Theorem 1idpr 5133

Description: 1 is an identity element for positive real multiplication. Theorem 9-3.7(iv) of [Gleason] p. 124.

Assertion

Ref	Expression
1idpr	|- (A e. P. -> (A .P. 1P) = A)

Proof of Theorem 1idpr

Step	Hyp	Ref	Expression
1		breq1 2622	. . . . . . . . . . . 12 |- (x = (f .Q g) -> (x <Q f <-> (f .Q g) <Q f))
2		visset 1813	. . . . . . . . . . . . . . 15 |- g e. V
3		1q 5057	. . . . . . . . . . . . . . . 16 |- 1Q e. Q.
4	3	elisseti 1818	. . . . . . . . . . . . . . 15 |- 1Q e. V
5	2, 4	ltmpq 5077	. . . . . . . . . . . . . 14 |- (f e. Q. -> (g <Q 1Q <-> (f .Q g) <Q (f .Q 1Q)))
6		mulidpq 5069	. . . . . . . . . . . . . . 15 |- (f e. Q. -> (f .Q 1Q) = f)
7	6	breq2d 2630	. . . . . . . . . . . . . 14 |- (f e. Q. -> ((f .Q g) <Q (f .Q 1Q) <-> (f .Q g) <Q f))
8	5, 7	bitrd 528	. . . . . . . . . . . . 13 |- (f e. Q. -> (g <Q 1Q <-> (f .Q g) <Q f))
9		df-1p 5087	. . . . . . . . . . . . . 14 |- 1P = {g | g <Q 1Q}
10	9	abeq2i 1570	. . . . . . . . . . . . 13 |- (g e. 1P <-> g <Q 1Q)
11	8, 10	syl5rbb 533	. . . . . . . . . . . 12 |- (f e. Q. -> ((f .Q g) <Q f <-> g e. 1P))
12	1, 11	sylan9bbr 541	. . . . . . . . . . 11 |- ((f e. Q. /\ x = (f .Q g)) -> (x <Q f <-> g e. 1P))
13		elprpq 5095	. . . . . . . . . . 11 |- ((A e. P. /\ f e. A) -> f e. Q.)
14	12, 13	sylan 448	. . . . . . . . . 10 |- (((A e. P. /\ f e. A) /\ x = (f .Q g)) -> (x <Q f <-> g e. 1P))
15	14	exp31 376	. . . . . . . . 9 |- (A e. P. -> (f e. A -> (x = (f .Q g) -> (x <Q f <-> g e. 1P))))
16	15	imp3a 361	. . . . . . . 8 |- (A e. P. -> ((f e. A /\ x = (f .Q g)) -> (x <Q f <-> g e. 1P)))
17	16	pm5.32d 647	. . . . . . 7 |- (A e. P. -> (((f e. A /\ x = (f .Q g)) /\ x <Q f) <-> ((f e. A /\ x = (f .Q g)) /\ g e. 1P)))
18		an23 485	. . . . . . 7 |- (((f e. A /\ x <Q f) /\ x = (f .Q g)) <-> ((f e. A /\ x = (f .Q g)) /\ x <Q f))
19		an23 485	. . . . . . 7 |- (((f e. A /\ g e. 1P) /\ x = (f .Q g)) <-> ((f e. A /\ x = (f .Q g)) /\ g e. 1P))
20	17, 18, 19	3bitr4g 555	. . . . . 6 |- (A e. P. -> (((f e. A /\ x <Q f) /\ x = (f .Q g)) <-> ((f e. A /\ g e. 1P) /\ x = (f .Q g))))
21	20	exbidv 1279	. . . . 5 |- (A e. P. -> (E.g((f e. A /\ x <Q f) /\ x = (f .Q g)) <-> E.g((f e. A /\ g e. 1P) /\ x = (f .Q g))))
22		19.42v 1308	. . . . 5 |- (E.g((f e. A /\ x <Q f) /\ x = (f .Q g)) <-> ((f e. A /\ x <Q f) /\ E.g x = (f .Q g)))
23	21, 22	syl5rbbr 535	. . . 4 |- (A e. P. -> (E.g((f e. A /\ g e. 1P) /\ x = (f .Q g)) <-> ((f e. A /\ x <Q f) /\ E.g x = (f .Q g))))
24	23	exbidv 1279	. . 3 |- (A e. P. -> (E.fE.g((f e. A /\ g e. 1P) /\ x = (f .Q g)) <-> E.f((f e. A /\ x <Q f) /\ E.g x = (f .Q g))))
25		1pr 5117	. . . 4 |- 1P e. P.
26		df-mp 5089	. . . . 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)})}
27		visset 1813	. . . . 5 |- x e. V
28	26, 27	genpelv 5103	. . . 4 |- ((A e. P. /\ 1P e. P.) -> (x e. (A .P. 1P) <-> E.fE.g((f e. A /\ g e. 1P) /\ x = (f .Q g))))
29	25, 28	mpan2 696	. . 3 |- (A e. P. -> (x e. (A .P. 1P) <-> E.fE.g((f e. A /\ g e. 1P) /\ x = (f .Q g))))
30		prnmax 5099	. . . . . 6 |- ((A e. P. /\ x e. A) -> E.f(f e. A /\ x <Q f))
31		visset 1813	. . . . . . . . . . 11 |- f e. V
32		ltrelpq 5051	. . . . . . . . . . 11 |- <Q (_ (Q. X. Q.)
33	31, 32	brel 3223	. . . . . . . . . 10 |- (x <Q f -> (x e. Q. /\ f e. Q.))
34		recidpq 5071	. . . . . . . . . . . . . 14 |- (f e. Q. -> (f .Q (*Q` f)) = 1Q)
35	34	opreq2d 3976	. . . . . . . . . . . . 13 |- (f e. Q. -> (x .Q (f .Q (*Q` f))) = (x .Q 1Q))
36		fvex 3732	. . . . . . . . . . . . . 14 |- (*Q` f) e. V
37		visset 1813	. . . . . . . . . . . . . . 15 |- y e. V
38		visset 1813	. . . . . . . . . . . . . . 15 |- z e. V
39	37, 38	mulcompq 5064	. . . . . . . . . . . . . 14 |- (y .Q z) = (z .Q y)
40		visset 1813	. . . . . . . . . . . . . . 15 |- w e. V
41	38, 40	mulasspq 5065	. . . . . . . . . . . . . 14 |- ((y .Q z) .Q w) = (y .Q (z .Q w))
42	31, 27, 36, 39, 41	caopr12 4061	. . . . . . . . . . . . 13 |- (f .Q (x .Q (*Q` f))) = (x .Q (f .Q (*Q` f)))
43	35, 42	syl5eq 1519	. . . . . . . . . . . 12 |- (f e. Q. -> (f .Q (x .Q (*Q` f))) = (x .Q 1Q))
44		mulidpq 5069	. . . . . . . . . . . 12 |- (x e. Q. -> (x .Q 1Q) = x)
45	43, 44	sylan9eqr 1529	. . . . . . . . . . 11 |- ((x e. Q. /\ f e. Q.) -> (f .Q (x .Q (*Q` f))) = x)
46	45	eqcomd 1480	. . . . . . . . . 10 |- ((x e. Q. /\ f e. Q.) -> x = (f .Q (x .Q (*Q` f))))
47		oprex 3983	. . . . . . . . . . 11 |- (x .Q (*Q` f)) e. V
48		opreq2 3969	. . . . . . . . . . . 12 |- (g = (x .Q (*Q` f)) -> (f .Q g) = (f .Q (x .Q (*Q` f))))
49	48	eqeq2d 1486	. . . . . . . . . . 11 |- (g = (x .Q (*Q` f)) -> (x = (f .Q g) <-> x = (f .Q (x .Q (*Q` f)))))
50	47, 49	cla4ev 1869	. . . . . . . . . 10 |- (x = (f .Q (x .Q (*Q` f))) -> E.g x = (f .Q g))
51	33, 46, 50	3syl 20	. . . . . . . . 9 |- (x <Q f -> E.g x = (f .Q g))
52	51	adantl 388	. . . . . . . 8 |- ((f e. A /\ x <Q f) -> E.g x = (f .Q g))
53	52	ancli 296	. . . . . . 7 |- ((f e. A /\ x <Q f) -> ((f e. A /\ x <Q f) /\ E.g x = (f .Q g)))
54	53	19.22i 1040	. . . . . 6 |- (E.f(f e. A /\ x <Q f) -> E.f((f e. A /\ x <Q f) /\ E.g x = (f .Q g)))
55	30, 54	syl 10	. . . . 5 |- ((A e. P. /\ x e. A) -> E.f((f e. A /\ x <Q f) /\ E.g x = (f .Q g)))
56	55	ex 373	. . . 4 |- (A e. P. -> (x e. A -> E.f((f e. A /\ x <Q f) /\ E.g x = (f .Q g))))
57		prcdpq 5097	. . . . . . . 8 |- ((A e. P. /\ f e. A) -> (x <Q f -> x e. A))
58	57	ex 373	. . . . . . 7 |- (A e. P. -> (f e. A -> (x <Q f -> x e. A)))
59	58	imp3a 361	. . . . . 6 |- (A e. P. -> ((f e. A /\ x <Q f) -> x e. A))
60	59	adantrd 391	. . . . 5 |- (A e. P. -> (((f e. A /\ x <Q f) /\ E.g x = (f .Q g)) -> x e. A))
61	60	19.23adv 1214	. . . 4 |- (A e. P. -> (E.f((f e. A /\ x <Q f) /\ E.g x = (f .Q g)) -> x e. A))
62	56, 61	impbid 516	. . 3 |- (A e. P. -> (x e. A <-> E.f((f e. A /\ x <Q f) /\ E.g x = (f .Q g))))
63	24, 29, 62	3bitr4d 550	. 2 |- (A e. P. -> (x e. (A .P. 1P) <-> x e. A))
64	63	eqrdv 1473	1 |- (A e. P. -> (A .P. 1P) = A)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980   class class class wbr 2619  ` cfv 3182  (class class class)co 3963  Q.cnq 4979  1Qc1q 4980   .Q cmq 4982  *Qcrq 4983   <Q cltq 4984  P.cnp 4985  1Pc1p 4986   .P. cmp 4988

This theorem is referenced by:  m1m1sr 5202  1idsr 5207

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-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-mp 5089

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