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Theorem ltasrg 7217
Description: Ordering property of addition. (Contributed by NM, 10-May-1996.)
Assertion
Ref Expression
ltasrg ((𝐴R𝐵R𝐶R) → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))

Proof of Theorem ltasrg
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑠 𝑟 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 7174 . . 3 R = ((P × P) / ~R )
2 oveq1 5596 . . . . 5 ([⟨𝑣, 𝑢⟩] ~R = 𝐶 → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑥, 𝑦⟩] ~R ) = (𝐶 +R [⟨𝑥, 𝑦⟩] ~R ))
3 oveq1 5596 . . . . 5 ([⟨𝑣, 𝑢⟩] ~R = 𝐶 → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = (𝐶 +R [⟨𝑧, 𝑤⟩] ~R ))
42, 3breq12d 3824 . . . 4 ([⟨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 )))
54bibi2d 230 . . 3 ([⟨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 ))))
6 breq1 3814 . . . 4 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R𝐴 <R [⟨𝑧, 𝑤⟩] ~R ))
7 oveq2 5597 . . . . 5 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (𝐶 +R [⟨𝑥, 𝑦⟩] ~R ) = (𝐶 +R 𝐴))
87breq1d 3821 . . . 4 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ((𝐶 +R [⟨𝑥, 𝑦⟩] ~R ) <R (𝐶 +R [⟨𝑧, 𝑤⟩] ~R ) ↔ (𝐶 +R 𝐴) <R (𝐶 +R [⟨𝑧, 𝑤⟩] ~R )))
96, 8bibi12d 233 . . 3 ([⟨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 ))))
10 breq2 3815 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 <R [⟨𝑧, 𝑤⟩] ~R𝐴 <R 𝐵))
11 oveq2 5597 . . . . 5 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐶 +R [⟨𝑧, 𝑤⟩] ~R ) = (𝐶 +R 𝐵))
1211breq2d 3823 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((𝐶 +R 𝐴) <R (𝐶 +R [⟨𝑧, 𝑤⟩] ~R ) ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
1310, 12bibi12d 233 . . 3 ([⟨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))))
14 simp2l 965 . . . . . . 7 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → 𝑥P)
15 simp3r 968 . . . . . . 7 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → 𝑤P)
16 addclpr 6997 . . . . . . 7 ((𝑥P𝑤P) → (𝑥 +P 𝑤) ∈ P)
1714, 15, 16syl2anc 403 . . . . . 6 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (𝑥 +P 𝑤) ∈ P)
18 simp2r 966 . . . . . . 7 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → 𝑦P)
19 simp3l 967 . . . . . . 7 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → 𝑧P)
20 addclpr 6997 . . . . . . 7 ((𝑦P𝑧P) → (𝑦 +P 𝑧) ∈ P)
2118, 19, 20syl2anc 403 . . . . . 6 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (𝑦 +P 𝑧) ∈ P)
22 addclpr 6997 . . . . . . 7 ((𝑣P𝑢P) → (𝑣 +P 𝑢) ∈ P)
23223ad2ant1 960 . . . . . 6 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (𝑣 +P 𝑢) ∈ P)
24 ltaprg 7079 . . . . . 6 (((𝑥 +P 𝑤) ∈ P ∧ (𝑦 +P 𝑧) ∈ P ∧ (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))))
2517, 21, 23, 24syl3anc 1170 . . . . 5 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ↔ ((𝑣 +P 𝑢) +P (𝑥 +P 𝑤))<P ((𝑣 +P 𝑢) +P (𝑦 +P 𝑧))))
26 ltsrprg 7194 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧)))
27263adant1 957 . . . . 5 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧)))
28 simp1l 963 . . . . . . . 8 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → 𝑣P)
29 addclpr 6997 . . . . . . . 8 ((𝑣P𝑥P) → (𝑣 +P 𝑥) ∈ P)
3028, 14, 29syl2anc 403 . . . . . . 7 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (𝑣 +P 𝑥) ∈ P)
31 simp1r 964 . . . . . . . 8 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → 𝑢P)
32 addclpr 6997 . . . . . . . 8 ((𝑢P𝑦P) → (𝑢 +P 𝑦) ∈ P)
3331, 18, 32syl2anc 403 . . . . . . 7 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (𝑢 +P 𝑦) ∈ P)
34 addclpr 6997 . . . . . . . 8 ((𝑣P𝑧P) → (𝑣 +P 𝑧) ∈ P)
3528, 19, 34syl2anc 403 . . . . . . 7 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (𝑣 +P 𝑧) ∈ P)
36 addclpr 6997 . . . . . . . 8 ((𝑢P𝑤P) → (𝑢 +P 𝑤) ∈ P)
3731, 15, 36syl2anc 403 . . . . . . 7 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (𝑢 +P 𝑤) ∈ P)
38 ltsrprg 7194 . . . . . . 7 ((((𝑣 +P 𝑥) ∈ P ∧ (𝑢 +P 𝑦) ∈ P) ∧ ((𝑣 +P 𝑧) ∈ P ∧ (𝑢 +P 𝑤) ∈ P)) → ([⟨(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))))
3930, 33, 35, 37, 38syl22anc 1171 . . . . . 6 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨(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))))
40 addcomprg 7038 . . . . . . . . 9 ((𝑟P𝑠P) → (𝑟 +P 𝑠) = (𝑠 +P 𝑟))
4140adantl 271 . . . . . . . 8 ((((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑟P𝑠P)) → (𝑟 +P 𝑠) = (𝑠 +P 𝑟))
42 addassprg 7039 . . . . . . . . 9 ((𝑟P𝑠P𝑡P) → ((𝑟 +P 𝑠) +P 𝑡) = (𝑟 +P (𝑠 +P 𝑡)))
4342adantl 271 . . . . . . . 8 ((((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑟P𝑠P𝑡P)) → ((𝑟 +P 𝑠) +P 𝑡) = (𝑟 +P (𝑠 +P 𝑡)))
44 addclpr 6997 . . . . . . . . 9 ((𝑟P𝑠P) → (𝑟 +P 𝑠) ∈ P)
4544adantl 271 . . . . . . . 8 ((((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) ∧ (𝑟P𝑠P)) → (𝑟 +P 𝑠) ∈ P)
4628, 14, 31, 41, 43, 15, 45caov4d 5762 . . . . . . 7 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑣 +P 𝑥) +P (𝑢 +P 𝑤)) = ((𝑣 +P 𝑢) +P (𝑥 +P 𝑤)))
4741, 33, 35caovcomd 5734 . . . . . . . 8 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑢 +P 𝑦) +P (𝑣 +P 𝑧)) = ((𝑣 +P 𝑧) +P (𝑢 +P 𝑦)))
4828, 19, 31, 41, 43, 18, 45caov42d 5764 . . . . . . . 8 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑣 +P 𝑧) +P (𝑢 +P 𝑦)) = ((𝑣 +P 𝑢) +P (𝑦 +P 𝑧)))
4947, 48eqtrd 2115 . . . . . . 7 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑢 +P 𝑦) +P (𝑣 +P 𝑧)) = ((𝑣 +P 𝑢) +P (𝑦 +P 𝑧)))
5046, 49breq12d 3824 . . . . . 6 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (((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))))
5139, 50bitrd 186 . . . . 5 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨(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))))
5225, 27, 513bitr4d 218 . . . 4 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ [⟨(𝑣 +P 𝑥), (𝑢 +P 𝑦)⟩] ~R <R [⟨(𝑣 +P 𝑧), (𝑢 +P 𝑤)⟩] ~R ))
53 addsrpr 7192 . . . . . 6 (((𝑣P𝑢P) ∧ (𝑥P𝑦P)) → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑥, 𝑦⟩] ~R ) = [⟨(𝑣 +P 𝑥), (𝑢 +P 𝑦)⟩] ~R )
54533adant3 959 . . . . 5 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑥, 𝑦⟩] ~R ) = [⟨(𝑣 +P 𝑥), (𝑢 +P 𝑦)⟩] ~R )
55 addsrpr 7192 . . . . . 6 (((𝑣P𝑢P) ∧ (𝑧P𝑤P)) → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = [⟨(𝑣 +P 𝑧), (𝑢 +P 𝑤)⟩] ~R )
56553adant2 958 . . . . 5 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = [⟨(𝑣 +P 𝑧), (𝑢 +P 𝑤)⟩] ~R )
5754, 56breq12d 3824 . . . 4 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (([⟨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 ))
5852, 57bitr4d 189 . . 3 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑥, 𝑦⟩] ~R ) <R ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R )))
591, 5, 9, 13, 583ecoptocl 6309 . 2 ((𝐶R𝐴R𝐵R) → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
60593coml 1146 1 ((𝐴R𝐵R𝐶R) → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920   = wceq 1285  wcel 1434  cop 3425   class class class wbr 3811  (class class class)co 5589  [cec 6218  Pcnp 6751   +P cpp 6753  <P cltp 6755   ~R cer 6756  Rcnr 6757   +R cplr 6761   <R cltr 6763
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4079  df-id 4083  df-po 4086  df-iso 4087  df-iord 4156  df-on 4158  df-suc 4161  df-iom 4368  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-ov 5592  df-oprab 5593  df-mpt2 5594  df-1st 5844  df-2nd 5845  df-recs 6000  df-irdg 6065  df-1o 6111  df-2o 6112  df-oadd 6115  df-omul 6116  df-er 6220  df-ec 6222  df-qs 6226  df-ni 6764  df-pli 6765  df-mi 6766  df-lti 6767  df-plpq 6804  df-mpq 6805  df-enq 6807  df-nqqs 6808  df-plqqs 6809  df-mqqs 6810  df-1nqqs 6811  df-rq 6812  df-ltnqqs 6813  df-enq0 6884  df-nq0 6885  df-0nq0 6886  df-plq0 6887  df-mq0 6888  df-inp 6926  df-iplp 6928  df-iltp 6930  df-enr 7173  df-nr 7174  df-plr 7175  df-ltr 7177
This theorem is referenced by:  addgt0sr  7222  ltadd1sr  7223  caucvgsrlemoffcau  7244  caucvgsrlemoffgt1  7245  caucvgsrlemoffres  7246  caucvgsr  7248  axpre-ltadd  7322
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