Theorem addclprlem2 5102

Description: Lemma to prove downward closure in positive real addition. Part of proof of Proposition 9-3.5 of [Gleason] p. 123.

Assertion

Ref	Expression
addclprlem2	|- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> (x <Q (g +Q h) -> x e. (A +P. B)))

Distinct variable groups:   x,g,h   x,A   x,B

Proof of Theorem addclprlem2

Step	Hyp	Ref	Expression
1		addclprlem1 5101	. . . . 5 |- (((A e. P. /\ g e. A) /\ x e. Q.) -> (x <Q (g +Q h) -> ((x .Q (*Q` (g +Q h))) .Q g) e. A))
2	1	adantlr 393	. . . 4 |- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> (x <Q (g +Q h) -> ((x .Q (*Q` (g +Q h))) .Q g) e. A))
3		addclprlem1 5101	. . . . . 6 |- (((B e. P. /\ h e. B) /\ x e. Q.) -> (x <Q (h +Q g) -> ((x .Q (*Q` (h +Q g))) .Q h) e. B))
4		visset 1810	. . . . . . . 8 |- g e. V
5		visset 1810	. . . . . . . 8 |- h e. V
6	4, 5	addcompq 5045	. . . . . . 7 |- (g +Q h) = (h +Q g)
7	6	breq2i 2623	. . . . . 6 |- (x <Q (g +Q h) <-> x <Q (h +Q g))
8	6	fveq2i 3722	. . . . . . . . 9 |- (*Q` (g +Q h)) = (*Q` (h +Q g))
9	8	opreq2i 3967	. . . . . . . 8 |- (x .Q (*Q` (g +Q h))) = (x .Q (*Q` (h +Q g)))
10	9	opreq1i 3966	. . . . . . 7 |- ((x .Q (*Q` (g +Q h))) .Q h) = ((x .Q (*Q` (h +Q g))) .Q h)
11	10	eleq1i 1535	. . . . . 6 |- (((x .Q (*Q` (g +Q h))) .Q h) e. B <-> ((x .Q (*Q` (h +Q g))) .Q h) e. B)
12	3, 7, 11	3imtr4g 552	. . . . 5 |- (((B e. P. /\ h e. B) /\ x e. Q.) -> (x <Q (g +Q h) -> ((x .Q (*Q` (g +Q h))) .Q h) e. B))
13	12	adantll 392	. . . 4 |- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> (x <Q (g +Q h) -> ((x .Q (*Q` (g +Q h))) .Q h) e. B))
14	2, 13	jcad 599	. . 3 |- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> (x <Q (g +Q h) -> (((x .Q (*Q` (g +Q h))) .Q g) e. A /\ ((x .Q (*Q` (g +Q h))) .Q h) e. B)))
15		pm3.26 319	. . . 4 |- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> ((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)))
16		pm3.26 319	. . . . 5 |- ((A e. P. /\ g e. A) -> A e. P.)
17		pm3.26 319	. . . . 5 |- ((B e. P. /\ h e. B) -> B e. P.)
18	16, 17	anim12i 333	. . . 4 |- (((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) -> (A e. P. /\ B e. P.))
19		df-plp 5071	. . . . 5 |- +P. = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (y +Q z)})}
20	19	genpprecl 5087	. . . 4 |- ((A e. P. /\ B e. P.) -> ((((x .Q (*Q` (g +Q h))) .Q g) e. A /\ ((x .Q (*Q` (g +Q h))) .Q h) e. B) -> (((x .Q (*Q` (g +Q h))) .Q g) +Q ((x .Q (*Q` (g +Q h))) .Q h)) e. (A +P. B)))
21	15, 18, 20	3syl 20	. . 3 |- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> ((((x .Q (*Q` (g +Q h))) .Q g) e. A /\ ((x .Q (*Q` (g +Q h))) .Q h) e. B) -> (((x .Q (*Q` (g +Q h))) .Q g) +Q ((x .Q (*Q` (g +Q h))) .Q h)) e. (A +P. B)))
22	14, 21	syld 27	. 2 |- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> (x <Q (g +Q h) -> (((x .Q (*Q` (g +Q h))) .Q g) +Q ((x .Q (*Q` (g +Q h))) .Q h)) e. (A +P. B)))
23		elprpq 5078	. . . . . . . . 9 |- ((A e. P. /\ g e. A) -> g e. Q.)
24		elprpq 5078	. . . . . . . . 9 |- ((B e. P. /\ h e. B) -> h e. Q.)
25	23, 24	anim12i 333	. . . . . . . 8 |- (((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) -> (g e. Q. /\ h e. Q.))
26		addclpq 5041	. . . . . . . 8 |- ((g e. Q. /\ h e. Q.) -> (g +Q h) e. Q.)
27		recidpq 5054	. . . . . . . 8 |- ((g +Q h) e. Q. -> ((g +Q h) .Q (*Q` (g +Q h))) = 1Q)
28	25, 26, 27	3syl 20	. . . . . . 7 |- (((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) -> ((g +Q h) .Q (*Q` (g +Q h))) = 1Q)
29		fvex 3727	. . . . . . . 8 |- (*Q` (g +Q h)) e. V
30		oprex 3978	. . . . . . . 8 |- (g +Q h) e. V
31	29, 30	mulcompq 5047	. . . . . . 7 |- ((*Q` (g +Q h)) .Q (g +Q h)) = ((g +Q h) .Q (*Q` (g +Q h)))
32	28, 31	syl5eq 1517	. . . . . 6 |- (((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) -> ((*Q` (g +Q h)) .Q (g +Q h)) = 1Q)
33	32	opreq2d 3971	. . . . 5 |- (((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) -> (x .Q ((*Q` (g +Q h)) .Q (g +Q h))) = (x .Q 1Q))
34		mulidpq 5052	. . . . 5 |- (x e. Q. -> (x .Q 1Q) = x)
35	33, 34	sylan9eq 1525	. . . 4 |- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> (x .Q ((*Q` (g +Q h)) .Q (g +Q h))) = x)
36	4, 5	distrpq 5050	. . . . 5 |- ((x .Q (*Q` (g +Q h))) .Q (g +Q h)) = (((x .Q (*Q` (g +Q h))) .Q g) +Q ((x .Q (*Q` (g +Q h))) .Q h))
37	29, 30	mulasspq 5048	. . . . 5 |- ((x .Q (*Q` (g +Q h))) .Q (g +Q h)) = (x .Q ((*Q` (g +Q h)) .Q (g +Q h)))
38	36, 37	eqtr3 1495	. . . 4 |- (((x .Q (*Q` (g +Q h))) .Q g) +Q ((x .Q (*Q` (g +Q h))) .Q h)) = (x .Q ((*Q` (g +Q h)) .Q (g +Q h)))
39	35, 38	syl5eq 1517	. . 3 |- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> (((x .Q (*Q` (g +Q h))) .Q g) +Q ((x .Q (*Q` (g +Q h))) .Q h)) = x)
40	39	eleq1d 1538	. 2 |- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> ((((x .Q (*Q` (g +Q h))) .Q g) +Q ((x .Q (*Q` (g +Q h))) .Q h)) e. (A +P. B) <-> x e. (A +P. B)))
41	22, 40	sylibd 202	1 |- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> (x <Q (g +Q h) -> x e. (A +P. B)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957   class class class wbr 2615  ` cfv 3178  (class class class)co 3958  Q.cnq 4962  1Qc1q 4963   +Q cplq 4964   .Q cmq 4965  *Qcrq 4966   <Q cltq 4967  P.cnp 4968   +P. cpp 4970

This theorem is referenced by:  addclpr 5103

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-inf2 4608

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1st 4072  df-2nd 4073  df-1o 4126  df-oadd 4128  df-omul 4129  df-er 4254  df-ec 4256  df-qs 4259  df-ni 4983  df-pli 4984  df-mi 4985  df-lti 4986  df-plpq 5018  df-mpq 5019  df-enq 5020  df-nq 5021  df-plq 5022  df-mq 5023  df-rq 5024  df-ltq 5025  df-1q 5026  df-np 5069  df-plp 5071

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