Theorem ltsrpr 10691

Description: Ordering of signed reals in terms of positive reals. (Contributed by NM, 20-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)

Assertion

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

Proof of Theorem ltsrpr

Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		enrer 10677	. . 3 ⊢ ~R Er (P × P)
2		erdm 8401	. . 3 ⊢ ( ~R Er (P × P) → dom ~R = (P × P))
3	1, 2	ax-mp 5	. 2 ⊢ dom ~R = (P × P)
4		df-nr 10670	. 2 ⊢ R = ((P × P) / ~R )
5		ltrelsr 10682	. 2 ⊢ <R ⊆ (R × R)
6		ltrelpr 10612	. 2 ⊢ <P ⊆ (P × P)
7		0npr 10606	. 2 ⊢ ¬ ∅ ∈ P
8		dmplp 10626	. 2 ⊢ dom +P = (P × P)
9		enrex 10681	. . 3 ⊢ ~R ∈ V
10		df-ltr 10673	. . 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)))}
11		addclpr 10632	. . . . . . 7 ⊢ ((𝑤 ∈ P ∧ 𝑣 ∈ P) → (𝑤 +P 𝑣) ∈ P)
12	11	ad2ant2lr 748	. . . . . 6 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑤 +P 𝑣) ∈ P)
13		addclpr 10632	. . . . . . 7 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +P 𝐶) ∈ P)
14	13	ad2ant2lr 748	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → (𝐵 +P 𝐶) ∈ P)
15	12, 14	anim12ci 617	. . . . 5 ⊢ ((((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) ∧ ((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P))) → ((𝐵 +P 𝐶) ∈ P ∧ (𝑤 +P 𝑣) ∈ P))
16	15	an4s 660	. . . 4 ⊢ ((((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝐴 ∈ P ∧ 𝐵 ∈ P)) ∧ ((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P))) → ((𝐵 +P 𝐶) ∈ P ∧ (𝑤 +P 𝑣) ∈ P))
17		enreceq 10680	. . . . . 6 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝐴 ∈ P ∧ 𝐵 ∈ P)) → ([⟨𝑧, 𝑤⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ↔ (𝑧 +P 𝐵) = (𝑤 +P 𝐴)))
18		enreceq 10680	. . . . . . 7 ⊢ (((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨𝑣, 𝑢⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ↔ (𝑣 +P 𝐷) = (𝑢 +P 𝐶)))
19		eqcom 2744	. . . . . . 7 ⊢ ((𝑣 +P 𝐷) = (𝑢 +P 𝐶) ↔ (𝑢 +P 𝐶) = (𝑣 +P 𝐷))
20	18, 19	bitrdi 290	. . . . . 6 ⊢ (((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨𝑣, 𝑢⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ↔ (𝑢 +P 𝐶) = (𝑣 +P 𝐷)))
21	17, 20	bi2anan9 639	. . . . 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))))
22		oveq12 7222	. . . . . 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)))
23		addcompr 10635	. . . . . . . . . 10 ⊢ (𝑢 +P 𝐵) = (𝐵 +P 𝑢)
24	23	oveq1i 7223	. . . . . . . . 9 ⊢ ((𝑢 +P 𝐵) +P 𝐶) = ((𝐵 +P 𝑢) +P 𝐶)
25		addasspr 10636	. . . . . . . . 9 ⊢ ((𝑢 +P 𝐵) +P 𝐶) = (𝑢 +P (𝐵 +P 𝐶))
26		addasspr 10636	. . . . . . . . 9 ⊢ ((𝐵 +P 𝑢) +P 𝐶) = (𝐵 +P (𝑢 +P 𝐶))
27	24, 25, 26	3eqtr3i 2773	. . . . . . . 8 ⊢ (𝑢 +P (𝐵 +P 𝐶)) = (𝐵 +P (𝑢 +P 𝐶))
28	27	oveq2i 7224	. . . . . . 7 ⊢ (𝑧 +P (𝑢 +P (𝐵 +P 𝐶))) = (𝑧 +P (𝐵 +P (𝑢 +P 𝐶)))
29		addasspr 10636	. . . . . . 7 ⊢ ((𝑧 +P 𝑢) +P (𝐵 +P 𝐶)) = (𝑧 +P (𝑢 +P (𝐵 +P 𝐶)))
30		addasspr 10636	. . . . . . 7 ⊢ ((𝑧 +P 𝐵) +P (𝑢 +P 𝐶)) = (𝑧 +P (𝐵 +P (𝑢 +P 𝐶)))
31	28, 29, 30	3eqtr4i 2775	. . . . . 6 ⊢ ((𝑧 +P 𝑢) +P (𝐵 +P 𝐶)) = ((𝑧 +P 𝐵) +P (𝑢 +P 𝐶))
32		addcompr 10635	. . . . . . . . . 10 ⊢ (𝑣 +P 𝐴) = (𝐴 +P 𝑣)
33	32	oveq1i 7223	. . . . . . . . 9 ⊢ ((𝑣 +P 𝐴) +P 𝐷) = ((𝐴 +P 𝑣) +P 𝐷)
34		addasspr 10636	. . . . . . . . 9 ⊢ ((𝑣 +P 𝐴) +P 𝐷) = (𝑣 +P (𝐴 +P 𝐷))
35		addasspr 10636	. . . . . . . . 9 ⊢ ((𝐴 +P 𝑣) +P 𝐷) = (𝐴 +P (𝑣 +P 𝐷))
36	33, 34, 35	3eqtr3i 2773	. . . . . . . 8 ⊢ (𝑣 +P (𝐴 +P 𝐷)) = (𝐴 +P (𝑣 +P 𝐷))
37	36	oveq2i 7224	. . . . . . 7 ⊢ (𝑤 +P (𝑣 +P (𝐴 +P 𝐷))) = (𝑤 +P (𝐴 +P (𝑣 +P 𝐷)))
38		addasspr 10636	. . . . . . 7 ⊢ ((𝑤 +P 𝑣) +P (𝐴 +P 𝐷)) = (𝑤 +P (𝑣 +P (𝐴 +P 𝐷)))
39		addasspr 10636	. . . . . . 7 ⊢ ((𝑤 +P 𝐴) +P (𝑣 +P 𝐷)) = (𝑤 +P (𝐴 +P (𝑣 +P 𝐷)))
40	37, 38, 39	3eqtr4i 2775	. . . . . 6 ⊢ ((𝑤 +P 𝑣) +P (𝐴 +P 𝐷)) = ((𝑤 +P 𝐴) +P (𝑣 +P 𝐷))
41	22, 31, 40	3eqtr4g 2803	. . . . 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)))
42	21, 41	syl6bi 256	. . . 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))))
43		ovex 7246	. . . . 5 ⊢ (𝑧 +P 𝑢) ∈ V
44		ovex 7246	. . . . 5 ⊢ (𝐵 +P 𝐶) ∈ V
45		ltapr 10659	. . . . 5 ⊢ (𝑓 ∈ P → (𝑥<P 𝑦 ↔ (𝑓 +P 𝑥)<P (𝑓 +P 𝑦)))
46		ovex 7246	. . . . 5 ⊢ (𝑤 +P 𝑣) ∈ V
47		addcompr 10635	. . . . 5 ⊢ (𝑥 +P 𝑦) = (𝑦 +P 𝑥)
48		ovex 7246	. . . . 5 ⊢ (𝐴 +P 𝐷) ∈ V
49	43, 44, 45, 46, 47, 48	caovord3 7421	. . . 4 ⊢ ((((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)))
50	16, 42, 49	syl6an 684	. . 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	9, 1, 4, 10, 50	brecop 8492	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨𝐴, 𝐵⟩] ~R <R [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷)<P (𝐵 +P 𝐶)))
52	3, 4, 5, 6, 7, 8, 51	brecop2 8493	1 ⊢ ([⟨𝐴, 𝐵⟩] ~R <R [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷)<P (𝐵 +P 𝐶))

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ⟨cop 4547   class class class wbr 5053   × cxp 5549  dom cdm 5551  (class class class)co 7213   Er wer 8388  [cec 8389  Pcnp 10473   +_P cpp 10475  <_P cltp 10477   ~_R cer 10478  Rcnr 10479   <_R cltr 10485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-oadd 8206  df-omul 8207  df-er 8391  df-ec 8393  df-qs 8397  df-ni 10486  df-pli 10487  df-mi 10488  df-lti 10489  df-plpq 10522  df-mpq 10523  df-ltpq 10524  df-enq 10525  df-nq 10526  df-erq 10527  df-plq 10528  df-mq 10529  df-1nq 10530  df-rq 10531  df-ltnq 10532  df-np 10595  df-plp 10597  df-ltp 10599  df-enr 10669  df-nr 10670  df-ltr 10673

This theorem is referenced by:  gt0srpr  10692  ltsosr  10708  0lt1sr  10709  ltasr  10714  mappsrpr  10722  ltpsrpr  10723  map2psrpr  10724