Theorem addclprlem2 5091

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 5090	. . . . 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 5090	. . . . . 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 1804	. . . . . . . 8 |- g e. V
5		visset 1804	. . . . . . . 8 |- h e. V
6	4, 5	addcompq 5034	. . . . . . 7 |- (g +Q h) = (h +Q g)
7	6	breq2i 2617	. . . . . 6 |- (x <Q (g +Q h) <-> x <Q (h +Q g))
8	6	fveq2i 3712	. . . . . . . . 9 |- (*Q` (g +Q h)) = (*Q` (h +Q g))
9	8	opreq2i 3957	. . . . . . . 8 |- (x .Q (*Q` (g +Q h))) = (x .Q (*Q` (h +Q g)))
10	9	opreq1i 3956	. . . . . . 7 |- ((x .Q (*Q` (g +Q h))) .Q h) = ((x .Q (*Q` (h +Q g))) .Q h)
11	10	eleq1i 1529	. . . . . 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 551	. . . . 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 598	. . 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 5060	. . . . 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 5076	. . . 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 5067	. . . . . . . . 9 |- ((A e. P. /\ g e. A) -> g e. Q.)
24		elprpq 5067	. . . . . . . . 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 5030	. . . . . . . 8 |- ((g e. Q. /\ h e. Q.) -> (g +Q h) e. Q.)
27		recidpq 5043	. . . . . . . 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 3717	. . . . . . . 8 |- (*Q` (g +Q h)) e. V
30		oprex 3968	. . . . . . . 8 |- (g +Q h) e. V
31	29, 30	mulcompq 5036	. . . . . . 7 |- ((*Q` (g +Q h)) .Q (g +Q h)) = ((g +Q h) .Q (*Q` (g +Q h)))
32	28, 31	syl5eq 1511	. . . . . 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 3961	. . . . 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 5041	. . . . 5 |- (x e. Q. -> (x .Q 1Q) = x)
35	33, 34	sylan9eq 1519	. . . 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 5039	. . . . 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 5037	. . . . 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 1489	. . . 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 1511	. . 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 1532	. 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 953   e. wcel 955   class class class wbr 2609  ` cfv 3172  (class class class)co 3948  Q.cnq 4951  1Qc1q 4952   +Q cplq 4953   .Q cmq 4954  *Qcrq 4955   <Q cltq 4956  P.cnp 4957   +P. cpp 4959

This theorem is referenced by:  addclpr 5092

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  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  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-reu 1643  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-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  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-fn 3183  df-f 3184  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-plp 5060

Copyright terms: Public domain