Theorem distrlem5prl 7773

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 7759	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )
2	1	3adant3 1041	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. B ) e. P. )
3		mulclpr 7759	. . . . 5 |- ( ( A e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
4	3	3adant2 1040	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
5		df-iplp 7655	. . . . 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 7562	. . . . 5 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
7	5, 6	genpelvl 7699	. . . 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 411	. . 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 7656	. . . . . . . 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 7563	. . . . . . . 8 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
11	9, 10	genpelvl 7699	. . . . . . 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 1040	. . . . . 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 464	. . . . 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 7656	. . . . . . . . 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 7699	. . . . . . . 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 1041	. . . . . . 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 7771	. . . . . . . . . . . . . . 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 6010	. . . . . . . . . . . . . . . . . 18 |- ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( v +Q u ) = ( ( x .Q y ) +Q ( f .Q z ) ) )
19	18	eqeq2d 2241	. . . . . . . . . . . . . . . . 17 |- ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( w = ( v +Q u ) <-> w = ( ( x .Q y ) +Q ( f .Q z ) ) ) )
20		eleq1 2292	. . . . . . . . . . . . . . . . 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	biimtrdi 163	. . . . . . . . . . . . . . . 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 124	. . . . . . . . . . . . . . 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 157	. . . . . . . . . . . . . 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 367	. . . . . . . . . . . . 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 81	. . . . . . . . . . . 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 367	. . . . . . . . . . 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 78	. . . . . . . . . 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 2654	. . . . . . . . 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 2655	. . . . . . . 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 79	. . . . . . 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 150	. . . . . 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 254	. . . . 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 150	. . . 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 2655	. . 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 150	. 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 3230	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 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   C_ wss 3197   ` cfv 5318  (class class class)co 6001   1stc1st 6284   +Q cplq 7469   .Q cmq 7470   P.cnp 7478   +P. cpp 7480   .P. cmp 7481

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-1o 6562  df-2o 6563  df-oadd 6566  df-omul 6567  df-er 6680  df-ec 6682  df-qs 6686  df-ni 7491  df-pli 7492  df-mi 7493  df-lti 7494  df-plpq 7531  df-mpq 7532  df-enq 7534  df-nqqs 7535  df-plqqs 7536  df-mqqs 7537  df-1nqqs 7538  df-rq 7539  df-ltnqqs 7540  df-enq0 7611  df-nq0 7612  df-0nq0 7613  df-plq0 7614  df-mq0 7615  df-inp 7653  df-iplp 7655  df-imp 7656

This theorem is referenced by:  distrprg  7775