ) <-> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
21	19, 20	biimtrdi 163	. . . . . . . . . . . . . . . 16 |- ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( w = ( v +Q u ) -> ( w e. ( 2nd ` ( A .P. ( B +P. C ) ) ) <-> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) ) )
22	21	imp 124	. . . . . . . . . . . . . . 15 |- ( ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) /\ w = ( v +Q u ) ) -> ( w e. ( 2nd ` ( A .P. ( B +P. C ) ) ) <-> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
23	17, 22	syl5ibrcom 157	. . . . . . . . . . . . . 14 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) ) -> ( ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) /\ w = ( v +Q u ) ) -> w e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
24	23	exp4b 367	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) -> ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( w = ( v +Q u ) -> w e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) ) ) )
25	24	com3l 81	. . . . . . . . . . . 12 |- ( ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) /\ ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) ) -> ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w = ( v +Q u ) -> w e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) ) ) )
26	25	exp4b 367	. . . . . . . . . . 11 |- ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) -> ( ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) -> ( v = ( x .Q y ) -> ( u = ( f .Q z ) -> ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w = ( v +Q u ) -> w e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) ) ) ) ) )
27	26	com23 78	. . . . . . . . . 10 |- ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) -> ( v = ( x .Q y ) -> ( ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) -> ( u = ( f .Q z ) -> ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w = ( v +Q u ) -> w e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) ) ) ) ) )
28	27	rexlimivv 2668	. . . . . . . . 9 |- ( E. x e. ( 2nd ` A ) E. y e. ( 2nd ` B ) v = ( x .Q y ) -> ( ( f e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) -> ( u = ( f .Q z ) -> ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w = ( v +Q u ) -> w e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) ) ) ) )
29	28	rexlimdvv 2669	. . . . . . . 8 |- ( E. x e. ( 2nd ` A ) E. y e. ( 2nd ` B ) v = ( x .Q y ) -> ( E. f e. ( 2nd ` A ) E. z e. ( 2nd ` C ) u = ( f .Q z ) -> ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w = ( v +Q u ) -> w e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) ) ) )
30	29	com3r 79	. . . . . . 7 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( E. x e. ( 2nd ` A ) E. y e. ( 2nd ` B ) v = ( x .Q y ) -> ( E. f e. ( 2nd ` A ) E. z e. ( 2nd ` C ) u = ( f .Q z ) -> ( w = ( v +Q u ) -> w e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) ) ) )
31	16, 30	sylbid 150	. . . . . 6 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( v e. ( 2nd ` ( A .P. B ) ) -> ( E. f e. ( 2nd ` A ) E. z e. ( 2nd ` C ) u = ( f .Q z ) -> ( w = ( v +Q u ) -> w e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) ) ) )
32	31	impd 254	. . . . 5 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( v e. ( 2nd ` ( A .P. B ) ) /\ E. f e. ( 2nd ` A ) E. z e. ( 2nd ` C ) u = ( f .Q z ) ) -> ( w = ( v +Q u ) -> w e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) ) )
33	13, 32	sylbid 150	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( v e. ( 2nd ` ( A .P. B ) ) /\ u e. ( 2nd ` ( A .P. C ) ) ) -> ( w = ( v +Q u ) -> w e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) ) )
34	33	rexlimdvv 2669	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( E. v e. ( 2nd ` ( A .P. B ) ) E. u e. ( 2nd ` ( A .P. C ) ) w = ( v +Q u ) -> w e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
35	8, 34	sylbid 150	. 2 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w e. ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) -> w e. ( 2nd ` ( A .P. ( B +P. C ) ) ) ) )
36	35	ssrdv 3246	1 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) C_ ( 2nd ` ( A .P. ( B +P. C ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   E.wrex 2523   C_ wss 3213   ` cfv 5354  (class class class)co 6052   2ndc2nd 6335   +Q cplq 7599   .Q cmq 7600   P.cnp 7608   +P. cpp 7610   .P. cmp 7611

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-eprel 4412  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-1o 6649  df-2o 6650  df-oadd 6653  df-omul 6654  df-er 6769  df-ec 6771  df-qs 6775  df-ni 7621  df-pli 7622  df-mi 7623  df-lti 7624  df-plpq 7661  df-mpq 7662  df-enq 7664  df-nqqs 7665  df-plqqs 7666  df-mqqs 7667  df-1nqqs 7668  df-rq 7669  df-ltnqqs 7670  df-enq0 7741  df-nq0 7742  df-0nq0 7743  df-plq0 7744  df-mq0 7745  df-inp 7783  df-iplp 7785  df-imp 7786

This theorem is referenced by:  distrprg  7905