Theorem distrlem5pru 7947

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 7932	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )
2	1	3adant3 1048	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. B ) e. P. )
3		mulclpr 7932	. . . . 5 |- ( ( A e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
4	3	3adant2 1047	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
5		df-iplp 7828	. . . . 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 7735	. . . . 5 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
7	5, 6	genpelvu 7873	. . . 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 415	. . 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 7829	. . . . . . . 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 7736	. . . . . . . 8 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
11	9, 10	genpelvu 7873	. . . . . . 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 1047	. . . . . 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 468	. . . . 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 7829	. . . . . . . . 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 7873	. . . . . . . 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 1048	. . . . . . 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 7945	. . . . . . . . . . . . . . 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 6087	. . . . . . . . . . . . . . . . . 18 |- ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( v +Q u ) = ( ( x .Q y ) +Q ( f .Q z ) ) )
19	18	eqeq2d 2250	. . . . . . . . . . . . . . . . 17 |- ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( w = ( v +Q u ) <-> w = ( ( x .Q y ) +Q ( f .Q z ) ) ) )
20		eleq1 2301	. . . . . . . . . . . . . . . . 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 2674	. . . . . . . . 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 2675	. . . . . . . 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 2675	. . 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 3254	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 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   C_ wss 3220   ` cfv 5375  (class class class)co 6078   2ndc2nd 6366   +Q cplq 7642   .Q cmq 7643   P.cnp 7651   +P. cpp 7653   .P. cmp 7654

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-eprel 4432  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6081  df-oprab 6082  df-mpo 6083  df-1st 6367  df-2nd 6368  df-recs 6569  df-irdg 6634  df-1o 6680  df-2o 6681  df-oadd 6684  df-omul 6685  df-er 6800  df-ec 6802  df-qs 6806  df-ni 7664  df-pli 7665  df-mi 7666  df-lti 7667  df-plpq 7704  df-mpq 7705  df-enq 7707  df-nqqs 7708  df-plqqs 7709  df-mqqs 7710  df-1nqqs 7711  df-rq 7712  df-ltnqqs 7713  df-enq0 7784  df-nq0 7785  df-0nq0 7786  df-plq0 7787  df-mq0 7788  df-inp 7826  df-iplp 7828  df-imp 7829

This theorem is referenced by:  distrprg  7948