Theorem ltasr 10856

Description: Ordering property of addition. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
ltasr	⊢ (𝐶 ∈ R → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))

Proof of Theorem ltasr

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

Step	Hyp	Ref	Expression
1		dmaddsr 10841	. 2 ⊢ dom +R = (R × R)
2		ltrelsr 10824	. 2 ⊢ <R ⊆ (R × R)
3		0nsr 10835	. 2 ⊢ ¬ ∅ ∈ R
4		df-nr 10812	. . . 4 ⊢ R = ((P × P) / ~R )
5		oveq1 7282	. . . . . 6 ⊢ ([⟨𝑣, 𝑢⟩] ~R = 𝐶 → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑥, 𝑦⟩] ~R ) = (𝐶 +R [⟨𝑥, 𝑦⟩] ~R ))
6		oveq1 7282	. . . . . 6 ⊢ ([⟨𝑣, 𝑢⟩] ~R = 𝐶 → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = (𝐶 +R [⟨𝑧, 𝑤⟩] ~R ))
7	5, 6	breq12d 5087	. . . . 5 ⊢ ([⟨𝑣, 𝑢⟩] ~R = 𝐶 → (([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑥, 𝑦⟩] ~R ) <R ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) ↔ (𝐶 +R [⟨𝑥, 𝑦⟩] ~R ) <R (𝐶 +R [⟨𝑧, 𝑤⟩] ~R )))
8	7	bibi2d 343	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ~R = 𝐶 → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑥, 𝑦⟩] ~R ) <R ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R )) ↔ ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝐶 +R [⟨𝑥, 𝑦⟩] ~R ) <R (𝐶 +R [⟨𝑧, 𝑤⟩] ~R ))))
9		breq1 5077	. . . . 5 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝐴 <R [⟨𝑧, 𝑤⟩] ~R ))
10		oveq2 7283	. . . . . 6 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (𝐶 +R [⟨𝑥, 𝑦⟩] ~R ) = (𝐶 +R 𝐴))
11	10	breq1d 5084	. . . . 5 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ((𝐶 +R [⟨𝑥, 𝑦⟩] ~R ) <R (𝐶 +R [⟨𝑧, 𝑤⟩] ~R ) ↔ (𝐶 +R 𝐴) <R (𝐶 +R [⟨𝑧, 𝑤⟩] ~R )))
12	9, 11	bibi12d 346	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝐶 +R [⟨𝑥, 𝑦⟩] ~R ) <R (𝐶 +R [⟨𝑧, 𝑤⟩] ~R )) ↔ (𝐴 <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝐶 +R 𝐴) <R (𝐶 +R [⟨𝑧, 𝑤⟩] ~R ))))
13		breq2 5078	. . . . 5 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝐴 <R 𝐵))
14		oveq2 7283	. . . . . 6 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐶 +R [⟨𝑧, 𝑤⟩] ~R ) = (𝐶 +R 𝐵))
15	14	breq2d 5086	. . . . 5 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((𝐶 +R 𝐴) <R (𝐶 +R [⟨𝑧, 𝑤⟩] ~R ) ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
16	13, 15	bibi12d 346	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((𝐴 <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝐶 +R 𝐴) <R (𝐶 +R [⟨𝑧, 𝑤⟩] ~R )) ↔ (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵))))
17		addclpr 10774	. . . . . . 7 ⊢ ((𝑣 ∈ P ∧ 𝑢 ∈ P) → (𝑣 +P 𝑢) ∈ P)
18	17	3ad2ant1 1132	. . . . . 6 ⊢ (((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑣 +P 𝑢) ∈ P)
19		ltapr 10801	. . . . . . 7 ⊢ ((𝑣 +P 𝑢) ∈ P → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ↔ ((𝑣 +P 𝑢) +P (𝑥 +P 𝑤))<P ((𝑣 +P 𝑢) +P (𝑦 +P 𝑧))))
20		ltsrpr 10833	. . . . . . 7 ⊢ ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧))
21		ltsrpr 10833	. . . . . . . 8 ⊢ ([⟨(𝑣 +P 𝑥), (𝑢 +P 𝑦)⟩] ~R <R [⟨(𝑣 +P 𝑧), (𝑢 +P 𝑤)⟩] ~R ↔ ((𝑣 +P 𝑥) +P (𝑢 +P 𝑤))<P ((𝑢 +P 𝑦) +P (𝑣 +P 𝑧)))
22		vex 3436	. . . . . . . . . 10 ⊢ 𝑣 ∈ V
23		vex 3436	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
24		vex 3436	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
25		addcompr 10777	. . . . . . . . . 10 ⊢ (𝑦 +P 𝑧) = (𝑧 +P 𝑦)
26		addasspr 10778	. . . . . . . . . 10 ⊢ ((𝑦 +P 𝑧) +P 𝑓) = (𝑦 +P (𝑧 +P 𝑓))
27		vex 3436	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
28	22, 23, 24, 25, 26, 27	caov4 7503	. . . . . . . . 9 ⊢ ((𝑣 +P 𝑥) +P (𝑢 +P 𝑤)) = ((𝑣 +P 𝑢) +P (𝑥 +P 𝑤))
29		addcompr 10777	. . . . . . . . . 10 ⊢ ((𝑢 +P 𝑦) +P (𝑣 +P 𝑧)) = ((𝑣 +P 𝑧) +P (𝑢 +P 𝑦))
30		vex 3436	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
31		addcompr 10777	. . . . . . . . . . 11 ⊢ (𝑥 +P 𝑤) = (𝑤 +P 𝑥)
32		addasspr 10778	. . . . . . . . . . 11 ⊢ ((𝑥 +P 𝑤) +P 𝑓) = (𝑥 +P (𝑤 +P 𝑓))
33		vex 3436	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
34	22, 30, 24, 31, 32, 33	caov42 7505	. . . . . . . . . 10 ⊢ ((𝑣 +P 𝑧) +P (𝑢 +P 𝑦)) = ((𝑣 +P 𝑢) +P (𝑦 +P 𝑧))
35	29, 34	eqtri 2766	. . . . . . . . 9 ⊢ ((𝑢 +P 𝑦) +P (𝑣 +P 𝑧)) = ((𝑣 +P 𝑢) +P (𝑦 +P 𝑧))
36	28, 35	breq12i 5083	. . . . . . . 8 ⊢ (((𝑣 +P 𝑥) +P (𝑢 +P 𝑤))<P ((𝑢 +P 𝑦) +P (𝑣 +P 𝑧)) ↔ ((𝑣 +P 𝑢) +P (𝑥 +P 𝑤))<P ((𝑣 +P 𝑢) +P (𝑦 +P 𝑧)))
37	21, 36	bitri 274	. . . . . . 7 ⊢ ([⟨(𝑣 +P 𝑥), (𝑢 +P 𝑦)⟩] ~R <R [⟨(𝑣 +P 𝑧), (𝑢 +P 𝑤)⟩] ~R ↔ ((𝑣 +P 𝑢) +P (𝑥 +P 𝑤))<P ((𝑣 +P 𝑢) +P (𝑦 +P 𝑧)))
38	19, 20, 37	3bitr4g 314	. . . . . 6 ⊢ ((𝑣 +P 𝑢) ∈ P → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ [⟨(𝑣 +P 𝑥), (𝑢 +P 𝑦)⟩] ~R <R [⟨(𝑣 +P 𝑧), (𝑢 +P 𝑤)⟩] ~R ))
39	18, 38	syl 17	. . . . 5 ⊢ (((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ [⟨(𝑣 +P 𝑥), (𝑢 +P 𝑦)⟩] ~R <R [⟨(𝑣 +P 𝑧), (𝑢 +P 𝑤)⟩] ~R ))
40		addsrpr 10831	. . . . . . 7 ⊢ (((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑥 ∈ P ∧ 𝑦 ∈ P)) → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑥, 𝑦⟩] ~R ) = [⟨(𝑣 +P 𝑥), (𝑢 +P 𝑦)⟩] ~R )
41	40	3adant3 1131	. . . . . 6 ⊢ (((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑥, 𝑦⟩] ~R ) = [⟨(𝑣 +P 𝑥), (𝑢 +P 𝑦)⟩] ~R )
42		addsrpr 10831	. . . . . . 7 ⊢ (((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = [⟨(𝑣 +P 𝑧), (𝑢 +P 𝑤)⟩] ~R )
43	42	3adant2 1130	. . . . . 6 ⊢ (((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = [⟨(𝑣 +P 𝑧), (𝑢 +P 𝑤)⟩] ~R )
44	41, 43	breq12d 5087	. . . . 5 ⊢ (((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑥, 𝑦⟩] ~R ) <R ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) ↔ [⟨(𝑣 +P 𝑥), (𝑢 +P 𝑦)⟩] ~R <R [⟨(𝑣 +P 𝑧), (𝑢 +P 𝑤)⟩] ~R ))
45	39, 44	bitr4d 281	. . . 4 ⊢ (((𝑣 ∈ P ∧ 𝑢 ∈ P) ∧ (𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑥, 𝑦⟩] ~R ) <R ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R )))
46	4, 8, 12, 16, 45	3ecoptocl 8598	. . 3 ⊢ ((𝐶 ∈ R ∧ 𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
47	46	3coml 1126	. 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
48	1, 2, 3, 47	ndmovord 7462	1 ⊢ (𝐶 ∈ R → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ⟨cop 4567   class class class wbr 5074  (class class class)co 7275  [cec 8496  Pcnp 10615   +_P cpp 10617  <_P cltp 10619   ~_R cer 10620  Rcnr 10621   +_R cplr 10625   <_R cltr 10627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-omul 8302  df-er 8498  df-ec 8500  df-qs 8504  df-ni 10628  df-pli 10629  df-mi 10630  df-lti 10631  df-plpq 10664  df-mpq 10665  df-ltpq 10666  df-enq 10667  df-nq 10668  df-erq 10669  df-plq 10670  df-mq 10671  df-1nq 10672  df-rq 10673  df-ltnq 10674  df-np 10737  df-plp 10739  df-ltp 10741  df-enr 10811  df-nr 10812  df-plr 10813  df-ltr 10815

This theorem is referenced by:  addgt0sr  10860  sqgt0sr  10862  mappsrpr  10864  ltpsrpr  10865  map2psrpr  10866  supsrlem  10867  axpre-ltadd  10923