Theorem genpprecl 5104

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 1475	. 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 5102	. . . . 5 |- ((A e. P. /\ B e. P.) -> (AFB) = {f | E.gE.h((g e. A /\ h e. B) /\ f = (gGh))})
4	3	eleq2d 1541	. . . 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 3983	. . . . 5 |- (CGD) e. V
6		eqeq1 1481	. . . . . . 7 |- (f = (CGD) -> (f = (gGh) <-> (CGD) = (gGh)))
7	6	anbi2d 616	. . . . . 6 |- (f = (CGD) -> (((g e. A /\ h e. B) /\ f = (gGh)) <-> ((g e. A /\ h e. B) /\ (CGD) = (gGh))))
8	7	2exbidv 1281	. . . . 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 1897	. . . 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 536	. . 3 |- ((A e. P. /\ B e. P.) -> ((CGD) e. (AFB) <-> E.gE.h((g e. A /\ h e. B) /\ (CGD) = (gGh))))
11		eleq1 1534	. . . . . . 7 |- (g = C -> (g e. A <-> C e. A))
12		eleq1 1534	. . . . . . 7 |- (h = D -> (h e. B <-> D e. B))
13	11, 12	bi2anan9 632	. . . . . 6 |- ((g = C /\ h = D) -> ((g e. A /\ h e. B) <-> (C e. A /\ D e. B)))
14		opreq12 3970	. . . . . . 7 |- ((g = C /\ h = D) -> (gGh) = (CGD))
15	14	eqeq2d 1486	. . . . . 6 |- ((g = C /\ h = D) -> ((CGD) = (gGh) <-> (CGD) = (CGD)))
16	13, 15	anbi12d 628	. . . . 5 |- ((g = C /\ h = D) -> (((g e. A /\ h e. B) /\ (CGD) = (gGh)) <-> ((C e. A /\ D e. B) /\ (CGD) = (CGD))))
17	16	cla42egv 1864	. . . 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 495	. . 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 699	1 |- ((A e. P. /\ B e. P.) -> ((C e. A /\ D e. B) -> (CGD) e. (AFB)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  E.wrex 1646  (class class class)co 3963  {copab2 3964  P.cnp 4985

This theorem is referenced by:  genpnmax 5110  addclprlem2 5119  mulclprlem 5121  distrlem1pr 5127  distrlem2pr 5128  ltaddpr 5140  ltexprlem7 5148

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-opr 3965  df-oprab 3966

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