Theorem addclprlem2 5131

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 5130	. . . . 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 395	. . . 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 5130	. . . . . 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 1816	. . . . . . . 8 |- g e. V
5		visset 1816	. . . . . . . 8 |- h e. V
6	4, 5	addcompq 5074	. . . . . . 7 |- (g +Q h) = (h +Q g)
7	6	breq2i 2632	. . . . . 6 |- (x <Q (g +Q h) <-> x <Q (h +Q g))
8	6	fveq2i 3733	. . . . . . . . 9 |- (*Q` (g +Q h)) = (*Q` (h +Q g))
9	8	opreq2i 3978	. . . . . . . 8 |- (x .Q (*Q` (g +Q h))) = (x .Q (*Q` (h +Q g)))
10	9	opreq1i 3977	. . . . . . 7 |- ((x .Q (*Q` (g +Q h))) .Q h) = ((x .Q (*Q` (h +Q g))) .Q h)
11	10	eleq1i 1540	. . . . . 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 555	. . . . 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 394	. . . 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 602	. . 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 5100	. . . . 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 5116	. . . 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 5107	. . . . . . . . 9 |- ((A e. P. /\ g e. A) -> g e. Q.)
24		elprpq 5107	. . . . . . . . 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 5070	. . . . . . . 8 |- ((g e. Q. /\ h e. Q.) -> (g +Q h) e. Q.)
27		recidpq 5083	. . . . . . . 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 3738	. . . . . . . 8 |- (*Q` (g +Q h)) e. V
30		oprex 3989	. . . . . . . 8 |- (g +Q h) e. V
31	29, 30	mulcompq 5076	. . . . . . 7 |- ((*Q` (g +Q h)) .Q (g +Q h)) = ((g +Q h) .Q (*Q` (g +Q h)))
32	28, 31	syl5eq 1522	. . . . . 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 3982	. . . . 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 5081	. . . . 5 |- (x e. Q. -> (x .Q 1Q) = x)
35	33, 34	sylan9eq 1530	. . . 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 5079	. . . . 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 5077	. . . . 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 1500	. . . 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 1522	. . 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 1543	. 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 958   e. wcel 960   class class class wbr 2624  ` cfv 3188  (class class class)co 3969  Q.cnq 4991  1Qc1q 4992   +Q cplq 4993   .Q cmq 4994  *Qcrq 4995   <Q cltq 4996  P.cnp 4997   +P. cpp 4999

This theorem is referenced by:  addclpr 5132

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  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  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-reu 1654  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-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  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-fn 3199  df-f 3200  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-plp 5100

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