Theorem distrlem2pr 5140

Description: Lemma for distributive law for positive reals.

Assertion

Ref	Expression
distrlem2pr	|- ((A e. P. /\ (B e. P. /\ C e. P.)) -> ((x e. A /\ (y e. B /\ z e. C)) -> ((x .Q y) +Q (x .Q z)) e. (A .P. (B +P. C))))

Distinct variable groups:   x,y,z,A   x,B,y,z   x,C,y,z

Proof of Theorem distrlem2pr

Step	Hyp	Ref	Expression
1		df-plp 5100	. . . . . 6 |- +P. = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {f | E.g e. w E.h e. v f = (g +Q h)})}
2	1	genpprecl 5116	. . . . 5 |- ((B e. P. /\ C e. P.) -> ((y e. B /\ z e. C) -> (y +Q z) e. (B +P. C)))
3	2	anim2d 563	. . . 4 |- ((B e. P. /\ C e. P.) -> ((x e. A /\ (y e. B /\ z e. C)) -> (x e. A /\ (y +Q z) e. (B +P. C))))
4	3	adantl 390	. . 3 |- ((A e. P. /\ (B e. P. /\ C e. P.)) -> ((x e. A /\ (y e. B /\ z e. C)) -> (x e. A /\ (y +Q z) e. (B +P. C))))
5		df-mp 5101	. . . . 5 |- .P. = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {f | E.g e. w E.h e. v f = (g .Q h)})}
6	5	genpprecl 5116	. . . 4 |- ((A e. P. /\ (B +P. C) e. P.) -> ((x e. A /\ (y +Q z) e. (B +P. C)) -> (x .Q (y +Q z)) e. (A .P. (B +P. C))))
7		addclpr 5132	. . . 4 |- ((B e. P. /\ C e. P.) -> (B +P. C) e. P.)
8	6, 7	sylan2 453	. . 3 |- ((A e. P. /\ (B e. P. /\ C e. P.)) -> ((x e. A /\ (y +Q z) e. (B +P. C)) -> (x .Q (y +Q z)) e. (A .P. (B +P. C))))
9	4, 8	syld 27	. 2 |- ((A e. P. /\ (B e. P. /\ C e. P.)) -> ((x e. A /\ (y e. B /\ z e. C)) -> (x .Q (y +Q z)) e. (A .P. (B +P. C))))
10		visset 1816	. . . 4 |- y e. V
11		visset 1816	. . . 4 |- z e. V
12	10, 11	distrpq 5079	. . 3 |- (x .Q (y +Q z)) = ((x .Q y) +Q (x .Q z))
13	12	eleq1i 1540	. 2 |- ((x .Q (y +Q z)) e. (A .P. (B +P. C)) <-> ((x .Q y) +Q (x .Q z)) e. (A .P. (B +P. C)))
14	9, 13	syl6ib 212	1 |- ((A e. P. /\ (B e. P. /\ C e. P.)) -> ((x e. A /\ (y e. B /\ z e. C)) -> ((x .Q y) +Q (x .Q z)) e. (A .P. (B +P. C))))

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 960  (class class class)co 3969   +Q cplq 4993   .Q cmq 4994  P.cnp 4997   +P. cpp 4999   .P. cmp 5000

This theorem is referenced by:  distrlem4pr 5142

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-plp 5100  df-mp 5101

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