Theorem addclprlem1 5118

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 3732	. . . . . . 7 |- (*Q` (g +Q h)) e. V
2		fvex 3732	. . . . . . 7 |- (*Q` x) e. V
3	1, 2	ltmpq 5077	. . . . . 6 |- (x e. Q. -> ((*Q` (g +Q h)) <Q (*Q` x) <-> (x .Q (*Q` (g +Q h))) <Q (x .Q (*Q` x))))
4		oprex 3983	. . . . . . 7 |- (x .Q (*Q` (g +Q h))) e. V
5		oprex 3983	. . . . . . 7 |- (x .Q (*Q` x)) e. V
6		visset 1813	. . . . . . . 8 |- y e. V
7		visset 1813	. . . . . . . 8 |- z e. V
8	6, 7	ltmpq 5077	. . . . . . 7 |- (w e. Q. -> (y <Q z <-> (w .Q y) <Q (w .Q z)))
9		visset 1813	. . . . . . 7 |- g e. V
10	6, 7	mulcompq 5064	. . . . . . 7 |- (y .Q z) = (z .Q y)
11	4, 5, 8, 9, 10	caoprord2 4057	. . . . . 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 541	. . . . 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 1813	. . . . . 6 |- x e. V
14		oprex 3983	. . . . . 6 |- (g +Q h) e. V
15	13, 14	ltrpq 5085	. . . . 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 5071	. . . . . . 7 |- (x e. Q. -> (x .Q (*Q` x)) = 1Q)
18	17	opreq1d 3975	. . . . . 6 |- (x e. Q. -> ((x .Q (*Q` x)) .Q g) = (1Q .Q g))
19		mulidpq 5069	. . . . . . 7 |- (g e. Q. -> (g .Q 1Q) = g)
20		1q 5057	. . . . . . . . 9 |- 1Q e. Q.
21	20	elisseti 1818	. . . . . . . 8 |- 1Q e. V
22	21, 9	mulcompq 5064	. . . . . . 7 |- (1Q .Q g) = (g .Q 1Q)
23	19, 22	syl5eq 1519	. . . . . 6 |- (g e. Q. -> (1Q .Q g) = g)
24	18, 23	sylan9eqr 1529	. . . . 5 |- ((g e. Q. /\ x e. Q.) -> ((x .Q (*Q` x)) .Q g) = g)
25	24	breq2d 2630	. . . 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 5095	. . 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 5097	. . 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 958   class class class wbr 2619  ` cfv 3182  (class class class)co 3963  Q.cnq 4979  1Qc1q 4980   +Q cplq 4981   .Q cmq 4982  *Qcrq 4983   <Q cltq 4984  P.cnp 4985

This theorem is referenced by:  addclprlem2 5119

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-ni 5000  df-mi 5002  df-lti 5003  df-mpq 5036  df-enq 5037  df-nq 5038  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086

Copyright terms: Public domain