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Theorem ltapr 9905
Description: Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.)
Assertion
Ref Expression
ltapr (𝐶P → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))

Proof of Theorem ltapr
StepHypRef Expression
1 dmplp 9872 . 2 dom +P = (P × P)
2 ltrelpr 9858 . 2 <P ⊆ (P × P)
3 0npr 9852 . 2 ¬ ∅ ∈ P
4 ltaprlem 9904 . . . . . 6 (𝐶P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
54adantr 480 . . . . 5 ((𝐶P ∧ (𝐵P𝐴P)) → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
6 olc 398 . . . . . . . . 9 ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → ((𝐶 +P 𝐵) = (𝐶 +P 𝐴) ∨ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
7 ltaprlem 9904 . . . . . . . . . . . 12 (𝐶P → (𝐵<P 𝐴 → (𝐶 +P 𝐵)<P (𝐶 +P 𝐴)))
87adantr 480 . . . . . . . . . . 11 ((𝐶P ∧ (𝐵P𝐴P)) → (𝐵<P 𝐴 → (𝐶 +P 𝐵)<P (𝐶 +P 𝐴)))
9 ltsopr 9892 . . . . . . . . . . . . 13 <P Or P
10 sotric 5090 . . . . . . . . . . . . 13 ((<P Or P ∧ (𝐵P𝐴P)) → (𝐵<P 𝐴 ↔ ¬ (𝐵 = 𝐴𝐴<P 𝐵)))
119, 10mpan 706 . . . . . . . . . . . 12 ((𝐵P𝐴P) → (𝐵<P 𝐴 ↔ ¬ (𝐵 = 𝐴𝐴<P 𝐵)))
1211adantl 481 . . . . . . . . . . 11 ((𝐶P ∧ (𝐵P𝐴P)) → (𝐵<P 𝐴 ↔ ¬ (𝐵 = 𝐴𝐴<P 𝐵)))
13 addclpr 9878 . . . . . . . . . . . . 13 ((𝐶P𝐵P) → (𝐶 +P 𝐵) ∈ P)
14 addclpr 9878 . . . . . . . . . . . . 13 ((𝐶P𝐴P) → (𝐶 +P 𝐴) ∈ P)
1513, 14anim12dan 900 . . . . . . . . . . . 12 ((𝐶P ∧ (𝐵P𝐴P)) → ((𝐶 +P 𝐵) ∈ P ∧ (𝐶 +P 𝐴) ∈ P))
16 sotric 5090 . . . . . . . . . . . 12 ((<P Or P ∧ ((𝐶 +P 𝐵) ∈ P ∧ (𝐶 +P 𝐴) ∈ P)) → ((𝐶 +P 𝐵)<P (𝐶 +P 𝐴) ↔ ¬ ((𝐶 +P 𝐵) = (𝐶 +P 𝐴) ∨ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))))
179, 15, 16sylancr 696 . . . . . . . . . . 11 ((𝐶P ∧ (𝐵P𝐴P)) → ((𝐶 +P 𝐵)<P (𝐶 +P 𝐴) ↔ ¬ ((𝐶 +P 𝐵) = (𝐶 +P 𝐴) ∨ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))))
188, 12, 173imtr3d 282 . . . . . . . . . 10 ((𝐶P ∧ (𝐵P𝐴P)) → (¬ (𝐵 = 𝐴𝐴<P 𝐵) → ¬ ((𝐶 +P 𝐵) = (𝐶 +P 𝐴) ∨ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))))
1918con4d 114 . . . . . . . . 9 ((𝐶P ∧ (𝐵P𝐴P)) → (((𝐶 +P 𝐵) = (𝐶 +P 𝐴) ∨ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)) → (𝐵 = 𝐴𝐴<P 𝐵)))
206, 19syl5 34 . . . . . . . 8 ((𝐶P ∧ (𝐵P𝐴P)) → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → (𝐵 = 𝐴𝐴<P 𝐵)))
21 df-or 384 . . . . . . . 8 ((𝐵 = 𝐴𝐴<P 𝐵) ↔ (¬ 𝐵 = 𝐴𝐴<P 𝐵))
2220, 21syl6ib 241 . . . . . . 7 ((𝐶P ∧ (𝐵P𝐴P)) → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → (¬ 𝐵 = 𝐴𝐴<P 𝐵)))
2322com23 86 . . . . . 6 ((𝐶P ∧ (𝐵P𝐴P)) → (¬ 𝐵 = 𝐴 → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → 𝐴<P 𝐵)))
249, 2soirri 5557 . . . . . . . 8 ¬ (𝐶 +P 𝐴)<P (𝐶 +P 𝐴)
25 oveq2 6698 . . . . . . . . 9 (𝐵 = 𝐴 → (𝐶 +P 𝐵) = (𝐶 +P 𝐴))
2625breq2d 4697 . . . . . . . 8 (𝐵 = 𝐴 → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐴)))
2724, 26mtbiri 316 . . . . . . 7 (𝐵 = 𝐴 → ¬ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))
2827pm2.21d 118 . . . . . 6 (𝐵 = 𝐴 → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → 𝐴<P 𝐵))
2923, 28pm2.61d2 172 . . . . 5 ((𝐶P ∧ (𝐵P𝐴P)) → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → 𝐴<P 𝐵))
305, 29impbid 202 . . . 4 ((𝐶P ∧ (𝐵P𝐴P)) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
31303impb 1279 . . 3 ((𝐶P𝐵P𝐴P) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
32313com13 1289 . 2 ((𝐴P𝐵P𝐶P) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
331, 2, 3, 32ndmovord 6866 1 (𝐶P → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030   class class class wbr 4685   Or wor 5063  (class class class)co 6690  Pcnp 9719   +P cpp 9721  <P cltp 9723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ni 9732  df-pli 9733  df-mi 9734  df-lti 9735  df-plpq 9768  df-mpq 9769  df-ltpq 9770  df-enq 9771  df-nq 9772  df-erq 9773  df-plq 9774  df-mq 9775  df-1nq 9776  df-rq 9777  df-ltnq 9778  df-np 9841  df-plp 9843  df-ltp 9845
This theorem is referenced by:  addcanpr  9906  ltsrpr  9936  gt0srpr  9937  ltsosr  9953  ltasr  9959  ltpsrpr  9968  map2psrpr  9969
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