Theorem distrlem5pru 7419

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 7404	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )
2	1	3adant3 1002	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. B ) e. P. )
3		mulclpr 7404	. . . . 5 |- ( ( A e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
4	3	3adant2 1001	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
5		df-iplp 7300	. . . . 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 7207	. . . . 5 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
7	5, 6	genpelvu 7345	. . . 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 409	. . 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 7301	. . . . . . . 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 7208	. . . . . . . 8 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
11	9, 10	genpelvu 7345	. . . . . . 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 1001	. . . . . 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 460	. . . . 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 7301	. . . . . . . . 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 7345	. . . . . . . 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 1002	. . . . . . 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 7417	. . . . . . . . . . . . . . 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 5791	. . . . . . . . . . . . . . . . . 18 |- ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( v +Q u ) = ( ( x .Q y ) +Q ( f .Q z ) ) )
19	18	eqeq2d 2152	. . . . . . . . . . . . . . . . 17 |- ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( w = ( v +Q u ) <-> w = ( ( x .Q y ) +Q ( f .Q z ) ) ) )
20		eleq1 2203	. . . . . . . . . . . . . . . . 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	syl6bi 162	. . . . . . . . . . . . . . . 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 123	. . . . . . . . . . . . . . 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 156	. . . . . . . . . . . . . 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 365	. . . . . . . . . . . . 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 365	. . . . . . . . . . 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 2558	. . . . . . . . 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 2559	. . . . . . . 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 149	. . . . . 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 252	. . . . 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 149	. . . 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 2559	. . 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 149	. 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 3108	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 103   <-> wb 104   /\ w3a 963   = wceq 1332   e. wcel 1481   E.wrex 2418   C_ wss 3076   ` cfv 5131  (class class class)co 5782   2ndc2nd 6045   +Q cplq 7114   .Q cmq 7115   P.cnp 7123   +P. cpp 7125   .P. cmp 7126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-2o 6322  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-enq0 7256  df-nq0 7257  df-0nq0 7258  df-plq0 7259  df-mq0 7260  df-inp 7298  df-iplp 7300  df-imp 7301

This theorem is referenced by:  distrprg  7420