Theorem genpv 5114

Description: Value of general operation (addition or multiplication) on positive reals.

Hypothesis

Ref	Expression
genp.1	|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}

Assertion

Ref	Expression
genpv	|- ((A e. P. /\ B e. P.) -> (AFB) = {f | E.gE.h((g e. A /\ h e. B) /\ f = (gGh))})

Distinct variable groups:   x,y,z,f,g,h,A   x,B,y,z,f,g,h   x,w,v,u,G,y,z,f,g,h   f,F,g

Proof of Theorem genpv

Step	Hyp	Ref	Expression
1		opreq1 3974	. . . 4 |- (f = A -> (fFg) = (AFg))
2		rexeq1 1790	. . . . 5 |- (f = A -> (E.y e. f E.z e. g x = (yGz) <-> E.y e. A E.z e. g x = (yGz)))
3	2	abbidv 1580	. . . 4 |- (f = A -> {x | E.y e. f E.z e. g x = (yGz)} = {x | E.y e. A E.z e. g x = (yGz)})
4	1, 3	eqeq12d 1492	. . 3 |- (f = A -> ((fFg) = {x | E.y e. f E.z e. g x = (yGz)} <-> (AFg) = {x | E.y e. A E.z e. g x = (yGz)}))
5		opreq2 3975	. . . 4 |- (g = B -> (AFg) = (AFB))
6		rexeq1 1790	. . . . . 6 |- (g = B -> (E.z e. g x = (yGz) <-> E.z e. B x = (yGz)))
7	6	rexbidv 1667	. . . . 5 |- (g = B -> (E.y e. A E.z e. g x = (yGz) <-> E.y e. A E.z e. B x = (yGz)))
8	7	abbidv 1580	. . . 4 |- (g = B -> {x | E.y e. A E.z e. g x = (yGz)} = {x | E.y e. A E.z e. B x = (yGz)})
9	5, 8	eqeq12d 1492	. . 3 |- (g = B -> ((AFg) = {x | E.y e. A E.z e. g x = (yGz)} <-> (AFB) = {x | E.y e. A E.z e. B x = (yGz)}))
10		visset 1816	. . . . 5 |- f e. V
11		visset 1816	. . . . 5 |- g e. V
12	10, 11	oprvalex 4047	. . . 4 |- {x | E.y e. f E.z e. g x = (yGz)} e. V
13		rexeq1 1790	. . . . 5 |- (w = f -> (E.y e. w E.z e. v x = (yGz) <-> E.y e. f E.z e. v x = (yGz)))
14	13	abbidv 1580	. . . 4 |- (w = f -> {x | E.y e. w E.z e. v x = (yGz)} = {x | E.y e. f E.z e. v x = (yGz)})
15		rexeq1 1790	. . . . . 6 |- (v = g -> (E.z e. v x = (yGz) <-> E.z e. g x = (yGz)))
16	15	rexbidv 1667	. . . . 5 |- (v = g -> (E.y e. f E.z e. v x = (yGz) <-> E.y e. f E.z e. g x = (yGz)))
17	16	abbidv 1580	. . . 4 |- (v = g -> {x | E.y e. f E.z e. v x = (yGz)} = {x | E.y e. f E.z e. g x = (yGz)})
18		genp.1	. . . 4 |- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}
19	12, 14, 17, 18	oprabval2 4034	. . 3 |- ((f e. P. /\ g e. P.) -> (fFg) = {x | E.y e. f E.z e. g x = (yGz)})
20	4, 9, 19	vtocl2ga 1856	. 2 |- ((A e. P. /\ B e. P.) -> (AFB) = {x | E.y e. A E.z e. B x = (yGz)})
21		eqeq1 1484	. . . . . 6 |- (x = f -> (x = (gGh) <-> f = (gGh)))
22	21	anbi2d 618	. . . . 5 |- (x = f -> (((g e. A /\ h e. B) /\ x = (gGh)) <-> ((g e. A /\ h e. B) /\ f = (gGh))))
23	22	2exbidv 1283	. . . 4 |- (x = f -> (E.gE.h((g e. A /\ h e. B) /\ x = (gGh)) <-> E.gE.h((g e. A /\ h e. B) /\ f = (gGh))))
24		r2ex 1694	. . . . 5 |- (E.y e. A E.z e. B x = (yGz) <-> E.yE.z((y e. A /\ z e. B) /\ x = (yGz)))
25		eleq1 1537	. . . . . . . 8 |- (y = g -> (y e. A <-> g e. A))
26		eleq1 1537	. . . . . . . 8 |- (z = h -> (z e. B <-> h e. B))
27	25, 26	bi2anan9 634	. . . . . . 7 |- ((y = g /\ z = h) -> ((y e. A /\ z e. B) <-> (g e. A /\ h e. B)))
28		opreq12 3976	. . . . . . . 8 |- ((y = g /\ z = h) -> (yGz) = (gGh))
29	28	eqeq2d 1489	. . . . . . 7 |- ((y = g /\ z = h) -> (x = (yGz) <-> x = (gGh)))
30	27, 29	anbi12d 630	. . . . . 6 |- ((y = g /\ z = h) -> (((y e. A /\ z e. B) /\ x = (yGz)) <-> ((g e. A /\ h e. B) /\ x = (gGh))))
31	30	cbvex2v 1321	. . . . 5 |- (E.yE.z((y e. A /\ z e. B) /\ x = (yGz)) <-> E.gE.h((g e. A /\ h e. B) /\ x = (gGh)))
32	24, 31	bitr 173	. . . 4 |- (E.y e. A E.z e. B x = (yGz) <-> E.gE.h((g e. A /\ h e. B) /\ x = (gGh)))
33	23, 32	syl5bb 534	. . 3 |- (x = f -> (E.y e. A E.z e. B x = (yGz) <-> E.gE.h((g e. A /\ h e. B) /\ f = (gGh))))
34	33	cbvabv 1912	. 2 |- {x | E.y e. A E.z e. B x = (yGz)} = {f | E.gE.h((g e. A /\ h e. B) /\ f = (gGh))}
35	20, 34	syl6eq 1526	1 |- ((A e. P. /\ B e. P.) -> (AFB) = {f | E.gE.h((g e. A /\ h e. B) /\ f = (gGh))})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982  {cab 1466  E.wrex 1649  (class class class)co 3969  {copab2 3970  P.cnp 4997

This theorem is referenced by:  genpelv 5115  genpprecl 5116  genpn0 5118  genpss 5119  genpnnp 5120  genpnmax 5122  plpv 5125  mpv 5126

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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  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-fv 3204  df-opr 3971  df-oprab 3972

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