Theorem ltsrpr 10764

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 10750	. . 3 ⊢ ~R Er (P × P)
2		erdm 8466	. . 3 ⊢ ( ~R Er (P × P) → dom ~R = (P × P))
3	1, 2	ax-mp 5	. 2 ⊢ dom ~R = (P × P)
4		df-nr 10743	. 2 ⊢ R = ((P × P) / ~R )
5		ltrelsr 10755	. 2 ⊢ <R ⊆ (R × R)
6		ltrelpr 10685	. 2 ⊢ <P ⊆ (P × P)
7		0npr 10679	. 2 ⊢ ¬ ∅ ∈ P
8		dmplp 10699	. 2 ⊢ dom +P = (P × P)
9		enrex 10754	. . 3 ⊢ ~R ∈ V
10		df-ltr 10746	. . 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 10705	. . . . . . 7 ⊢ ((𝑤 ∈ P ∧ 𝑣 ∈ P) → (𝑤 +P 𝑣) ∈ P)
12	11	ad2ant2lr 744	. . . . . 6 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑤 +P 𝑣) ∈ P)
13		addclpr 10705	. . . . . . 7 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +P 𝐶) ∈ P)
14	13	ad2ant2lr 744	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → (𝐵 +P 𝐶) ∈ P)
15	12, 14	anim12ci 613	. . . . 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 656	. . . 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 10753	. . . . . 6 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝐴 ∈ P ∧ 𝐵 ∈ P)) → ([⟨𝑧, 𝑤⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ↔ (𝑧 +P 𝐵) = (𝑤 +P 𝐴)))
18		enreceq 10753	. . . . . . 7 ⊢ (((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨𝑣, 𝑢⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ↔ (𝑣 +P 𝐷) = (𝑢 +P 𝐶)))
19		eqcom 2745	. . . . . . 7 ⊢ ((𝑣 +P 𝐷) = (𝑢 +P 𝐶) ↔ (𝑢 +P 𝐶) = (𝑣 +P 𝐷))
20	18, 19	bitrdi 286	. . . . . 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 635	. . . . 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 7264	. . . . . 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 10708	. . . . . . . . . 10 ⊢ (𝑢 +P 𝐵) = (𝐵 +P 𝑢)
24	23	oveq1i 7265	. . . . . . . . 9 ⊢ ((𝑢 +P 𝐵) +P 𝐶) = ((𝐵 +P 𝑢) +P 𝐶)
25		addasspr 10709	. . . . . . . . 9 ⊢ ((𝑢 +P 𝐵) +P 𝐶) = (𝑢 +P (𝐵 +P 𝐶))
26		addasspr 10709	. . . . . . . . 9 ⊢ ((𝐵 +P 𝑢) +P 𝐶) = (𝐵 +P (𝑢 +P 𝐶))
27	24, 25, 26	3eqtr3i 2774	. . . . . . . 8 ⊢ (𝑢 +P (𝐵 +P 𝐶)) = (𝐵 +P (𝑢 +P 𝐶))
28	27	oveq2i 7266	. . . . . . 7 ⊢ (𝑧 +P (𝑢 +P (𝐵 +P 𝐶))) = (𝑧 +P (𝐵 +P (𝑢 +P 𝐶)))
29		addasspr 10709	. . . . . . 7 ⊢ ((𝑧 +P 𝑢) +P (𝐵 +P 𝐶)) = (𝑧 +P (𝑢 +P (𝐵 +P 𝐶)))
30		addasspr 10709	. . . . . . 7 ⊢ ((𝑧 +P 𝐵) +P (𝑢 +P 𝐶)) = (𝑧 +P (𝐵 +P (𝑢 +P 𝐶)))
31	28, 29, 30	3eqtr4i 2776	. . . . . 6 ⊢ ((𝑧 +P 𝑢) +P (𝐵 +P 𝐶)) = ((𝑧 +P 𝐵) +P (𝑢 +P 𝐶))
32		addcompr 10708	. . . . . . . . . 10 ⊢ (𝑣 +P 𝐴) = (𝐴 +P 𝑣)
33	32	oveq1i 7265	. . . . . . . . 9 ⊢ ((𝑣 +P 𝐴) +P 𝐷) = ((𝐴 +P 𝑣) +P 𝐷)
34		addasspr 10709	. . . . . . . . 9 ⊢ ((𝑣 +P 𝐴) +P 𝐷) = (𝑣 +P (𝐴 +P 𝐷))
35		addasspr 10709	. . . . . . . . 9 ⊢ ((𝐴 +P 𝑣) +P 𝐷) = (𝐴 +P (𝑣 +P 𝐷))
36	33, 34, 35	3eqtr3i 2774	. . . . . . . 8 ⊢ (𝑣 +P (𝐴 +P 𝐷)) = (𝐴 +P (𝑣 +P 𝐷))
37	36	oveq2i 7266	. . . . . . 7 ⊢ (𝑤 +P (𝑣 +P (𝐴 +P 𝐷))) = (𝑤 +P (𝐴 +P (𝑣 +P 𝐷)))
38		addasspr 10709	. . . . . . 7 ⊢ ((𝑤 +P 𝑣) +P (𝐴 +P 𝐷)) = (𝑤 +P (𝑣 +P (𝐴 +P 𝐷)))
39		addasspr 10709	. . . . . . 7 ⊢ ((𝑤 +P 𝐴) +P (𝑣 +P 𝐷)) = (𝑤 +P (𝐴 +P (𝑣 +P 𝐷)))
40	37, 38, 39	3eqtr4i 2776	. . . . . 6 ⊢ ((𝑤 +P 𝑣) +P (𝐴 +P 𝐷)) = ((𝑤 +P 𝐴) +P (𝑣 +P 𝐷))
41	22, 31, 40	3eqtr4g 2804	. . . . 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 252	. . . 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 7288	. . . . 5 ⊢ (𝑧 +P 𝑢) ∈ V
44		ovex 7288	. . . . 5 ⊢ (𝐵 +P 𝐶) ∈ V
45		ltapr 10732	. . . . 5 ⊢ (𝑓 ∈ P → (𝑥<P 𝑦 ↔ (𝑓 +P 𝑥)<P (𝑓 +P 𝑦)))
46		ovex 7288	. . . . 5 ⊢ (𝑤 +P 𝑣) ∈ V
47		addcompr 10708	. . . . 5 ⊢ (𝑥 +P 𝑦) = (𝑦 +P 𝑥)
48		ovex 7288	. . . . 5 ⊢ (𝐴 +P 𝐷) ∈ V
49	43, 44, 45, 46, 47, 48	caovord3 7463	. . . 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 680	. . 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 8557	. 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 8558	1 ⊢ ([⟨𝐴, 𝐵⟩] ~R <R [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷)<P (𝐵 +P 𝐶))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ⟨cop 4564   class class class wbr 5070   × cxp 5578  dom cdm 5580  (class class class)co 7255   Er wer 8453  [cec 8454  Pcnp 10546   +_P cpp 10548  <_P cltp 10550   ~_R cer 10551  Rcnr 10552   <_R cltr 10558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-omul 8272  df-er 8456  df-ec 8458  df-qs 8462  df-ni 10559  df-pli 10560  df-mi 10561  df-lti 10562  df-plpq 10595  df-mpq 10596  df-ltpq 10597  df-enq 10598  df-nq 10599  df-erq 10600  df-plq 10601  df-mq 10602  df-1nq 10603  df-rq 10604  df-ltnq 10605  df-np 10668  df-plp 10670  df-ltp 10672  df-enr 10742  df-nr 10743  df-ltr 10746

This theorem is referenced by:  gt0srpr  10765  ltsosr  10781  0lt1sr  10782  ltasr  10787  mappsrpr  10795  ltpsrpr  10796  map2psrpr  10797