Theorem distrlem5pru 7577

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 7562	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )
2	1	3adant3 1017	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. B ) e. P. )
3		mulclpr 7562	. . . . 5 |- ( ( A e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
4	3	3adant2 1016	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
5		df-iplp 7458	. . . . 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 7365	. . . . 5 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
7	5, 6	genpelvu 7503	. . . 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 7459	. . . . . . . 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 7366	. . . . . . . 8 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
11	9, 10	genpelvu 7503	. . . . . . 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 1016	. . . . . 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 7459	. . . . . . . . 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 7503	. . . . . . . 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 1017	. . . . . . 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 7575	. . . . . . . . . . . . . . 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 5878	. . . . . . . . . . . . . . . . . 18 |- ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( v +Q u ) = ( ( x .Q y ) +Q ( f .Q z ) ) )
19	18	eqeq2d 2189	. . . . . . . . . . . . . . . . 17 |- ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( w = ( v +Q u ) <-> w = ( ( x .Q y ) +Q ( f .Q z ) ) ) )
20		eleq1 2240	. . . . . . . . . . . . . . . . 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 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 2600	. . . . . . . . 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 2601	. . . . . . . 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 2601	. . 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 3161	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 978   = wceq 1353   e. wcel 2148   E.wrex 2456   C_ wss 3129   ` cfv 5212  (class class class)co 5869   2ndc2nd 6134   +Q cplq 7272   .Q cmq 7273   P.cnp 7281   +P. cpp 7283   .P. cmp 7284

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-iplp 7458  df-imp 7459

This theorem is referenced by:  distrprg  7578