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Theorem ltapr 9723
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 9690 . 2 dom +P = (P × P)
2 ltrelpr 9676 . 2 <P ⊆ (P × P)
3 0npr 9670 . 2 ¬ ∅ ∈ P
4 ltaprlem 9722 . . . . . 6 (𝐶P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
54adantr 479 . . . . 5 ((𝐶P ∧ (𝐵P𝐴P)) → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
6 olc 397 . . . . . . . . 9 ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → ((𝐶 +P 𝐵) = (𝐶 +P 𝐴) ∨ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
7 ltaprlem 9722 . . . . . . . . . . . 12 (𝐶P → (𝐵<P 𝐴 → (𝐶 +P 𝐵)<P (𝐶 +P 𝐴)))
87adantr 479 . . . . . . . . . . 11 ((𝐶P ∧ (𝐵P𝐴P)) → (𝐵<P 𝐴 → (𝐶 +P 𝐵)<P (𝐶 +P 𝐴)))
9 ltsopr 9710 . . . . . . . . . . . . 13 <P Or P
10 sotric 4975 . . . . . . . . . . . . 13 ((<P Or P ∧ (𝐵P𝐴P)) → (𝐵<P 𝐴 ↔ ¬ (𝐵 = 𝐴𝐴<P 𝐵)))
119, 10mpan 701 . . . . . . . . . . . 12 ((𝐵P𝐴P) → (𝐵<P 𝐴 ↔ ¬ (𝐵 = 𝐴𝐴<P 𝐵)))
1211adantl 480 . . . . . . . . . . 11 ((𝐶P ∧ (𝐵P𝐴P)) → (𝐵<P 𝐴 ↔ ¬ (𝐵 = 𝐴𝐴<P 𝐵)))
13 addclpr 9696 . . . . . . . . . . . . 13 ((𝐶P𝐵P) → (𝐶 +P 𝐵) ∈ P)
14 addclpr 9696 . . . . . . . . . . . . 13 ((𝐶P𝐴P) → (𝐶 +P 𝐴) ∈ P)
1513, 14anim12dan 877 . . . . . . . . . . . 12 ((𝐶P ∧ (𝐵P𝐴P)) → ((𝐶 +P 𝐵) ∈ P ∧ (𝐶 +P 𝐴) ∈ P))
16 sotric 4975 . . . . . . . . . . . 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 693 . . . . . . . . . . 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 280 . . . . . . . . . 10 ((𝐶P ∧ (𝐵P𝐴P)) → (¬ (𝐵 = 𝐴𝐴<P 𝐵) → ¬ ((𝐶 +P 𝐵) = (𝐶 +P 𝐴) ∨ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))))
1918con4d 112 . . . . . . . . 9 ((𝐶P ∧ (𝐵P𝐴P)) → (((𝐶 +P 𝐵) = (𝐶 +P 𝐴) ∨ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)) → (𝐵 = 𝐴𝐴<P 𝐵)))
206, 19syl5 33 . . . . . . . 8 ((𝐶P ∧ (𝐵P𝐴P)) → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → (𝐵 = 𝐴𝐴<P 𝐵)))
21 df-or 383 . . . . . . . 8 ((𝐵 = 𝐴𝐴<P 𝐵) ↔ (¬ 𝐵 = 𝐴𝐴<P 𝐵))
2220, 21syl6ib 239 . . . . . . 7 ((𝐶P ∧ (𝐵P𝐴P)) → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → (¬ 𝐵 = 𝐴𝐴<P 𝐵)))
2322com23 83 . . . . . 6 ((𝐶P ∧ (𝐵P𝐴P)) → (¬ 𝐵 = 𝐴 → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → 𝐴<P 𝐵)))
249, 2soirri 5428 . . . . . . . 8 ¬ (𝐶 +P 𝐴)<P (𝐶 +P 𝐴)
25 oveq2 6535 . . . . . . . . 9 (𝐵 = 𝐴 → (𝐶 +P 𝐵) = (𝐶 +P 𝐴))
2625breq2d 4589 . . . . . . . 8 (𝐵 = 𝐴 → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐴)))
2724, 26mtbiri 315 . . . . . . 7 (𝐵 = 𝐴 → ¬ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))
2827pm2.21d 116 . . . . . 6 (𝐵 = 𝐴 → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → 𝐴<P 𝐵))
2923, 28pm2.61d2 170 . . . . 5 ((𝐶P ∧ (𝐵P𝐴P)) → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → 𝐴<P 𝐵))
305, 29impbid 200 . . . 4 ((𝐶P ∧ (𝐵P𝐴P)) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
31303impb 1251 . . 3 ((𝐶P𝐵P𝐴P) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
32313com13 1261 . 2 ((𝐴P𝐵P𝐶P) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
331, 2, 3, 32ndmovord 6699 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 194  wo 381  wa 382   = wceq 1474  wcel 1976   class class class wbr 4577   Or wor 4948  (class class class)co 6527  Pcnp 9537   +P cpp 9539  <P cltp 9541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-omul 7429  df-er 7606  df-ni 9550  df-pli 9551  df-mi 9552  df-lti 9553  df-plpq 9586  df-mpq 9587  df-ltpq 9588  df-enq 9589  df-nq 9590  df-erq 9591  df-plq 9592  df-mq 9593  df-1nq 9594  df-rq 9595  df-ltnq 9596  df-np 9659  df-plp 9661  df-ltp 9663
This theorem is referenced by:  addcanpr  9724  ltsrpr  9754  gt0srpr  9755  ltsosr  9771  ltasr  9777  ltpsrpr  9786  map2psrpr  9787
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