Theorem distrlem5prl 7548

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 7534	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )
2	1	3adant3 1012	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. B ) e. P. )
3		mulclpr 7534	. . . . 5 |- ( ( A e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
4	3	3adant2 1011	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
5		df-iplp 7430	. . . . 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 7337	. . . . 5 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
7	5, 6	genpelvl 7474	. . . 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 409	. . 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 7431	. . . . . . . 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 7338	. . . . . . . 8 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
11	9, 10	genpelvl 7474	. . . . . . 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 1011	. . . . . 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 461	. . . . 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 7431	. . . . . . . . 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 7474	. . . . . . . 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 1012	. . . . . . 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 7546	. . . . . . . . . . . . . . 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 5862	. . . . . . . . . . . . . . . . . 18 |- ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( v +Q u ) = ( ( x .Q y ) +Q ( f .Q z ) ) )
19	18	eqeq2d 2182	. . . . . . . . . . . . . . . . 17 |- ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( w = ( v +Q u ) <-> w = ( ( x .Q y ) +Q ( f .Q z ) ) ) )
20		eleq1 2233	. . . . . . . . . . . . . . . . 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 162	. . . . . . . . . . . . . . . 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 123	. . . . . . . . . . . . . . 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 156	. . . . . . . . . . . . . 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 365	. . . . . . . . . . . . 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 365	. . . . . . . . . . 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 2593	. . . . . . . . 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 2594	. . . . . . . 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 149	. . . . . 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 252	. . . . 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 149	. . . 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 2594	. . 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 149	. 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 3153	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 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   E.wrex 2449   C_ wss 3121   ` cfv 5198  (class class class)co 5853   1stc1st 6117   +Q cplq 7244   .Q cmq 7245   P.cnp 7253   +P. cpp 7255   .P. cmp 7256

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-imp 7431

This theorem is referenced by:  distrprg  7550