MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ltasr Structured version   Visualization version   GIF version

Theorem ltasr 10511
Description: Ordering property of addition. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
Assertion
Ref Expression
ltasr (𝐶R → (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.
StepHypRef Expression
1 dmaddsr 10496 . 2 dom +R = (R × R)
2 ltrelsr 10479 . 2 <R ⊆ (R × R)
3 0nsr 10490 . 2 ¬ ∅ ∈ R
4 df-nr 10467 . . . 4 R = ((P × P) / ~R )
5 oveq1 7142 . . . . . 6 ([⟨𝑣, 𝑢⟩] ~R = 𝐶 → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑥, 𝑦⟩] ~R ) = (𝐶 +R [⟨𝑥, 𝑦⟩] ~R ))
6 oveq1 7142 . . . . . 6 ([⟨𝑣, 𝑢⟩] ~R = 𝐶 → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = (𝐶 +R [⟨𝑧, 𝑤⟩] ~R ))
75, 6breq12d 5043 . . . . 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 )))
87bibi2d 346 . . . 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 5033 . . . . 5 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R𝐴 <R [⟨𝑧, 𝑤⟩] ~R ))
10 oveq2 7143 . . . . . 6 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (𝐶 +R [⟨𝑥, 𝑦⟩] ~R ) = (𝐶 +R 𝐴))
1110breq1d 5040 . . . . 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 )))
129, 11bibi12d 349 . . . 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 5034 . . . . 5 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 <R [⟨𝑧, 𝑤⟩] ~R𝐴 <R 𝐵))
14 oveq2 7143 . . . . . 6 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐶 +R [⟨𝑧, 𝑤⟩] ~R ) = (𝐶 +R 𝐵))
1514breq2d 5042 . . . . 5 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((𝐶 +R 𝐴) <R (𝐶 +R [⟨𝑧, 𝑤⟩] ~R ) ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
1613, 15bibi12d 349 . . . 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 10429 . . . . . . 7 ((𝑣P𝑢P) → (𝑣 +P 𝑢) ∈ P)
18173ad2ant1 1130 . . . . . 6 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (𝑣 +P 𝑢) ∈ P)
19 ltapr 10456 . . . . . . 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 10488 . . . . . . 7 ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧))
21 ltsrpr 10488 . . . . . . . 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 3444 . . . . . . . . . 10 𝑣 ∈ V
23 vex 3444 . . . . . . . . . 10 𝑥 ∈ V
24 vex 3444 . . . . . . . . . 10 𝑢 ∈ V
25 addcompr 10432 . . . . . . . . . 10 (𝑦 +P 𝑧) = (𝑧 +P 𝑦)
26 addasspr 10433 . . . . . . . . . 10 ((𝑦 +P 𝑧) +P 𝑓) = (𝑦 +P (𝑧 +P 𝑓))
27 vex 3444 . . . . . . . . . 10 𝑤 ∈ V
2822, 23, 24, 25, 26, 27caov4 7359 . . . . . . . . 9 ((𝑣 +P 𝑥) +P (𝑢 +P 𝑤)) = ((𝑣 +P 𝑢) +P (𝑥 +P 𝑤))
29 addcompr 10432 . . . . . . . . . 10 ((𝑢 +P 𝑦) +P (𝑣 +P 𝑧)) = ((𝑣 +P 𝑧) +P (𝑢 +P 𝑦))
30 vex 3444 . . . . . . . . . . 11 𝑧 ∈ V
31 addcompr 10432 . . . . . . . . . . 11 (𝑥 +P 𝑤) = (𝑤 +P 𝑥)
32 addasspr 10433 . . . . . . . . . . 11 ((𝑥 +P 𝑤) +P 𝑓) = (𝑥 +P (𝑤 +P 𝑓))
33 vex 3444 . . . . . . . . . . 11 𝑦 ∈ V
3422, 30, 24, 31, 32, 33caov42 7361 . . . . . . . . . 10 ((𝑣 +P 𝑧) +P (𝑢 +P 𝑦)) = ((𝑣 +P 𝑢) +P (𝑦 +P 𝑧))
3529, 34eqtri 2821 . . . . . . . . 9 ((𝑢 +P 𝑦) +P (𝑣 +P 𝑧)) = ((𝑣 +P 𝑢) +P (𝑦 +P 𝑧))
3628, 35breq12i 5039 . . . . . . . 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)))
3721, 36bitri 278 . . . . . . 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)))
3819, 20, 373bitr4g 317 . . . . . 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 ))
3918, 38syl 17 . . . . 5 (((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 ))
40 addsrpr 10486 . . . . . . 7 (((𝑣P𝑢P) ∧ (𝑥P𝑦P)) → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑥, 𝑦⟩] ~R ) = [⟨(𝑣 +P 𝑥), (𝑢 +P 𝑦)⟩] ~R )
41403adant3 1129 . . . . . 6 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑥, 𝑦⟩] ~R ) = [⟨(𝑣 +P 𝑥), (𝑢 +P 𝑦)⟩] ~R )
42 addsrpr 10486 . . . . . . 7 (((𝑣P𝑢P) ∧ (𝑧P𝑤P)) → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = [⟨(𝑣 +P 𝑧), (𝑢 +P 𝑤)⟩] ~R )
43423adant2 1128 . . . . . 6 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = [⟨(𝑣 +P 𝑧), (𝑢 +P 𝑤)⟩] ~R )
4441, 43breq12d 5043 . . . . 5 (((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 ))
4539, 44bitr4d 285 . . . 4 (((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 )))
464, 8, 12, 16, 453ecoptocl 8372 . . 3 ((𝐶R𝐴R𝐵R) → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
47463coml 1124 . 2 ((𝐴R𝐵R𝐶R) → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
481, 2, 3, 47ndmovord 7318 1 (𝐶R → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  cop 4531   class class class wbr 5030  (class class class)co 7135  [cec 8270  Pcnp 10270   +P cpp 10272  <P cltp 10274   ~R cer 10275  Rcnr 10276   +R cplr 10280   <R cltr 10282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-omul 8090  df-er 8272  df-ec 8274  df-qs 8278  df-ni 10283  df-pli 10284  df-mi 10285  df-lti 10286  df-plpq 10319  df-mpq 10320  df-ltpq 10321  df-enq 10322  df-nq 10323  df-erq 10324  df-plq 10325  df-mq 10326  df-1nq 10327  df-rq 10328  df-ltnq 10329  df-np 10392  df-plp 10394  df-ltp 10396  df-enr 10466  df-nr 10467  df-plr 10468  df-ltr 10470
This theorem is referenced by:  addgt0sr  10515  sqgt0sr  10517  mappsrpr  10519  ltpsrpr  10520  map2psrpr  10521  supsrlem  10522  axpre-ltadd  10578
  Copyright terms: Public domain W3C validator