Theorem addclprlem1 5090

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
addclprlem1	|- (((A e. P. /\ g e. A) /\ x e. Q.) -> (x <Q (g +Q h) -> ((x .Q (*Q` (g +Q h))) .Q g) e. A))

Proof of Theorem addclprlem1

Step	Hyp	Ref	Expression
1		fvex 3717	. . . . . . 7 |- (*Q` (g +Q h)) e. V
2		fvex 3717	. . . . . . 7 |- (*Q` x) e. V
3	1, 2	ltmpq 5049	. . . . . 6 |- (x e. Q. -> ((*Q` (g +Q h)) <Q (*Q` x) <-> (x .Q (*Q` (g +Q h))) <Q (x .Q (*Q` x))))
4		oprex 3968	. . . . . . 7 |- (x .Q (*Q` (g +Q h))) e. V
5		oprex 3968	. . . . . . 7 |- (x .Q (*Q` x)) e. V
6		visset 1804	. . . . . . . 8 |- y e. V
7		visset 1804	. . . . . . . 8 |- z e. V
8	6, 7	ltmpq 5049	. . . . . . 7 |- (w e. Q. -> (y <Q z <-> (w .Q y) <Q (w .Q z)))
9		visset 1804	. . . . . . 7 |- g e. V
10	6, 7	mulcompq 5036	. . . . . . 7 |- (y .Q z) = (z .Q y)
11	4, 5, 8, 9, 10	caoprord2 4043	. . . . . 6 |- (g e. Q. -> ((x .Q (*Q` (g +Q h))) <Q (x .Q (*Q` x)) <-> ((x .Q (*Q` (g +Q h))) .Q g) <Q ((x .Q (*Q` x)) .Q g)))
12	3, 11	sylan9bbr 539	. . . . 5 |- ((g e. Q. /\ x e. Q.) -> ((*Q` (g +Q h)) <Q (*Q` x) <-> ((x .Q (*Q` (g +Q h))) .Q g) <Q ((x .Q (*Q` x)) .Q g)))
13		visset 1804	. . . . . 6 |- x e. V
14		oprex 3968	. . . . . 6 |- (g +Q h) e. V
15	13, 14	ltrpq 5057	. . . . 5 |- (x <Q (g +Q h) -> (*Q` (g +Q h)) <Q (*Q` x))
16	12, 15	syl5bi 208	. . . 4 |- ((g e. Q. /\ x e. Q.) -> (x <Q (g +Q h) -> ((x .Q (*Q` (g +Q h))) .Q g) <Q ((x .Q (*Q` x)) .Q g)))
17		recidpq 5043	. . . . . . 7 |- (x e. Q. -> (x .Q (*Q` x)) = 1Q)
18	17	opreq1d 3960	. . . . . 6 |- (x e. Q. -> ((x .Q (*Q` x)) .Q g) = (1Q .Q g))
19		mulidpq 5041	. . . . . . 7 |- (g e. Q. -> (g .Q 1Q) = g)
20		1q 5029	. . . . . . . . 9 |- 1Q e. Q.
21	20	elisseti 1809	. . . . . . . 8 |- 1Q e. V
22	21, 9	mulcompq 5036	. . . . . . 7 |- (1Q .Q g) = (g .Q 1Q)
23	19, 22	syl5eq 1511	. . . . . 6 |- (g e. Q. -> (1Q .Q g) = g)
24	18, 23	sylan9eqr 1521	. . . . 5 |- ((g e. Q. /\ x e. Q.) -> ((x .Q (*Q` x)) .Q g) = g)
25	24	breq2d 2620	. . . 4 |- ((g e. Q. /\ x e. Q.) -> (((x .Q (*Q` (g +Q h))) .Q g) <Q ((x .Q (*Q` x)) .Q g) <-> ((x .Q (*Q` (g +Q h))) .Q g) <Q g))
26	16, 25	sylibd 202	. . 3 |- ((g e. Q. /\ x e. Q.) -> (x <Q (g +Q h) -> ((x .Q (*Q` (g +Q h))) .Q g) <Q g))
27		elprpq 5067	. . 3 |- ((A e. P. /\ g e. A) -> g e. Q.)
28	26, 27	sylan 448	. 2 |- (((A e. P. /\ g e. A) /\ x e. Q.) -> (x <Q (g +Q h) -> ((x .Q (*Q` (g +Q h))) .Q g) <Q g))
29		prcdpq 5069	. . 3 |- ((A e. P. /\ g e. A) -> (((x .Q (*Q` (g +Q h))) .Q g) <Q g -> ((x .Q (*Q` (g +Q h))) .Q g) e. A))
30	29	adantr 389	. 2 |- (((A e. P. /\ g e. A) /\ x e. Q.) -> (((x .Q (*Q` (g +Q h))) .Q g) <Q g -> ((x .Q (*Q` (g +Q h))) .Q g) e. A))
31	28, 30	syld 27	1 |- (((A e. P. /\ g e. A) /\ x e. Q.) -> (x <Q (g +Q h) -> ((x .Q (*Q` (g +Q h))) .Q g) e. A))

Syntax hints:   -> wi 3   /\ wa 223   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

This theorem is referenced by:  addclprlem2 5091

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-mi 4974  df-lti 4975  df-mpq 5008  df-enq 5009  df-nq 5010  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058

Copyright terms: Public domain