Theorem genpprecl 5076

Description: Pre-closure law for general operation 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
genpprecl	|- ((A e. P. /\ B e. P.) -> ((C e. A /\ D e. B) -> (CGD) e. (AFB)))

Distinct variable groups:   x,y,z,A   x,B,y,z   x,w,v,u,G,y,z

Proof of Theorem genpprecl

Step	Hyp	Ref	Expression
1		eqid 1468	. 2 |- (CGD) = (CGD)
2		genp.1	. . . . . 6 |- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}
3	2	genpv 5074	. . . . 5 |- ((A e. P. /\ B e. P.) -> (AFB) = {f | E.gE.h((g e. A /\ h e. B) /\ f = (gGh))})
4	3	eleq2d 1533	. . . 4 |- ((A e. P. /\ B e. P.) -> ((CGD) e. (AFB) <-> (CGD) e. {f | E.gE.h((g e. A /\ h e. B) /\ f = (gGh))}))
5		oprex 3968	. . . . 5 |- (CGD) e. V
6		eqeq1 1473	. . . . . . 7 |- (f = (CGD) -> (f = (gGh) <-> (CGD) = (gGh)))
7	6	anbi2d 614	. . . . . 6 |- (f = (CGD) -> (((g e. A /\ h e. B) /\ f = (gGh)) <-> ((g e. A /\ h e. B) /\ (CGD) = (gGh))))
8	7	2exbidv 1276	. . . . 5 |- (f = (CGD) -> (E.gE.h((g e. A /\ h e. B) /\ f = (gGh)) <-> E.gE.h((g e. A /\ h e. B) /\ (CGD) = (gGh))))
9	5, 8	elab 1888	. . . 4 |- ((CGD) e. {f | E.gE.h((g e. A /\ h e. B) /\ f = (gGh))} <-> E.gE.h((g e. A /\ h e. B) /\ (CGD) = (gGh)))
10	4, 9	syl6bb 534	. . 3 |- ((A e. P. /\ B e. P.) -> ((CGD) e. (AFB) <-> E.gE.h((g e. A /\ h e. B) /\ (CGD) = (gGh))))
11		eleq1 1526	. . . . . . 7 |- (g = C -> (g e. A <-> C e. A))
12		eleq1 1526	. . . . . . 7 |- (h = D -> (h e. B <-> D e. B))
13	11, 12	bi2anan9 630	. . . . . 6 |- ((g = C /\ h = D) -> ((g e. A /\ h e. B) <-> (C e. A /\ D e. B)))
14		opreq12 3955	. . . . . . 7 |- ((g = C /\ h = D) -> (gGh) = (CGD))
15	14	eqeq2d 1478	. . . . . 6 |- ((g = C /\ h = D) -> ((CGD) = (gGh) <-> (CGD) = (CGD)))
16	13, 15	anbi12d 626	. . . . 5 |- ((g = C /\ h = D) -> (((g e. A /\ h e. B) /\ (CGD) = (gGh)) <-> ((C e. A /\ D e. B) /\ (CGD) = (CGD))))
17	16	cla42egv 1855	. . . 4 |- ((C e. A /\ D e. B) -> (((C e. A /\ D e. B) /\ (CGD) = (CGD)) -> E.gE.h((g e. A /\ h e. B) /\ (CGD) = (gGh))))
18	17	anabsi5 494	. . 3 |- (((C e. A /\ D e. B) /\ (CGD) = (CGD)) -> E.gE.h((g e. A /\ h e. B) /\ (CGD) = (gGh)))
19	10, 18	syl5bir 210	. 2 |- ((A e. P. /\ B e. P.) -> (((C e. A /\ D e. B) /\ (CGD) = (CGD)) -> (CGD) e. (AFB)))
20	1, 19	mpan2i 697	1 |- ((A e. P. /\ B e. P.) -> ((C e. A /\ D e. B) -> (CGD) e. (AFB)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  {cab 1456  E.wrex 1638  (class class class)co 3948  {copab2 3949  P.cnp 4957

This theorem is referenced by:  genpnmax 5082  addclprlem2 5091  mulclprlem 5093  distrlem1pr 5099  distrlem2pr 5100  ltaddpr 5112  ltexprlem7 5120

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fv 3188  df-opr 3950  df-oprab 3951

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