Theorem distrlem5pr 9849

Description: Lemma for distributive law for positive reals. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
distrlem5pr	|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( A .P. B ) +P. ( A .P. C ) ) C_ ( A .P. ( B +P. C ) ) )

Proof of Theorem distrlem5pr

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 9842	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )
2	1	3adant3 1081	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. B ) e. P. )
3		mulclpr 9842	. . . . 5 |- ( ( A e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
4	3	3adant2 1080	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
5		df-plp 9805	. . . . 5 |- +P. = ( x e. P. , y e. P. |-> { f | E. g e. x E. h e. y f = ( g +Q h ) } )
6		addclnq 9767	. . . . 5 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
7	5, 6	genpelv 9822	. . . 4 |- ( ( ( A .P. B ) e. P. /\ ( A .P. C ) e. P. ) -> ( w e. ( ( A .P. B ) +P. ( A .P. C ) ) <-> E. v e. ( A .P. B ) E. u e. ( A .P. C ) w = ( v +Q u ) ) )
8	2, 4, 7	syl2anc 693	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w e. ( ( A .P. B ) +P. ( A .P. C ) ) <-> E. v e. ( A .P. B ) E. u e. ( A .P. C ) w = ( v +Q u ) ) )
9		df-mp 9806	. . . . . . . 8 |- .P. = ( w e. P. , v e. P. |-> { x | E. g e. w E. h e. v x = ( g .Q h ) } )
10		mulclnq 9769	. . . . . . . 8 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
11	9, 10	genpelv 9822	. . . . . . 7 |- ( ( A e. P. /\ C e. P. ) -> ( u e. ( A .P. C ) <-> E. f e. A E. z e. C u = ( f .Q z ) ) )
12	11	3adant2 1080	. . . . . 6 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( u e. ( A .P. C ) <-> E. f e. A E. z e. C u = ( f .Q z ) ) )
13	12	anbi2d 740	. . . . 5 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( v e. ( A .P. B ) /\ u e. ( A .P. C ) ) <-> ( v e. ( A .P. B ) /\ E. f e. A E. z e. C u = ( f .Q z ) ) ) )
14		df-mp 9806	. . . . . . . . 9 |- .P. = ( w e. P. , v e. P. |-> { f | E. g e. w E. h e. v f = ( g .Q h ) } )
15	14, 10	genpelv 9822	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( v e. ( A .P. B ) <-> E. x e. A E. y e. B v = ( x .Q y ) ) )
16	15	3adant3 1081	. . . . . . 7 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( v e. ( A .P. B ) <-> E. x e. A E. y e. B v = ( x .Q y ) ) )
17		distrlem4pr 9848	. . . . . . . . . . . . . . 15 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. A /\ y e. B ) /\ ( f e. A /\ z e. C ) ) ) -> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( A .P. ( B +P. C ) ) )
18		oveq12 6659	. . . . . . . . . . . . . . . . . 18 |- ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( v +Q u ) = ( ( x .Q y ) +Q ( f .Q z ) ) )
19	18	eqeq2d 2632	. . . . . . . . . . . . . . . . 17 |- ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( w = ( v +Q u ) <-> w = ( ( x .Q y ) +Q ( f .Q z ) ) ) )
20		eleq1 2689	. . . . . . . . . . . . . . . . 17 |- ( w = ( ( x .Q y ) +Q ( f .Q z ) ) -> ( w e. ( A .P. ( B +P. C ) ) <-> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( A .P. ( B +P. C ) ) ) )
21	19, 20	syl6bi 243	. . . . . . . . . . . . . . . 16 |- ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( w = ( v +Q u ) -> ( w e. ( A .P. ( B +P. C ) ) <-> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( A .P. ( B +P. C ) ) ) ) )
22	21	imp 445	. . . . . . . . . . . . . . 15 |- ( ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) /\ w = ( v +Q u ) ) -> ( w e. ( A .P. ( B +P. C ) ) <-> ( ( x .Q y ) +Q ( f .Q z ) ) e. ( A .P. ( B +P. C ) ) ) )
23	17, 22	syl5ibrcom 237	. . . . . . . . . . . . . 14 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( ( x e. A /\ y e. B ) /\ ( f e. A /\ z e. C ) ) ) -> ( ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) /\ w = ( v +Q u ) ) -> w e. ( A .P. ( B +P. C ) ) ) )
24	23	exp4b 632	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( ( x e. A /\ y e. B ) /\ ( f e. A /\ z e. C ) ) -> ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( w = ( v +Q u ) -> w e. ( A .P. ( B +P. C ) ) ) ) ) )
25	24	com3l 89	. . . . . . . . . . . 12 |- ( ( ( x e. A /\ y e. B ) /\ ( f e. A /\ z e. C ) ) -> ( ( v = ( x .Q y ) /\ u = ( f .Q z ) ) -> ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w = ( v +Q u ) -> w e. ( A .P. ( B +P. C ) ) ) ) ) )
26	25	exp4b 632	. . . . . . . . . . 11 |- ( ( x e. A /\ y e. B ) -> ( ( f e. A /\ z e. C ) -> ( v = ( x .Q y ) -> ( u = ( f .Q z ) -> ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w = ( v +Q u ) -> w e. ( A .P. ( B +P. C ) ) ) ) ) ) ) )
27	26	com23 86	. . . . . . . . . 10 |- ( ( x e. A /\ y e. B ) -> ( v = ( x .Q y ) -> ( ( f e. A /\ z e. C ) -> ( u = ( f .Q z ) -> ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w = ( v +Q u ) -> w e. ( A .P. ( B +P. C ) ) ) ) ) ) ) )
28	27	rexlimivv 3036	. . . . . . . . 9 |- ( E. x e. A E. y e. B v = ( x .Q y ) -> ( ( f e. A /\ z e. C ) -> ( u = ( f .Q z ) -> ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w = ( v +Q u ) -> w e. ( A .P. ( B +P. C ) ) ) ) ) ) )
29	28	rexlimdvv 3037	. . . . . . . 8 |- ( E. x e. A E. y e. B v = ( x .Q y ) -> ( E. f e. A E. z e. C u = ( f .Q z ) -> ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w = ( v +Q u ) -> w e. ( A .P. ( B +P. C ) ) ) ) ) )
30	29	com3r 87	. . . . . . 7 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( E. x e. A E. y e. B v = ( x .Q y ) -> ( E. f e. A E. z e. C u = ( f .Q z ) -> ( w = ( v +Q u ) -> w e. ( A .P. ( B +P. C ) ) ) ) ) )
31	16, 30	sylbid 230	. . . . . 6 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( v e. ( A .P. B ) -> ( E. f e. A E. z e. C u = ( f .Q z ) -> ( w = ( v +Q u ) -> w e. ( A .P. ( B +P. C ) ) ) ) ) )
32	31	impd 447	. . . . 5 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( v e. ( A .P. B ) /\ E. f e. A E. z e. C u = ( f .Q z ) ) -> ( w = ( v +Q u ) -> w e. ( A .P. ( B +P. C ) ) ) ) )
33	13, 32	sylbid 230	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( v e. ( A .P. B ) /\ u e. ( A .P. C ) ) -> ( w = ( v +Q u ) -> w e. ( A .P. ( B +P. C ) ) ) ) )
34	33	rexlimdvv 3037	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( E. v e. ( A .P. B ) E. u e. ( A .P. C ) w = ( v +Q u ) -> w e. ( A .P. ( B +P. C ) ) ) )
35	8, 34	sylbid 230	. 2 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w e. ( ( A .P. B ) +P. ( A .P. C ) ) -> w e. ( A .P. ( B +P. C ) ) ) )
36	35	ssrdv 3609	1 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( A .P. B ) +P. ( A .P. C ) ) C_ ( A .P. ( B +P. C ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574  (class class class)co 6650   +Q cplq 9677   .Q cmq 9678   P.cnp 9681   +P. cpp 9683   .P. cmp 9684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-plp 9805  df-mp 9806

This theorem is referenced by:  distrpr  9850