Theorem ltsrpr 11115

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 11101	. . 3 ⊢ ~R Er (P × P)
2		erdm 8754	. . 3 ⊢ ( ~R Er (P × P) → dom ~R = (P × P))
3	1, 2	ax-mp 5	. 2 ⊢ dom ~R = (P × P)
4		df-nr 11094	. 2 ⊢ R = ((P × P) / ~R )
5		ltrelsr 11106	. 2 ⊢ <R ⊆ (R × R)
6		ltrelpr 11036	. 2 ⊢ <P ⊆ (P × P)
7		0npr 11030	. 2 ⊢ ¬ ∅ ∈ P
8		dmplp 11050	. 2 ⊢ dom +P = (P × P)
9		enrex 11105	. . 3 ⊢ ~R ∈ V
10		df-ltr 11097	. . 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 11056	. . . . . . 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 11056	. . . . . . 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 614	. . . . 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 11104	. . . . . 6 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝐴 ∈ P ∧ 𝐵 ∈ P)) → ([⟨𝑧, 𝑤⟩] ~R = [⟨𝐴, 𝐵⟩] ~R ↔ (𝑧 +P 𝐵) = (𝑤 +P 𝐴)))
18		enreceq 11104	. . . . . . 7 ⊢ (((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([⟨𝑣, 𝑢⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ↔ (𝑣 +P 𝐷) = (𝑢 +P 𝐶)))
19		eqcom 2742	. . . . . . 7 ⊢ ((𝑣 +P 𝐷) = (𝑢 +P 𝐶) ↔ (𝑢 +P 𝐶) = (𝑣 +P 𝐷))
20	18, 19	bitrdi 287	. . . . . 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 638	. . . . 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 7440	. . . . . 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 11059	. . . . . . . . . 10 ⊢ (𝑢 +P 𝐵) = (𝐵 +P 𝑢)
24	23	oveq1i 7441	. . . . . . . . 9 ⊢ ((𝑢 +P 𝐵) +P 𝐶) = ((𝐵 +P 𝑢) +P 𝐶)
25		addasspr 11060	. . . . . . . . 9 ⊢ ((𝑢 +P 𝐵) +P 𝐶) = (𝑢 +P (𝐵 +P 𝐶))
26		addasspr 11060	. . . . . . . . 9 ⊢ ((𝐵 +P 𝑢) +P 𝐶) = (𝐵 +P (𝑢 +P 𝐶))
27	24, 25, 26	3eqtr3i 2771	. . . . . . . 8 ⊢ (𝑢 +P (𝐵 +P 𝐶)) = (𝐵 +P (𝑢 +P 𝐶))
28	27	oveq2i 7442	. . . . . . 7 ⊢ (𝑧 +P (𝑢 +P (𝐵 +P 𝐶))) = (𝑧 +P (𝐵 +P (𝑢 +P 𝐶)))
29		addasspr 11060	. . . . . . 7 ⊢ ((𝑧 +P 𝑢) +P (𝐵 +P 𝐶)) = (𝑧 +P (𝑢 +P (𝐵 +P 𝐶)))
30		addasspr 11060	. . . . . . 7 ⊢ ((𝑧 +P 𝐵) +P (𝑢 +P 𝐶)) = (𝑧 +P (𝐵 +P (𝑢 +P 𝐶)))
31	28, 29, 30	3eqtr4i 2773	. . . . . 6 ⊢ ((𝑧 +P 𝑢) +P (𝐵 +P 𝐶)) = ((𝑧 +P 𝐵) +P (𝑢 +P 𝐶))
32		addcompr 11059	. . . . . . . . . 10 ⊢ (𝑣 +P 𝐴) = (𝐴 +P 𝑣)
33	32	oveq1i 7441	. . . . . . . . 9 ⊢ ((𝑣 +P 𝐴) +P 𝐷) = ((𝐴 +P 𝑣) +P 𝐷)
34		addasspr 11060	. . . . . . . . 9 ⊢ ((𝑣 +P 𝐴) +P 𝐷) = (𝑣 +P (𝐴 +P 𝐷))
35		addasspr 11060	. . . . . . . . 9 ⊢ ((𝐴 +P 𝑣) +P 𝐷) = (𝐴 +P (𝑣 +P 𝐷))
36	33, 34, 35	3eqtr3i 2771	. . . . . . . 8 ⊢ (𝑣 +P (𝐴 +P 𝐷)) = (𝐴 +P (𝑣 +P 𝐷))
37	36	oveq2i 7442	. . . . . . 7 ⊢ (𝑤 +P (𝑣 +P (𝐴 +P 𝐷))) = (𝑤 +P (𝐴 +P (𝑣 +P 𝐷)))
38		addasspr 11060	. . . . . . 7 ⊢ ((𝑤 +P 𝑣) +P (𝐴 +P 𝐷)) = (𝑤 +P (𝑣 +P (𝐴 +P 𝐷)))
39		addasspr 11060	. . . . . . 7 ⊢ ((𝑤 +P 𝐴) +P (𝑣 +P 𝐷)) = (𝑤 +P (𝐴 +P (𝑣 +P 𝐷)))
40	37, 38, 39	3eqtr4i 2773	. . . . . 6 ⊢ ((𝑤 +P 𝑣) +P (𝐴 +P 𝐷)) = ((𝑤 +P 𝐴) +P (𝑣 +P 𝐷))
41	22, 31, 40	3eqtr4g 2800	. . . . 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 253	. . . 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 7464	. . . . 5 ⊢ (𝑧 +P 𝑢) ∈ V
44		ovex 7464	. . . . 5 ⊢ (𝐵 +P 𝐶) ∈ V
45		ltapr 11083	. . . . 5 ⊢ (𝑓 ∈ P → (𝑥<P 𝑦 ↔ (𝑓 +P 𝑥)<P (𝑓 +P 𝑦)))
46		ovex 7464	. . . . 5 ⊢ (𝑤 +P 𝑣) ∈ V
47		addcompr 11059	. . . . 5 ⊢ (𝑥 +P 𝑦) = (𝑦 +P 𝑥)
48		ovex 7464	. . . . 5 ⊢ (𝐴 +P 𝐷) ∈ V
49	43, 44, 45, 46, 47, 48	caovord3 7646	. . . 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 8849	. 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 8850	1 ⊢ ([⟨𝐴, 𝐵⟩] ~R <R [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷)<P (𝐵 +P 𝐶))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ⟨cop 4637   class class class wbr 5148   × cxp 5687  dom cdm 5689  (class class class)co 7431   Er wer 8741  [cec 8742  Pcnp 10897   +_P cpp 10899  <_P cltp 10901   ~_R cer 10902  Rcnr 10903   <_R cltr 10909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510  df-er 8744  df-ec 8746  df-qs 8750  df-ni 10910  df-pli 10911  df-mi 10912  df-lti 10913  df-plpq 10946  df-mpq 10947  df-ltpq 10948  df-enq 10949  df-nq 10950  df-erq 10951  df-plq 10952  df-mq 10953  df-1nq 10954  df-rq 10955  df-ltnq 10956  df-np 11019  df-plp 11021  df-ltp 11023  df-enr 11093  df-nr 11094  df-ltr 11097

This theorem is referenced by:  gt0srpr  11116  ltsosr  11132  0lt1sr  11133  ltasr  11138  mappsrpr  11146  ltpsrpr  11147  map2psrpr  11148