Theorem map2psrpr 4014

Description: Equivalence for positive signed real.

Hypothesis

Ref	Expression
map2psrpr.1	⊢ A ∈ V

Assertion

Ref	Expression
map2psrpr	⊢ (0R <R A ↔ ∃x(x ∈ P ∧ [⟨(x +P 1P), 1P⟩] ~R = A))

Distinct variable group(s):   x,A

Proof of Theorem map2psrpr

Step	Hyp	Ref	Expression
1		map2psrpr.1	. . . . 5 ⊢ A ∈ V
2		ltrelsr 3974	. . . . 5 ⊢ <R ⊆ (R × R)
3	1, 2	brel 2459	. . . 4 ⊢ (0R <R A → (0R ∈ R ∧ A ∈ R))
4	3	pm3.27d 262	. . 3 ⊢ (0R <R A → A ∈ R)
5		df-nr 3961	. . . 4 ⊢ R = ((P × P) / ~R )
6		breq2 2066	. . . . 5 ⊢ ([⟨y, z⟩] ~R = A → (0R <R [⟨y, z⟩] ~R ↔ 0R <R A))
7		cleq2 1110	. . . . . . 7 ⊢ ([⟨y, z⟩] ~R = A → ([⟨(x +P 1P), 1P⟩] ~R = [⟨y, z⟩] ~R ↔ [⟨(x +P 1P), 1P⟩] ~R = A))
8	7	anbi2d 468	. . . . . 6 ⊢ ([⟨y, z⟩] ~R = A → ((x ∈ P ∧ [⟨(x +P 1P), 1P⟩] ~R = [⟨y, z⟩] ~R ) ↔ (x ∈ P ∧ [⟨(x +P 1P), 1P⟩] ~R = A)))
9	8	biexdv 936	. . . . 5 ⊢ ([⟨y, z⟩] ~R = A → (∃x(x ∈ P ∧ [⟨(x +P 1P), 1P⟩] ~R = [⟨y, z⟩] ~R ) ↔ ∃x(x ∈ P ∧ [⟨(x +P 1P), 1P⟩] ~R = A)))
10	6, 9	imbi12d 474	. . . 4 ⊢ ([⟨y, z⟩] ~R = A → ((0R <R [⟨y, z⟩] ~R → ∃x(x ∈ P ∧ [⟨(x +P 1P), 1P⟩] ~R = [⟨y, z⟩] ~R )) ↔ (0R <R A → ∃x(x ∈ P ∧ [⟨(x +P 1P), 1P⟩] ~R = A))))
11		enreceq 3971	. . . . . . . . . . 11 ⊢ ((((x +P 1P) ∈ P ∧ 1P ∈ P) ∧ (y ∈ P ∧ z ∈ P)) → ([⟨(x +P 1P), 1P⟩] ~R = [⟨y, z⟩] ~R ↔ ((x +P 1P) +P z) = (1P +P y)))
12		1pr 3911	. . . . . . . . . . . . 13 ⊢ 1P ∈ P
13		addclpr 3914	. . . . . . . . . . . . 13 ⊢ ((x ∈ P ∧ 1P ∈ P) → (x +P 1P) ∈ P)
14	12, 13	mpan2 519	. . . . . . . . . . . 12 ⊢ (x ∈ P → (x +P 1P) ∈ P)
15	14, 12	jctir 241	. . . . . . . . . . 11 ⊢ (x ∈ P → ((x +P 1P) ∈ P ∧ 1P ∈ P))
16	11, 15	sylan 343	. . . . . . . . . 10 ⊢ ((x ∈ P ∧ (y ∈ P ∧ z ∈ P)) → ([⟨(x +P 1P), 1P⟩] ~R = [⟨y, z⟩] ~R ↔ ((x +P 1P) +P z) = (1P +P y)))
17	12	elisseti 1355	. . . . . . . . . . . . 13 ⊢ 1P ∈ V
18		visset 1350	. . . . . . . . . . . . 13 ⊢ z ∈ V
19	17, 18	addasspr 3918	. . . . . . . . . . . 12 ⊢ ((x +P 1P) +P z) = (x +P (1P +P z))
20		visset 1350	. . . . . . . . . . . . 13 ⊢ x ∈ V
21		oprex 3018	. . . . . . . . . . . . 13 ⊢ (1P +P z) ∈ V
22	20, 21	addcompr 3917	. . . . . . . . . . . 12 ⊢ (x +P (1P +P z)) = ((1P +P z) +P x)
23	19, 22	eqtr 1119	. . . . . . . . . . 11 ⊢ ((x +P 1P) +P z) = ((1P +P z) +P x)
24	23	cleq1i 1108	. . . . . . . . . 10 ⊢ (((x +P 1P) +P z) = (1P +P y) ↔ ((1P +P z) +P x) = (1P +P y))
25	16, 24	syl6bb 414	. . . . . . . . 9 ⊢ ((x ∈ P ∧ (y ∈ P ∧ z ∈ P)) → ([⟨(x +P 1P), 1P⟩] ~R = [⟨y, z⟩] ~R ↔ ((1P +P z) +P x) = (1P +P y)))
26	25	exp 291	. . . . . . . 8 ⊢ (x ∈ P → ((y ∈ P ∧ z ∈ P) → ([⟨(x +P 1P), 1P⟩] ~R = [⟨y, z⟩] ~R ↔ ((1P +P z) +P x) = (1P +P y))))
27	26	com12 13	. . . . . . 7 ⊢ ((y ∈ P ∧ z ∈ P) → (x ∈ P → ([⟨(x +P 1P), 1P⟩] ~R = [⟨y, z⟩] ~R ↔ ((1P +P z) +P x) = (1P +P y))))
28	27	pm5.32d 491	. . . . . 6 ⊢ ((y ∈ P ∧ z ∈ P) → ((x ∈ P ∧ [⟨(x +P 1P), 1P⟩] ~R = [⟨y, z⟩] ~R ) ↔ (x ∈ P ∧ ((1P +P z) +P x) = (1P +P y))))
29	28	biexdv 936	. . . . 5 ⊢ ((y ∈ P ∧ z ∈ P) → (∃x(x ∈ P ∧ [⟨(x +P 1P), 1P⟩] ~R = [⟨y, z⟩] ~R ) ↔ ∃x(x ∈ P ∧ ((1P +P z) +P x) = (1P +P y))))
30		df-0r 3965	. . . . . . . 8 ⊢ 0R = [⟨1P, 1P⟩] ~R
31	30	breq1i 2068	. . . . . . 7 ⊢ (0R <R [⟨y, z⟩] ~R ↔ [⟨1P, 1P⟩] ~R <R [⟨y, z⟩] ~R )
32		visset 1350	. . . . . . . 8 ⊢ y ∈ V
33	17, 17, 32, 18	ltsrpr 3980	. . . . . . 7 ⊢ ([⟨1P, 1P⟩] ~R <R [⟨y, z⟩] ~R ↔ (1P +P z)<P (1P +P y))
34	31, 33	bitr 151	. . . . . 6 ⊢ (0R <R [⟨y, z⟩] ~R ↔ (1P +P z)<P (1P +P y))
35		oprex 3018	. . . . . . 7 ⊢ (1P +P y) ∈ V
36	35	ltexpri 3943	. . . . . 6 ⊢ ((1P +P z)<P (1P +P y) → ∃x(x ∈ P ∧ ((1P +P z) +P x) = (1P +P y)))
37	34, 36	sylbi 174	. . . . 5 ⊢ (0R <R [⟨y, z⟩] ~R → ∃x(x ∈ P ∧ ((1P +P z) +P x) = (1P +P y)))
38	29, 37	syl5bir 184	. . . 4 ⊢ ((y ∈ P ∧ z ∈ P) → (0R <R [⟨y, z⟩] ~R → ∃x(x ∈ P ∧ [⟨(x +P 1P), 1P⟩] ~R = [⟨y, z⟩] ~R )))
39	5, 10, 38	ecoptocl 3239	. . 3 ⊢ (A ∈ R → (0R <R A → ∃x(x ∈ P ∧ [⟨(x +P 1P), 1P⟩] ~R = A)))
40	4, 39	mpcom 49	. 2 ⊢ (0R <R A → ∃x(x ∈ P ∧ [⟨(x +P 1P), 1P⟩] ~R = A))
41		breq2 2066	. . . . 5 ⊢ ([⟨(x +P 1P), 1P⟩] ~R = A → (0R <R [⟨(x +P 1P), 1P⟩] ~R ↔ 0R <R A))
42	20	mappsrpr 4012	. . . . 5 ⊢ (0R <R [⟨(x +P 1P), 1P⟩] ~R ↔ x ∈ P)
43	41, 42	syl5bbr 412	. . . 4 ⊢ ([⟨(x +P 1P), 1P⟩] ~R = A → (x ∈ P ↔ 0R <R A))
44	43	biimpac 326	. . 3 ⊢ ((x ∈ P ∧ [⟨(x +P 1P), 1P⟩] ~R = A) → 0R <R A)
45	44	19.23aiv 952	. 2 ⊢ (∃x(x ∈ P ∧ [⟨(x +P 1P), 1P⟩] ~R = A) → 0R <R A)
46	40, 45	impbi 139	1 ⊢ (0R <R A ↔ ∃x(x ∈ P ∧ [⟨(x +P 1P), 1P⟩] ~R = A))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   class class class wbr 2054  (class class class)co 3001  [cec 3198  Pcnp 3779  1_Pc1p 3780   +_P cpp 3781  <_P cltp 3783   ~_R cer 3786  Rcnr 3787  0_Rc0r 3788   <_R cltr 3793

This theorem is referenced by:  suppsrlem 4015  suppsr 4016

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-plp 3882  df-ltp 3884  df-enr 3960  df-nr 3961  df-ltr 3964  df-0r 3965

metamath.org