Theorem ltsrpr 3980

Description: Ordering of signed reals in terms of positive reals.

Hypotheses

Ref	Expression
ltsrpr.1	⊢ A ∈ V
ltsrpr.2	⊢ B ∈ V
ltsrpr.3	⊢ C ∈ V
ltsrpr.4	⊢ D ∈ V

Assertion

Ref	Expression
ltsrpr	⊢ ([⟨A, B⟩] ~R <R [⟨C, D⟩] ~R ↔ (A +P D)<P (B +P C))

Proof of Theorem ltsrpr

Step	Hyp	Ref	Expression
1		enrex 3972	. 2 ⊢ ~R ∈ V
2		ltsrpr.2	. 2 ⊢ B ∈ V
3		ltsrpr.3	. 2 ⊢ C ∈ V
4		ltsrpr.4	. 2 ⊢ D ∈ V
5		enrer 3970	. 2 ⊢ Er ~R
6		dmenr 3969	. 2 ⊢ dom ~R = (P × P)
7		df-nr 3961	. 2 ⊢ R = ((P × P) / ~R )
8		ltrelsr 3974	. 2 ⊢ <R ⊆ (R × R)
9		ltrelpr 3895	. 2 ⊢ <P ⊆ (P × P)
10		0npr 3890	. 2 ⊢ ¬ ∅ ∈ P
11		dmplp 3909	. 2 ⊢ dom +P = (P × P)
12		df-ltr 3964	. . 3 ⊢ <R = {⟨x, y⟩∣((x ∈ R ∧ y ∈ R) ∧ ∃z∃w∃v∃u((x = [⟨z, w⟩] ~R ∧ y = [⟨v, u⟩] ~R ) ∧ (z +P u)<P (w +P v)))}
13		enreceq 3971	. . . . . 6 ⊢ (((z ∈ P ∧ w ∈ P) ∧ (A ∈ P ∧ B ∈ P)) → ([⟨z, w⟩] ~R = [⟨A, B⟩] ~R ↔ (z +P B) = (w +P A)))
14		enreceq 3971	. . . . . . 7 ⊢ (((v ∈ P ∧ u ∈ P) ∧ (C ∈ P ∧ D ∈ P)) → ([⟨v, u⟩] ~R = [⟨C, D⟩] ~R ↔ (v +P D) = (u +P C)))
15		cleqcom 1103	. . . . . . 7 ⊢ ((v +P D) = (u +P C) ↔ (u +P C) = (v +P D))
16	14, 15	syl6bb 414	. . . . . 6 ⊢ (((v ∈ P ∧ u ∈ P) ∧ (C ∈ P ∧ D ∈ P)) → ([⟨v, u⟩] ~R = [⟨C, D⟩] ~R ↔ (u +P C) = (v +P D)))
17	13, 16	bi2anan9 478	. . . . 5 ⊢ ((((z ∈ P ∧ w ∈ P) ∧ (A ∈ P ∧ B ∈ P)) ∧ ((v ∈ P ∧ u ∈ P) ∧ (C ∈ P ∧ D ∈ P))) → (([⟨z, w⟩] ~R = [⟨A, B⟩] ~R ∧ [⟨v, u⟩] ~R = [⟨C, D⟩] ~R ) ↔ ((z +P B) = (w +P A) ∧ (u +P C) = (v +P D))))
18		opreq12 3008	. . . . . 6 ⊢ (((z +P B) = (w +P A) ∧ (u +P C) = (v +P D)) → ((z +P B) +P (u +P C)) = ((w +P A) +P (v +P D)))
19		visset 1350	. . . . . . 7 ⊢ z ∈ V
20		visset 1350	. . . . . . 7 ⊢ u ∈ V
21		visset 1350	. . . . . . . 8 ⊢ x ∈ V
22		visset 1350	. . . . . . . 8 ⊢ y ∈ V
23	21, 22	addcompr 3917	. . . . . . 7 ⊢ (x +P y) = (y +P x)
24		visset 1350	. . . . . . . 8 ⊢ f ∈ V
25	22, 24	addasspr 3918	. . . . . . 7 ⊢ ((x +P y) +P f) = (x +P (y +P f))
26	19, 20, 2, 23, 25, 3	caopr4 3078	. . . . . 6 ⊢ ((z +P u) +P (B +P C)) = ((z +P B) +P (u +P C))
27		visset 1350	. . . . . . 7 ⊢ w ∈ V
28		visset 1350	. . . . . . 7 ⊢ v ∈ V
29		ltsrpr.1	. . . . . . 7 ⊢ A ∈ V
30	27, 28, 29, 23, 25, 4	caopr4 3078	. . . . . 6 ⊢ ((w +P v) +P (A +P D)) = ((w +P A) +P (v +P D))
31	18, 26, 30	3eqtr4g 1147	. . . . 5 ⊢ (((z +P B) = (w +P A) ∧ (u +P C) = (v +P D)) → ((z +P u) +P (B +P C)) = ((w +P v) +P (A +P D)))
32	17, 31	syl6bi 187	. . . 4 ⊢ ((((z ∈ P ∧ w ∈ P) ∧ (A ∈ P ∧ B ∈ P)) ∧ ((v ∈ P ∧ u ∈ P) ∧ (C ∈ P ∧ D ∈ P))) → (([⟨z, w⟩] ~R = [⟨A, B⟩] ~R ∧ [⟨v, u⟩] ~R = [⟨C, D⟩] ~R ) → ((z +P u) +P (B +P C)) = ((w +P v) +P (A +P D))))
33		addclpr 3914	. . . . . . . . . 10 ⊢ ((B ∈ P ∧ C ∈ P) → (B +P C) ∈ P)
34	33	adantrr 312	. . . . . . . . 9 ⊢ ((B ∈ P ∧ (C ∈ P ∧ D ∈ P)) → (B +P C) ∈ P)
35	34	adantll 309	. . . . . . . 8 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (C ∈ P ∧ D ∈ P)) → (B +P C) ∈ P)
36		addclpr 3914	. . . . . . . . . 10 ⊢ ((w ∈ P ∧ v ∈ P) → (w +P v) ∈ P)
37	36	adantrr 312	. . . . . . . . 9 ⊢ ((w ∈ P ∧ (v ∈ P ∧ u ∈ P)) → (w +P v) ∈ P)
38	37	adantll 309	. . . . . . . 8 ⊢ (((z ∈ P ∧ w ∈ P) ∧ (v ∈ P ∧ u ∈ P)) → (w +P v) ∈ P)
39	35, 38	anim12i 268	. . . . . . 7 ⊢ ((((A ∈ P ∧ B ∈ P) ∧ (C ∈ P ∧ D ∈ P)) ∧ ((z ∈ P ∧ w ∈ P) ∧ (v ∈ P ∧ u ∈ P))) → ((B +P C) ∈ P ∧ (w +P v) ∈ P))
40	39	ancoms 334	. . . . . 6 ⊢ ((((z ∈ P ∧ w ∈ P) ∧ (v ∈ P ∧ u ∈ P)) ∧ ((A ∈ P ∧ B ∈ P) ∧ (C ∈ P ∧ D ∈ P))) → ((B +P C) ∈ P ∧ (w +P v) ∈ P))
41	40	an4s 390	. . . . 5 ⊢ ((((z ∈ P ∧ w ∈ P) ∧ (A ∈ P ∧ B ∈ P)) ∧ ((v ∈ P ∧ u ∈ P) ∧ (C ∈ P ∧ D ∈ P))) → ((B +P C) ∈ P ∧ (w +P v) ∈ P))
42		oprex 3018	. . . . . . 7 ⊢ (z +P u) ∈ V
43		oprex 3018	. . . . . . 7 ⊢ (B +P C) ∈ V
44	21, 22	ltapr 3945	. . . . . . 7 ⊢ (f ∈ P → (x<P y ↔ (f +P x)<P (f +P y)))
45		oprex 3018	. . . . . . 7 ⊢ (w +P v) ∈ V
46		oprex 3018	. . . . . . 7 ⊢ (A +P D) ∈ V
47	42, 43, 44, 45, 23, 46	caoprord3 3072	. . . . . 6 ⊢ ((((B +P C) ∈ P ∧ (w +P v) ∈ P) ∧ ((z +P u) +P (B +P C)) = ((w +P v) +P (A +P D))) → ((z +P u)<P (w +P v) ↔ (A +P D)<P (B +P C)))
48	47	exp 291	. . . . 5 ⊢ (((B +P C) ∈ P ∧ (w +P v) ∈ P) → (((z +P u) +P (B +P C)) = ((w +P v) +P (A +P D)) → ((z +P u)<P (w +P v) ↔ (A +P D)<P (B +P C))))
49	41, 48	syl 12	. . . 4 ⊢ ((((z ∈ P ∧ w ∈ P) ∧ (A ∈ P ∧ B ∈ P)) ∧ ((v ∈ P ∧ u ∈ P) ∧ (C ∈ P ∧ D ∈ P))) → (((z +P u) +P (B +P C)) = ((w +P v) +P (A +P D)) → ((z +P u)<P (w +P v) ↔ (A +P D)<P (B +P C))))
50	32, 49	syld 27	. . 3 ⊢ ((((z ∈ P ∧ w ∈ P) ∧ (A ∈ P ∧ B ∈ P)) ∧ ((v ∈ P ∧ u ∈ P) ∧ (C ∈ P ∧ D ∈ P))) → (([⟨z, w⟩] ~R = [⟨A, B⟩] ~R ∧ [⟨v, u⟩] ~R = [⟨C, D⟩] ~R ) → ((z +P u)<P (w +P v) ↔ (A +P D)<P (B +P C))))
51	1, 5, 6, 7, 12, 50	brecop 3242	. 2 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (C ∈ P ∧ D ∈ P)) → ([⟨A, B⟩] ~R <R [⟨C, D⟩] ~R ↔ (A +P D)<P (B +P C)))
52	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 51	brecop2 3243	1 ⊢ ([⟨A, B⟩] ~R <R [⟨C, D⟩] ~R ↔ (A +P D)<P (B +P C))

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

This theorem is referenced by:  gt0srpr 3981  ltsosr 3997  0lt1sr 3998  ltasr 4003  mappsrpr 4012  ltpsrpr 4013  map2psrpr 4014

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

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