Theorem addclprlem2 6271

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 6270	. . . . 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 429	. . . 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 6270	. . . . . 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 2295	. . . . . . . 8 |- g e. _V
5		visset 2295	. . . . . . . 8 |- h e. _V
6	4, 5	addcompq 6214	. . . . . . 7 |- (g +Q h) = (h +Q g)
7	6	breq2i 3346	. . . . . 6 |- (x <Q (g +Q h) <-> x <Q (h +Q g))
8	6	fveq2i 4684	. . . . . . . . 9 |- (*Q` (g +Q h)) = (*Q` (h +Q g))
9	8	opreq2i 4893	. . . . . . . 8 |- (x .Q (*Q` (g +Q h))) = (x .Q (*Q` (h +Q g)))
10	9	opreq1i 4892	. . . . . . 7 |- ((x .Q (*Q` (g +Q h))) .Q h) = ((x .Q (*Q` (h +Q g))) .Q h)
11	10	eleq1i 1960	. . . . . 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 612	. . . . 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 428	. . . 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 661	. . 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		simpl 346	. . . 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		simpl 346	. . . . 5 |- ((A e. P. /\ g e. A) -> A e. P.)
17		simpl 346	. . . . 5 |- ((B e. P. /\ h e. B) -> B e. P.)
18	16, 17	anim12i 360	. . . 4 |- (((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) -> (A e. P. /\ B e. P.))
19		df-plp 6240	. . . . 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 6256	. . . 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 24	. . 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 30	. 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 6247	. . . . . . . . 9 |- ((A e. P. /\ g e. A) -> g e. Q.)
24		elprpq 6247	. . . . . . . . 9 |- ((B e. P. /\ h e. B) -> h e. Q.)
25	23, 24	anim12i 360	. . . . . . . 8 |- (((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) -> (g e. Q. /\ h e. Q.))
26		addclpq 6210	. . . . . . . 8 |- ((g e. Q. /\ h e. Q.) -> (g +Q h) e. Q.)
27		recidpq 6223	. . . . . . . 8 |- ((g +Q h) e. Q. -> ((g +Q h) .Q (*Q` (g +Q h))) = 1Q)
28	25, 26, 27	3syl 24	. . . . . . 7 |- (((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) -> ((g +Q h) .Q (*Q` (g +Q h))) = 1Q)
29		fvex 4689	. . . . . . . 8 |- (*Q` (g +Q h)) e. _V
30		oprex 4907	. . . . . . . 8 |- (g +Q h) e. _V
31	29, 30	mulcompq 6216	. . . . . . 7 |- ((*Q` (g +Q h)) .Q (g +Q h)) = ((g +Q h) .Q (*Q` (g +Q h)))
32	28, 31	syl5eq 1940	. . . . . 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 4898	. . . . 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 6221	. . . . 5 |- (x e. Q. -> (x .Q 1Q) = x)
35	33, 34	sylan9eq 1948	. . . 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 6219	. . . . 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 6217	. . . . 5 |- ((x .Q (*Q` (g +Q h))) .Q (g +Q h)) = (x .Q ((*Q` (g +Q h)) .Q (g +Q h)))
38	36, 37	eqtr3i 1910	. . . 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 1940	. . 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 1963	. 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 219	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 240   = wceq 1298   e. wcel 1300   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  Q.cnq 6131  1Qc1q 6132   +Q cplq 6133   .Q cmq 6134  *Qcrq 6135   <Q cltq 6136  P.cnp 6137   +P. cpp 6139

This theorem is referenced by:  addclpr 6272

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-plp 6240

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