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

Theorem ltasr 11036
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 11021 . 2 dom +R = (R × R)
2 ltrelsr 11004 . 2 <R ⊆ (R × R)
3 0nsr 11015 . 2 ¬ ∅ ∈ R
4 df-nr 10992 . . . 4 R = ((P × P) / ~R )
5 oveq1 7364 . . . . . 6 ([⟨𝑣, 𝑢⟩] ~R = 𝐶 → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑥, 𝑦⟩] ~R ) = (𝐶 +R [⟨𝑥, 𝑦⟩] ~R ))
6 oveq1 7364 . . . . . 6 ([⟨𝑣, 𝑢⟩] ~R = 𝐶 → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = (𝐶 +R [⟨𝑧, 𝑤⟩] ~R ))
75, 6breq12d 5118 . . . . 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 342 . . . 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 5108 . . . . 5 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R𝐴 <R [⟨𝑧, 𝑤⟩] ~R ))
10 oveq2 7365 . . . . . 6 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (𝐶 +R [⟨𝑥, 𝑦⟩] ~R ) = (𝐶 +R 𝐴))
1110breq1d 5115 . . . . 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 345 . . . 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 5109 . . . . 5 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 <R [⟨𝑧, 𝑤⟩] ~R𝐴 <R 𝐵))
14 oveq2 7365 . . . . . 6 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐶 +R [⟨𝑧, 𝑤⟩] ~R ) = (𝐶 +R 𝐵))
1514breq2d 5117 . . . . 5 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((𝐶 +R 𝐴) <R (𝐶 +R [⟨𝑧, 𝑤⟩] ~R ) ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
1613, 15bibi12d 345 . . . 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 10954 . . . . . . 7 ((𝑣P𝑢P) → (𝑣 +P 𝑢) ∈ P)
18173ad2ant1 1133 . . . . . 6 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (𝑣 +P 𝑢) ∈ P)
19 ltapr 10981 . . . . . . 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 11013 . . . . . . 7 ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧))
21 ltsrpr 11013 . . . . . . . 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 3449 . . . . . . . . . 10 𝑣 ∈ V
23 vex 3449 . . . . . . . . . 10 𝑥 ∈ V
24 vex 3449 . . . . . . . . . 10 𝑢 ∈ V
25 addcompr 10957 . . . . . . . . . 10 (𝑦 +P 𝑧) = (𝑧 +P 𝑦)
26 addasspr 10958 . . . . . . . . . 10 ((𝑦 +P 𝑧) +P 𝑓) = (𝑦 +P (𝑧 +P 𝑓))
27 vex 3449 . . . . . . . . . 10 𝑤 ∈ V
2822, 23, 24, 25, 26, 27caov4 7585 . . . . . . . . 9 ((𝑣 +P 𝑥) +P (𝑢 +P 𝑤)) = ((𝑣 +P 𝑢) +P (𝑥 +P 𝑤))
29 addcompr 10957 . . . . . . . . . 10 ((𝑢 +P 𝑦) +P (𝑣 +P 𝑧)) = ((𝑣 +P 𝑧) +P (𝑢 +P 𝑦))
30 vex 3449 . . . . . . . . . . 11 𝑧 ∈ V
31 addcompr 10957 . . . . . . . . . . 11 (𝑥 +P 𝑤) = (𝑤 +P 𝑥)
32 addasspr 10958 . . . . . . . . . . 11 ((𝑥 +P 𝑤) +P 𝑓) = (𝑥 +P (𝑤 +P 𝑓))
33 vex 3449 . . . . . . . . . . 11 𝑦 ∈ V
3422, 30, 24, 31, 32, 33caov42 7587 . . . . . . . . . 10 ((𝑣 +P 𝑧) +P (𝑢 +P 𝑦)) = ((𝑣 +P 𝑢) +P (𝑦 +P 𝑧))
3529, 34eqtri 2764 . . . . . . . . 9 ((𝑢 +P 𝑦) +P (𝑣 +P 𝑧)) = ((𝑣 +P 𝑢) +P (𝑦 +P 𝑧))
3628, 35breq12i 5114 . . . . . . . 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 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)))
3819, 20, 373bitr4g 313 . . . . . 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 11011 . . . . . . 7 (((𝑣P𝑢P) ∧ (𝑥P𝑦P)) → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑥, 𝑦⟩] ~R ) = [⟨(𝑣 +P 𝑥), (𝑢 +P 𝑦)⟩] ~R )
41403adant3 1132 . . . . . 6 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑥, 𝑦⟩] ~R ) = [⟨(𝑣 +P 𝑥), (𝑢 +P 𝑦)⟩] ~R )
42 addsrpr 11011 . . . . . . 7 (((𝑣P𝑢P) ∧ (𝑧P𝑤P)) → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = [⟨(𝑣 +P 𝑧), (𝑢 +P 𝑤)⟩] ~R )
43423adant2 1131 . . . . . 6 (((𝑣P𝑢P) ∧ (𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑣, 𝑢⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = [⟨(𝑣 +P 𝑧), (𝑢 +P 𝑤)⟩] ~R )
4441, 43breq12d 5118 . . . . 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 281 . . . 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 8748 . . 3 ((𝐶R𝐴R𝐵R) → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
47463coml 1127 . 2 ((𝐴R𝐵R𝐶R) → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
481, 2, 3, 47ndmovord 7544 1 (𝐶R → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  cop 4592   class class class wbr 5105  (class class class)co 7357  [cec 8646  Pcnp 10795   +P cpp 10797  <P cltp 10799   ~R cer 10800  Rcnr 10801   +R cplr 10805   <R cltr 10807
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-omul 8417  df-er 8648  df-ec 8650  df-qs 8654  df-ni 10808  df-pli 10809  df-mi 10810  df-lti 10811  df-plpq 10844  df-mpq 10845  df-ltpq 10846  df-enq 10847  df-nq 10848  df-erq 10849  df-plq 10850  df-mq 10851  df-1nq 10852  df-rq 10853  df-ltnq 10854  df-np 10917  df-plp 10919  df-ltp 10921  df-enr 10991  df-nr 10992  df-plr 10993  df-ltr 10995
This theorem is referenced by:  addgt0sr  11040  sqgt0sr  11042  mappsrpr  11044  ltpsrpr  11045  map2psrpr  11046  supsrlem  11047  axpre-ltadd  11103
  Copyright terms: Public domain W3C validator