Theorem mulclprlem 6273

Description: Lemma to prove downward closure in positive real multiplication. Part of proof of Proposition 9-3.7 of [Gleason] p. 124.

Assertion

Ref	Expression
mulclprlem	|- ((((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 mulclprlem

Step	Hyp	Ref	Expression
1		recclpq 6224	. . . . . . . . 9 |- (h e. Q. -> (*Q` h) e. Q.)
2	1	adantl 424	. . . . . . . 8 |- ((g e. Q. /\ h e. Q.) -> (*Q` h) e. Q.)
3		visset 2295	. . . . . . . . 9 |- x e. _V
4		oprex 4907	. . . . . . . . 9 |- (g .Q h) e. _V
5		visset 2295	. . . . . . . . . 10 |- y e. _V
6		visset 2295	. . . . . . . . . 10 |- z e. _V
7	5, 6	ltmpq 6229	. . . . . . . . 9 |- (w e. Q. -> (y <Q z <-> (w .Q y) <Q (w .Q z)))
8		fvex 4689	. . . . . . . . 9 |- (*Q` h) e. _V
9	5, 6	mulcompq 6216	. . . . . . . . 9 |- (y .Q z) = (z .Q y)
10	3, 4, 7, 8, 9	caoprord2 4990	. . . . . . . 8 |- ((*Q` h) e. Q. -> (x <Q (g .Q h) <-> (x .Q (*Q` h)) <Q ((g .Q h) .Q (*Q` h))))
11	2, 10	syl 12	. . . . . . 7 |- ((g e. Q. /\ h e. Q.) -> (x <Q (g .Q h) <-> (x .Q (*Q` h)) <Q ((g .Q h) .Q (*Q` h))))
12		recidpq 6223	. . . . . . . . . . 11 |- (h e. Q. -> (h .Q (*Q` h)) = 1Q)
13	12	opreq2d 4898	. . . . . . . . . 10 |- (h e. Q. -> (g .Q (h .Q (*Q` h))) = (g .Q 1Q))
14		visset 2295	. . . . . . . . . . 11 |- h e. _V
15	14, 8	mulasspq 6217	. . . . . . . . . 10 |- ((g .Q h) .Q (*Q` h)) = (g .Q (h .Q (*Q` h)))
16	13, 15	syl5eq 1940	. . . . . . . . 9 |- (h e. Q. -> ((g .Q h) .Q (*Q` h)) = (g .Q 1Q))
17		mulidpq 6221	. . . . . . . . 9 |- (g e. Q. -> (g .Q 1Q) = g)
18	16, 17	sylan9eqr 1951	. . . . . . . 8 |- ((g e. Q. /\ h e. Q.) -> ((g .Q h) .Q (*Q` h)) = g)
19	18	breq2d 3350	. . . . . . 7 |- ((g e. Q. /\ h e. Q.) -> ((x .Q (*Q` h)) <Q ((g .Q h) .Q (*Q` h)) <-> (x .Q (*Q` h)) <Q g))
20	11, 19	bitrd 587	. . . . . 6 |- ((g e. Q. /\ h e. Q.) -> (x <Q (g .Q h) <-> (x .Q (*Q` h)) <Q g))
21		elprpq 6247	. . . . . 6 |- ((A e. P. /\ g e. A) -> g e. Q.)
22		elprpq 6247	. . . . . 6 |- ((B e. P. /\ h e. B) -> h e. Q.)
23	20, 21, 22	syl2an 503	. . . . 5 |- (((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) -> (x <Q (g .Q h) <-> (x .Q (*Q` h)) <Q g))
24		prcdpq 6249	. . . . . 6 |- ((A e. P. /\ g e. A) -> ((x .Q (*Q` h)) <Q g -> (x .Q (*Q` h)) e. A))
25	24	adantr 425	. . . . 5 |- (((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) -> ((x .Q (*Q` h)) <Q g -> (x .Q (*Q` h)) e. A))
26	23, 25	sylbid 220	. . . 4 |- (((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) -> (x <Q (g .Q h) -> (x .Q (*Q` h)) e. A))
27		df-mp 6241	. . . . . . . . 9 |- .P. = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (y .Q z)})}
28	27	genpprecl 6256	. . . . . . . 8 |- ((A e. P. /\ B e. P.) -> (((x .Q (*Q` h)) e. A /\ h e. B) -> ((x .Q (*Q` h)) .Q h) e. (A .P. B)))
29	28	exp4b 410	. . . . . . 7 |- (A e. P. -> (B e. P. -> ((x .Q (*Q` h)) e. A -> (h e. B -> ((x .Q (*Q` h)) .Q h) e. (A .P. B)))))
30	29	com34 40	. . . . . 6 |- (A e. P. -> (B e. P. -> (h e. B -> ((x .Q (*Q` h)) e. A -> ((x .Q (*Q` h)) .Q h) e. (A .P. B)))))
31	30	imp32 390	. . . . 5 |- ((A e. P. /\ (B e. P. /\ h e. B)) -> ((x .Q (*Q` h)) e. A -> ((x .Q (*Q` h)) .Q h) e. (A .P. B)))
32	31	adantlr 429	. . . 4 |- (((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) -> ((x .Q (*Q` h)) e. A -> ((x .Q (*Q` h)) .Q h) e. (A .P. B)))
33	26, 32	syld 30	. . 3 |- (((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) -> (x <Q (g .Q h) -> ((x .Q (*Q` h)) .Q h) e. (A .P. B)))
34	33	adantr 425	. 2 |- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> (x <Q (g .Q h) -> ((x .Q (*Q` h)) .Q h) e. (A .P. B)))
35	8, 14	mulcompq 6216	. . . . . . . 8 |- ((*Q` h) .Q h) = (h .Q (*Q` h))
36	12, 35	syl5eq 1940	. . . . . . 7 |- (h e. Q. -> ((*Q` h) .Q h) = 1Q)
37	36	opreq2d 4898	. . . . . 6 |- (h e. Q. -> (x .Q ((*Q` h) .Q h)) = (x .Q 1Q))
38	8, 14	mulasspq 6217	. . . . . 6 |- ((x .Q (*Q` h)) .Q h) = (x .Q ((*Q` h) .Q h))
39	37, 38	syl5eq 1940	. . . . 5 |- (h e. Q. -> ((x .Q (*Q` h)) .Q h) = (x .Q 1Q))
40		mulidpq 6221	. . . . 5 |- (x e. Q. -> (x .Q 1Q) = x)
41	39, 40	sylan9eq 1948	. . . 4 |- ((h e. Q. /\ x e. Q.) -> ((x .Q (*Q` h)) .Q h) = x)
42	41	eleq1d 1963	. . 3 |- ((h e. Q. /\ x e. Q.) -> (((x .Q (*Q` h)) .Q h) e. (A .P. B) <-> x e. (A .P. B)))
43	22	adantl 424	. . 3 |- (((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) -> h e. Q.)
44	42, 43	sylan 497	. 2 |- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> (((x .Q (*Q` h)) .Q h) e. (A .P. B) <-> x e. (A .P. B)))
45	34, 44	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   <-> wb 163   /\ wa 240   e. wcel 1300   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  Q.cnq 6131  1Qc1q 6132   .Q cmq 6134  *Qcrq 6135   <Q cltq 6136  P.cnp 6137   .P. cmp 6140

This theorem is referenced by:  mulclpr 6274

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-mi 6154  df-lti 6155  df-mpq 6188  df-enq 6189  df-nq 6190  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-mp 6241

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