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Theorem ltapr 10460
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 10427 . 2 dom +P = (P × P)
2 ltrelpr 10413 . 2 <P ⊆ (P × P)
3 0npr 10407 . 2 ¬ ∅ ∈ P
4 ltaprlem 10459 . . . . . 6 (𝐶P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
54adantr 484 . . . . 5 ((𝐶P ∧ (𝐵P𝐴P)) → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
6 olc 865 . . . . . . . . 9 ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → ((𝐶 +P 𝐵) = (𝐶 +P 𝐴) ∨ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
7 ltaprlem 10459 . . . . . . . . . . . 12 (𝐶P → (𝐵<P 𝐴 → (𝐶 +P 𝐵)<P (𝐶 +P 𝐴)))
87adantr 484 . . . . . . . . . . 11 ((𝐶P ∧ (𝐵P𝐴P)) → (𝐵<P 𝐴 → (𝐶 +P 𝐵)<P (𝐶 +P 𝐴)))
9 ltsopr 10447 . . . . . . . . . . . . 13 <P Or P
10 sotric 5469 . . . . . . . . . . . . 13 ((<P Or P ∧ (𝐵P𝐴P)) → (𝐵<P 𝐴 ↔ ¬ (𝐵 = 𝐴𝐴<P 𝐵)))
119, 10mpan 689 . . . . . . . . . . . 12 ((𝐵P𝐴P) → (𝐵<P 𝐴 ↔ ¬ (𝐵 = 𝐴𝐴<P 𝐵)))
1211adantl 485 . . . . . . . . . . 11 ((𝐶P ∧ (𝐵P𝐴P)) → (𝐵<P 𝐴 ↔ ¬ (𝐵 = 𝐴𝐴<P 𝐵)))
13 addclpr 10433 . . . . . . . . . . . . 13 ((𝐶P𝐵P) → (𝐶 +P 𝐵) ∈ P)
14 addclpr 10433 . . . . . . . . . . . . 13 ((𝐶P𝐴P) → (𝐶 +P 𝐴) ∈ P)
1513, 14anim12dan 621 . . . . . . . . . . . 12 ((𝐶P ∧ (𝐵P𝐴P)) → ((𝐶 +P 𝐵) ∈ P ∧ (𝐶 +P 𝐴) ∈ P))
16 sotric 5469 . . . . . . . . . . . 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 590 . . . . . . . . . . 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 296 . . . . . . . . . 10 ((𝐶P ∧ (𝐵P𝐴P)) → (¬ (𝐵 = 𝐴𝐴<P 𝐵) → ¬ ((𝐶 +P 𝐵) = (𝐶 +P 𝐴) ∨ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))))
1918con4d 115 . . . . . . . . 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 845 . . . . . . . 8 ((𝐵 = 𝐴𝐴<P 𝐵) ↔ (¬ 𝐵 = 𝐴𝐴<P 𝐵))
2220, 21syl6ib 254 . . . . . . 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 5957 . . . . . . . 8 ¬ (𝐶 +P 𝐴)<P (𝐶 +P 𝐴)
25 oveq2 7147 . . . . . . . . 9 (𝐵 = 𝐴 → (𝐶 +P 𝐵) = (𝐶 +P 𝐴))
2625breq2d 5045 . . . . . . . 8 (𝐵 = 𝐴 → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐴)))
2724, 26mtbiri 330 . . . . . . 7 (𝐵 = 𝐴 → ¬ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))
2827pm2.21d 121 . . . . . 6 (𝐵 = 𝐴 → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → 𝐴<P 𝐵))
2923, 28pm2.61d2 184 . . . . 5 ((𝐶P ∧ (𝐵P𝐴P)) → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → 𝐴<P 𝐵))
305, 29impbid 215 . . . 4 ((𝐶P ∧ (𝐵P𝐴P)) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
31303impb 1112 . . 3 ((𝐶P𝐵P𝐴P) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
32313com13 1121 . 2 ((𝐴P𝐵P𝐶P) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
331, 2, 3, 32ndmovord 7322 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 209  wa 399  wo 844   = wceq 1538  wcel 2112   class class class wbr 5033   Or wor 5441  (class class class)co 7139  Pcnp 10274   +P cpp 10276  <P cltp 10278
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-omul 8094  df-er 8276  df-ni 10287  df-pli 10288  df-mi 10289  df-lti 10290  df-plpq 10323  df-mpq 10324  df-ltpq 10325  df-enq 10326  df-nq 10327  df-erq 10328  df-plq 10329  df-mq 10330  df-1nq 10331  df-rq 10332  df-ltnq 10333  df-np 10396  df-plp 10398  df-ltp 10400
This theorem is referenced by:  addcanpr  10461  ltsrpr  10492  gt0srpr  10493  ltsosr  10509  ltasr  10515  ltpsrpr  10524  map2psrpr  10525
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