Theorem 1idpr 3927

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

Assertion

Ref	Expression
1idpr	⊢ (A ∈ P → (A ·P 1P) = A)

Proof of Theorem 1idpr

Step	Hyp	Ref	Expression
1		breq1 2065	. . . . . . . . . . . 12 ⊢ (x = (f ·Q g) → (x <Q f ↔ (f ·Q g) <Q f))
2		visset 1350	. . . . . . . . . . . . . . 15 ⊢ g ∈ V
3		1q 3851	. . . . . . . . . . . . . . . 16 ⊢ 1Q ∈ Q
4	3	elisseti 1355	. . . . . . . . . . . . . . 15 ⊢ 1Q ∈ V
5	2, 4	ltmpq 3871	. . . . . . . . . . . . . 14 ⊢ (f ∈ Q → (g <Q 1Q ↔ (f ·Q g) <Q (f ·Q 1Q)))
6		mulidpq 3863	. . . . . . . . . . . . . . 15 ⊢ (f ∈ Q → (f ·Q 1Q) = f)
7	6	breq2d 2072	. . . . . . . . . . . . . 14 ⊢ (f ∈ Q → ((f ·Q g) <Q (f ·Q 1Q) ↔ (f ·Q g) <Q f))
8	5, 7	bitrd 406	. . . . . . . . . . . . 13 ⊢ (f ∈ Q → (g <Q 1Q ↔ (f ·Q g) <Q f))
9		df-1p 3881	. . . . . . . . . . . . . 14 ⊢ 1P = {g∣g <Q 1Q}
10	9	cleqabi 1176	. . . . . . . . . . . . 13 ⊢ (g ∈ 1P ↔ g <Q 1Q)
11	8, 10	syl5rbb 411	. . . . . . . . . . . 12 ⊢ (f ∈ Q → ((f ·Q g) <Q f ↔ g ∈ 1P))
12	1, 11	sylan9bbr 419	. . . . . . . . . . 11 ⊢ ((f ∈ Q ∧ x = (f ·Q g)) → (x <Q f ↔ g ∈ 1P))
13		elprpq 3889	. . . . . . . . . . 11 ⊢ ((A ∈ P ∧ f ∈ A) → f ∈ Q)
14	12, 13	sylan 343	. . . . . . . . . 10 ⊢ (((A ∈ P ∧ f ∈ A) ∧ x = (f ·Q g)) → (x <Q f ↔ g ∈ 1P))
15	14	exp31 293	. . . . . . . . 9 ⊢ (A ∈ P → (f ∈ A → (x = (f ·Q g) → (x <Q f ↔ g ∈ 1P))))
16	15	imp3a 279	. . . . . . . 8 ⊢ (A ∈ P → ((f ∈ A ∧ x = (f ·Q g)) → (x <Q f ↔ g ∈ 1P)))
17	16	pm5.32d 491	. . . . . . 7 ⊢ (A ∈ P → (((f ∈ A ∧ x = (f ·Q g)) ∧ x <Q f) ↔ ((f ∈ A ∧ x = (f ·Q g)) ∧ g ∈ 1P)))
18		an23 371	. . . . . . 7 ⊢ (((f ∈ A ∧ x <Q f) ∧ x = (f ·Q g)) ↔ ((f ∈ A ∧ x = (f ·Q g)) ∧ x <Q f))
19		an23 371	. . . . . . 7 ⊢ (((f ∈ A ∧ g ∈ 1P) ∧ x = (f ·Q g)) ↔ ((f ∈ A ∧ x = (f ·Q g)) ∧ g ∈ 1P))
20	17, 18, 19	3bitr4g 428	. . . . . 6 ⊢ (A ∈ P → (((f ∈ A ∧ x <Q f) ∧ x = (f ·Q g)) ↔ ((f ∈ A ∧ g ∈ 1P) ∧ x = (f ·Q g))))
21	20	biexdv 936	. . . . 5 ⊢ (A ∈ P → (∃g((f ∈ A ∧ x <Q f) ∧ x = (f ·Q g)) ↔ ∃g((f ∈ A ∧ g ∈ 1P) ∧ x = (f ·Q g))))
22		19.42v 966	. . . . 5 ⊢ (∃g((f ∈ A ∧ x <Q f) ∧ x = (f ·Q g)) ↔ ((f ∈ A ∧ x <Q f) ∧ ∃g x = (f ·Q g)))
23	21, 22	syl5rbbr 413	. . . 4 ⊢ (A ∈ P → (∃g((f ∈ A ∧ g ∈ 1P) ∧ x = (f ·Q g)) ↔ ((f ∈ A ∧ x <Q f) ∧ ∃g x = (f ·Q g))))
24	23	biexdv 936	. . 3 ⊢ (A ∈ P → (∃f∃g((f ∈ A ∧ g ∈ 1P) ∧ x = (f ·Q g)) ↔ ∃f((f ∈ A ∧ x <Q f) ∧ ∃g x = (f ·Q g))))
25		1pr 3911	. . . 4 ⊢ 1P ∈ P
26		df-mp 3883	. . . . 5 ⊢ ·P = {⟨⟨w, v⟩, u⟩∣((w ∈ P ∧ v ∈ P) ∧ u = {x∣∃y ∈ w ∃z ∈ v x = (y ·Q z)})}
27		visset 1350	. . . . 5 ⊢ x ∈ V
28	26, 27	genpelv 3897	. . . 4 ⊢ ((A ∈ P ∧ 1P ∈ P) → (x ∈ (A ·P 1P) ↔ ∃f∃g((f ∈ A ∧ g ∈ 1P) ∧ x = (f ·Q g))))
29	25, 28	mpan2 519	. . 3 ⊢ (A ∈ P → (x ∈ (A ·P 1P) ↔ ∃f∃g((f ∈ A ∧ g ∈ 1P) ∧ x = (f ·Q g))))
30		prnmax 3893	. . . . . 6 ⊢ ((A ∈ P ∧ x ∈ A) → ∃f(f ∈ A ∧ x <Q f))
31		visset 1350	. . . . . . . . . . 11 ⊢ f ∈ V
32		ltrelpq 3845	. . . . . . . . . . 11 ⊢ <Q ⊆ (Q × Q)
33	31, 32	brel 2459	. . . . . . . . . 10 ⊢ (x <Q f → (x ∈ Q ∧ f ∈ Q))
34		recidpq 3865	. . . . . . . . . . . . . 14 ⊢ (f ∈ Q → (f ·Q (*Q ‘f)) = 1Q)
35	34	opreq2d 3013	. . . . . . . . . . . . 13 ⊢ (f ∈ Q → (x ·Q (f ·Q (*Q ‘f))) = (x ·Q 1Q))
36		fvex 2838	. . . . . . . . . . . . . 14 ⊢ (*Q ‘f) ∈ V
37		visset 1350	. . . . . . . . . . . . . . 15 ⊢ y ∈ V
38		visset 1350	. . . . . . . . . . . . . . 15 ⊢ z ∈ V
39	37, 38	mulcompq 3858	. . . . . . . . . . . . . 14 ⊢ (y ·Q z) = (z ·Q y)
40		visset 1350	. . . . . . . . . . . . . . 15 ⊢ w ∈ V
41	38, 40	mulasspq 3859	. . . . . . . . . . . . . 14 ⊢ ((y ·Q z) ·Q w) = (y ·Q (z ·Q w))
42	31, 27, 36, 39, 41	caopr12 3075	. . . . . . . . . . . . 13 ⊢ (f ·Q (x ·Q (*Q ‘f))) = (x ·Q (f ·Q (*Q ‘f)))
43	35, 42	syl5eq 1136	. . . . . . . . . . . 12 ⊢ (f ∈ Q → (f ·Q (x ·Q (*Q ‘f))) = (x ·Q 1Q))
44		mulidpq 3863	. . . . . . . . . . . 12 ⊢ (x ∈ Q → (x ·Q 1Q) = x)
45	43, 44	sylan9eqr 1145	. . . . . . . . . . 11 ⊢ ((x ∈ Q ∧ f ∈ Q) → (f ·Q (x ·Q (*Q ‘f))) = x)
46	45	cleqcomd 1106	. . . . . . . . . 10 ⊢ ((x ∈ Q ∧ f ∈ Q) → x = (f ·Q (x ·Q (*Q ‘f))))
47		oprex 3018	. . . . . . . . . . 11 ⊢ (x ·Q (*Q ‘f)) ∈ V
48		opreq2 3007	. . . . . . . . . . . 12 ⊢ (g = (x ·Q (*Q ‘f)) → (f ·Q g) = (f ·Q (x ·Q (*Q ‘f))))
49	48	cleq2d 1112	. . . . . . . . . . 11 ⊢ (g = (x ·Q (*Q ‘f)) → (x = (f ·Q g) ↔ x = (f ·Q (x ·Q (*Q ‘f)))))
50	47, 49	cla4ev 1401	. . . . . . . . . 10 ⊢ (x = (f ·Q (x ·Q (*Q ‘f))) → ∃g x = (f ·Q g))
51	33, 46, 50	3syl 21	. . . . . . . . 9 ⊢ (x <Q f → ∃g x = (f ·Q g))
52	51	adantl 305	. . . . . . . 8 ⊢ ((f ∈ A ∧ x <Q f) → ∃g x = (f ·Q g))
53	52	ancli 244	. . . . . . 7 ⊢ ((f ∈ A ∧ x <Q f) → ((f ∈ A ∧ x <Q f) ∧ ∃g x = (f ·Q g)))
54	53	19.22i 723	. . . . . 6 ⊢ (∃f(f ∈ A ∧ x <Q f) → ∃f((f ∈ A ∧ x <Q f) ∧ ∃g x = (f ·Q g)))
55	30, 54	syl 12	. . . . 5 ⊢ ((A ∈ P ∧ x ∈ A) → ∃f((f ∈ A ∧ x <Q f) ∧ ∃g x = (f ·Q g)))
56	55	exp 291	. . . 4 ⊢ (A ∈ P → (x ∈ A → ∃f((f ∈ A ∧ x <Q f) ∧ ∃g x = (f ·Q g))))
57		prcdpq 3891	. . . . . . . 8 ⊢ ((A ∈ P ∧ f ∈ A) → (x <Q f → x ∈ A))
58	57	exp 291	. . . . . . 7 ⊢ (A ∈ P → (f ∈ A → (x <Q f → x ∈ A)))
59	58	imp3a 279	. . . . . 6 ⊢ (A ∈ P → ((f ∈ A ∧ x <Q f) → x ∈ A))
60	59	adantrd 308	. . . . 5 ⊢ (A ∈ P → (((f ∈ A ∧ x <Q f) ∧ ∃g x = (f ·Q g)) → x ∈ A))
61	60	19.23adv 954	. . . 4 ⊢ (A ∈ P → (∃f((f ∈ A ∧ x <Q f) ∧ ∃g x = (f ·Q g)) → x ∈ A))
62	56, 61	impbid 397	. . 3 ⊢ (A ∈ P → (x ∈ A ↔ ∃f((f ∈ A ∧ x <Q f) ∧ ∃g x = (f ·Q g))))
63	24, 29, 62	3bitr4d 424	. 2 ⊢ (A ∈ P → (x ∈ (A ·P 1P) ↔ x ∈ A))
64	63	cleqrd 1100	1 ⊢ (A ∈ P → (A ·P 1P) = A)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092   class class class wbr 2054   ‘cfv 2422  (class class class)co 3001  Qcnq 3773  1_Qc1q 3774   ·_Q cmq 3776  *_Qcrq 3777   <_Q cltq 3778  Pcnp 3779  1_Pc1p 3780   ·_P cmp 3782

This theorem is referenced by:  m1m1sr 3996  1idsr 4001

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-mp 3883

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