Theorem 1idpr 5113

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 2617	. . . . . . . . . . . 12 |- (x = (f .Q g) -> (x <Q f <-> (f .Q g) <Q f))
2		visset 1809	. . . . . . . . . . . . . . 15 |- g e. V
3		1q 5037	. . . . . . . . . . . . . . . 16 |- 1Q e. Q.
4	3	elisseti 1814	. . . . . . . . . . . . . . 15 |- 1Q e. V
5	2, 4	ltmpq 5057	. . . . . . . . . . . . . 14 |- (f e. Q. -> (g <Q 1Q <-> (f .Q g) <Q (f .Q 1Q)))
6		mulidpq 5049	. . . . . . . . . . . . . . 15 |- (f e. Q. -> (f .Q 1Q) = f)
7	6	breq2d 2625	. . . . . . . . . . . . . 14 |- (f e. Q. -> ((f .Q g) <Q (f .Q 1Q) <-> (f .Q g) <Q f))
8	5, 7	bitrd 527	. . . . . . . . . . . . 13 |- (f e. Q. -> (g <Q 1Q <-> (f .Q g) <Q f))
9		df-1p 5067	. . . . . . . . . . . . . 14 |- 1P = {g | g <Q 1Q}
10	9	abeq2i 1567	. . . . . . . . . . . . 13 |- (g e. 1P <-> g <Q 1Q)
11	8, 10	syl5rbb 532	. . . . . . . . . . . 12 |- (f e. Q. -> ((f .Q g) <Q f <-> g e. 1P))
12	1, 11	sylan9bbr 540	. . . . . . . . . . 11 |- ((f e. Q. /\ x = (f .Q g)) -> (x <Q f <-> g e. 1P))
13		elprpq 5075	. . . . . . . . . . 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 646	. . . . . . 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 554	. . . . . 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 1277	. . . . 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 1306	. . . . 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 534	. . . 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 1277	. . 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 5097	. . . 4 |- 1P e. P.
26		df-mp 5069	. . . . 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 1809	. . . . 5 |- x e. V
28	26, 27	genpelv 5083	. . . 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 695	. . 3 |- (A e. P. -> (x e. (A .P. 1P) <-> E.fE.g((f e. A /\ g e. 1P) /\ x = (f .Q g))))
30		prnmax 5079	. . . . . 6 |- ((A e. P. /\ x e. A) -> E.f(f e. A /\ x <Q f))
31		visset 1809	. . . . . . . . . . 11 |- f e. V
32		ltrelpq 5031	. . . . . . . . . . 11 |- <Q (_ (Q. X. Q.)
33	31, 32	brel 3218	. . . . . . . . . 10 |- (x <Q f -> (x e. Q. /\ f e. Q.))
34		recidpq 5051	. . . . . . . . . . . . . 14 |- (f e. Q. -> (f .Q (*Q` f)) = 1Q)
35	34	opreq2d 3967	. . . . . . . . . . . . 13 |- (f e. Q. -> (x .Q (f .Q (*Q` f))) = (x .Q 1Q))
36		fvex 3723	. . . . . . . . . . . . . 14 |- (*Q` f) e. V
37		visset 1809	. . . . . . . . . . . . . . 15 |- y e. V
38		visset 1809	. . . . . . . . . . . . . . 15 |- z e. V
39	37, 38	mulcompq 5044	. . . . . . . . . . . . . 14 |- (y .Q z) = (z .Q y)
40		visset 1809	. . . . . . . . . . . . . . 15 |- w e. V
41	38, 40	mulasspq 5045	. . . . . . . . . . . . . 14 |- ((y .Q z) .Q w) = (y .Q (z .Q w))
42	31, 27, 36, 39, 41	caopr12 4053	. . . . . . . . . . . . 13 |- (f .Q (x .Q (*Q` f))) = (x .Q (f .Q (*Q` f)))
43	35, 42	syl5eq 1516	. . . . . . . . . . . 12 |- (f e. Q. -> (f .Q (x .Q (*Q` f))) = (x .Q 1Q))
44		mulidpq 5049	. . . . . . . . . . . 12 |- (x e. Q. -> (x .Q 1Q) = x)
45	43, 44	sylan9eqr 1526	. . . . . . . . . . 11 |- ((x e. Q. /\ f e. Q.) -> (f .Q (x .Q (*Q` f))) = x)
46	45	eqcomd 1477	. . . . . . . . . 10 |- ((x e. Q. /\ f e. Q.) -> x = (f .Q (x .Q (*Q` f))))
47		oprex 3974	. . . . . . . . . . 11 |- (x .Q (*Q` f)) e. V
48		opreq2 3960	. . . . . . . . . . . 12 |- (g = (x .Q (*Q` f)) -> (f .Q g) = (f .Q (x .Q (*Q` f))))
49	48	eqeq2d 1483	. . . . . . . . . . 11 |- (g = (x .Q (*Q` f)) -> (x = (f .Q g) <-> x = (f .Q (x .Q (*Q` f)))))
50	47, 49	cla4ev 1865	. . . . . . . . . 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 1038	. . . . . 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 5077	. . . . . . . 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 1212	. . . 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 515	. . 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 549	. 2 |- (A e. P. -> (x e. (A .P. 1P) <-> x e. A))
64	63	eqrdv 1471	1 |- (A e. P. -> (A .P. 1P) = A)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978   class class class wbr 2614  ` cfv 3177  (class class class)co 3954  Q.cnq 4959  1Qc1q 4960   .Q cmq 4962  *Qcrq 4963   <Q cltq 4964  P.cnp 4965  1Pc1p 4966   .P. cmp 4968

This theorem is referenced by:  m1m1sr 5182  1idsr 5187

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-mp 5069

Copyright terms: Public domain