Theorem ltsrpr 11035

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 11021	. . 3 ⊢ ~R Er (P × P)
2		erdm 8689	. . 3 ⊢ ( ~R Er (P × P) → dom ~R = (P × P))
3	1, 2	ax-mp 5	. 2 ⊢ dom ~R = (P × P)
4		df-nr 11014	. 2 ⊢ R = ((P × P) / ~R )
5		ltrelsr 11026	. 2 ⊢ <R ⊆ (R × R)
6		ltrelpr 10956	. 2 ⊢ <P ⊆ (P × P)
7		0npr 10950	. 2 ⊢ ¬ ∅ ∈ P
8		dmplp 10970	. 2 ⊢ dom +P = (P × P)
9		enrex 11025	. . 3 ⊢ ~R ∈ V
10		df-ltr 11017	. . 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 10976	. . . . . . 7 ⊢ ((𝑤 ∈ P ∧ 𝑣 ∈ P) → (𝑤 +P 𝑣) ∈ P)
12	11	ad2ant2lr 758	. . . . . 6 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑣 ∈ P ∧ 𝑢 ∈ P)) → (𝑤 +P 𝑣) ∈ P)
13		addclpr 10976	. . . . . . 7 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +P 𝐶) ∈ P)
14	13	ad2ant2lr 758	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → (𝐵 +P 𝐶) ∈ P)
15	12, 14	anim12ci 623	. . . . 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 670	. . . 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 11024	. . . . . 6 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝐴 ∈ P ∧ 𝐵 ∈ P)) → ([⟨𝑧, 𝑤⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ↔ (𝑧 +P 𝐵) = (𝑤 +P 𝐴)))
18		enreceq 11024	. . . . . . 7 ⊢ (((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨𝑣, 𝑢⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ↔ (𝑣 +P 𝐷) = (𝑢 +P 𝐶)))
19		eqcom 2769	. . . . . . 7 ⊢ ((𝑣 +P 𝐷) = (𝑢 +P 𝐶) ↔ (𝑢 +P 𝐶) = (𝑣 +P 𝐷))
20	18, 19	bitrdi 289	. . . . . 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 647	. . . . 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 7405	. . . . . 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 10979	. . . . . . . . . 10 ⊢ (𝑢 +P 𝐵) = (𝐵 +P 𝑢)
24	23	oveq1i 7406	. . . . . . . . 9 ⊢ ((𝑢 +P 𝐵) +P 𝐶) = ((𝐵 +P 𝑢) +P 𝐶)
25		addasspr 10980	. . . . . . . . 9 ⊢ ((𝑢 +P 𝐵) +P 𝐶) = (𝑢 +P (𝐵 +P 𝐶))
26		addasspr 10980	. . . . . . . . 9 ⊢ ((𝐵 +P 𝑢) +P 𝐶) = (𝐵 +P (𝑢 +P 𝐶))
27	24, 25, 26	3eqtr3i 2793	. . . . . . . 8 ⊢ (𝑢 +P (𝐵 +P 𝐶)) = (𝐵 +P (𝑢 +P 𝐶))
28	27	oveq2i 7407	. . . . . . 7 ⊢ (𝑧 +P (𝑢 +P (𝐵 +P 𝐶))) = (𝑧 +P (𝐵 +P (𝑢 +P 𝐶)))
29		addasspr 10980	. . . . . . 7 ⊢ ((𝑧 +P 𝑢) +P (𝐵 +P 𝐶)) = (𝑧 +P (𝑢 +P (𝐵 +P 𝐶)))
30		addasspr 10980	. . . . . . 7 ⊢ ((𝑧 +P 𝐵) +P (𝑢 +P 𝐶)) = (𝑧 +P (𝐵 +P (𝑢 +P 𝐶)))
31	28, 29, 30	3eqtr4i 2795	. . . . . 6 ⊢ ((𝑧 +P 𝑢) +P (𝐵 +P 𝐶)) = ((𝑧 +P 𝐵) +P (𝑢 +P 𝐶))
32		addcompr 10979	. . . . . . . . . 10 ⊢ (𝑣 +P 𝐴) = (𝐴 +P 𝑣)
33	32	oveq1i 7406	. . . . . . . . 9 ⊢ ((𝑣 +P 𝐴) +P 𝐷) = ((𝐴 +P 𝑣) +P 𝐷)
34		addasspr 10980	. . . . . . . . 9 ⊢ ((𝑣 +P 𝐴) +P 𝐷) = (𝑣 +P (𝐴 +P 𝐷))
35		addasspr 10980	. . . . . . . . 9 ⊢ ((𝐴 +P 𝑣) +P 𝐷) = (𝐴 +P (𝑣 +P 𝐷))
36	33, 34, 35	3eqtr3i 2793	. . . . . . . 8 ⊢ (𝑣 +P (𝐴 +P 𝐷)) = (𝐴 +P (𝑣 +P 𝐷))
37	36	oveq2i 7407	. . . . . . 7 ⊢ (𝑤 +P (𝑣 +P (𝐴 +P 𝐷))) = (𝑤 +P (𝐴 +P (𝑣 +P 𝐷)))
38		addasspr 10980	. . . . . . 7 ⊢ ((𝑤 +P 𝑣) +P (𝐴 +P 𝐷)) = (𝑤 +P (𝑣 +P (𝐴 +P 𝐷)))
39		addasspr 10980	. . . . . . 7 ⊢ ((𝑤 +P 𝐴) +P (𝑣 +P 𝐷)) = (𝑤 +P (𝐴 +P (𝑣 +P 𝐷)))
40	37, 38, 39	3eqtr4i 2795	. . . . . 6 ⊢ ((𝑤 +P 𝑣) +P (𝐴 +P 𝐷)) = ((𝑤 +P 𝐴) +P (𝑣 +P 𝐷))
41	22, 31, 40	3eqtr4g 2822	. . . . 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	biimtrdi 255	. . . 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 7429	. . . . 5 ⊢ (𝑧 +P 𝑢) ∈ V
44		ovex 7429	. . . . 5 ⊢ (𝐵 +P 𝐶) ∈ V
45		ltapr 11003	. . . . 5 ⊢ (𝑓 ∈ P → (𝑥<P 𝑦 ↔ (𝑓 +P 𝑥)<P (𝑓 +P 𝑦)))
46		ovex 7429	. . . . 5 ⊢ (𝑤 +P 𝑣) ∈ V
47		addcompr 10979	. . . . 5 ⊢ (𝑥 +P 𝑦) = (𝑦 +P 𝑥)
48		ovex 7429	. . . . 5 ⊢ (𝐴 +P 𝐷) ∈ V
49	43, 44, 45, 46, 47, 48	caovord3 7609	. . . 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 694	. . 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 8792	. 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 8793	1 ⊢ ([⟨𝐴, 𝐵⟩] ~R <R [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷)<P (𝐵 +P 𝐶))

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ⟨cop 4588   class class class wbr 5100   × cxp 5645  dom cdm 5647  (class class class)co 7396   Er wer 8675  [cec 8676  Pcnp 10817   +_P cpp 10819  <_P cltp 10821   ~_R cer 10822  Rcnr 10823   <_R cltr 10829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-inf2 9596

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-oadd 8441  df-omul 8442  df-er 8678  df-ec 8680  df-qs 8684  df-ni 10830  df-pli 10831  df-mi 10832  df-lti 10833  df-plpq 10866  df-mpq 10867  df-ltpq 10868  df-enq 10869  df-nq 10870  df-erq 10871  df-plq 10872  df-mq 10873  df-1nq 10874  df-rq 10875  df-ltnq 10876  df-np 10939  df-plp 10941  df-ltp 10943  df-enr 11013  df-nr 11014  df-ltr 11017

This theorem is referenced by:  gt0srpr  11036  ltsosr  11052  0lt1sr  11053  ltasr  11058  mappsrpr  11066  ltpsrpr  11067  map2psrpr  11068