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Theorem nnaord 7864
Description: Ordering property of addition. Proposition 8.4 of [TakeutiZaring] p. 58, limited to natural numbers, and its converse. (Contributed by NM, 7-Mar-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nnaord ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 ↔ (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))

Proof of Theorem nnaord
StepHypRef Expression
1 nnaordi 7863 . . 3 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
213adant1 1125 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
3 oveq2 6817 . . . . . 6 (𝐴 = 𝐵 → (𝐶 +𝑜 𝐴) = (𝐶 +𝑜 𝐵))
43a1i 11 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 = 𝐵 → (𝐶 +𝑜 𝐴) = (𝐶 +𝑜 𝐵)))
5 nnaordi 7863 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵𝐴 → (𝐶 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐴)))
653adant2 1126 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵𝐴 → (𝐶 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐴)))
74, 6orim12d 919 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 = 𝐵𝐵𝐴) → ((𝐶 +𝑜 𝐴) = (𝐶 +𝑜 𝐵) ∨ (𝐶 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐴))))
87con3d 148 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (¬ ((𝐶 +𝑜 𝐴) = (𝐶 +𝑜 𝐵) ∨ (𝐶 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐴)) → ¬ (𝐴 = 𝐵𝐵𝐴)))
9 df-3an 1074 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐶 ∈ ω))
10 ancom 465 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐶 ∈ ω) ↔ (𝐶 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)))
11 anandi 906 . . . . . 6 ((𝐶 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) ↔ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐶 ∈ ω ∧ 𝐵 ∈ ω)))
129, 10, 113bitri 286 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ↔ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐶 ∈ ω ∧ 𝐵 ∈ ω)))
13 nnacl 7856 . . . . . . 7 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 +𝑜 𝐴) ∈ ω)
14 nnord 7234 . . . . . . 7 ((𝐶 +𝑜 𝐴) ∈ ω → Ord (𝐶 +𝑜 𝐴))
1513, 14syl 17 . . . . . 6 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → Ord (𝐶 +𝑜 𝐴))
16 nnacl 7856 . . . . . . 7 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 +𝑜 𝐵) ∈ ω)
17 nnord 7234 . . . . . . 7 ((𝐶 +𝑜 𝐵) ∈ ω → Ord (𝐶 +𝑜 𝐵))
1816, 17syl 17 . . . . . 6 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → Ord (𝐶 +𝑜 𝐵))
1915, 18anim12i 591 . . . . 5 (((𝐶 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐶 ∈ ω ∧ 𝐵 ∈ ω)) → (Ord (𝐶 +𝑜 𝐴) ∧ Ord (𝐶 +𝑜 𝐵)))
2012, 19sylbi 207 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (Ord (𝐶 +𝑜 𝐴) ∧ Ord (𝐶 +𝑜 𝐵)))
21 ordtri2 5915 . . . 4 ((Ord (𝐶 +𝑜 𝐴) ∧ Ord (𝐶 +𝑜 𝐵)) → ((𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵) ↔ ¬ ((𝐶 +𝑜 𝐴) = (𝐶 +𝑜 𝐵) ∨ (𝐶 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐴))))
2220, 21syl 17 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵) ↔ ¬ ((𝐶 +𝑜 𝐴) = (𝐶 +𝑜 𝐵) ∨ (𝐶 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐴))))
23 nnord 7234 . . . . . 6 (𝐴 ∈ ω → Ord 𝐴)
24 nnord 7234 . . . . . 6 (𝐵 ∈ ω → Ord 𝐵)
2523, 24anim12i 591 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (Ord 𝐴 ∧ Ord 𝐵))
26253adant3 1127 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (Ord 𝐴 ∧ Ord 𝐵))
27 ordtri2 5915 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
2826, 27syl 17 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
298, 22, 283imtr4d 283 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵) → 𝐴𝐵))
302, 29impbid 202 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 ↔ (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1072   = wceq 1628  wcel 2135  Ord word 5879  (class class class)co 6809  ωcom 7226   +𝑜 coa 7722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-om 7227  df-wrecs 7572  df-recs 7633  df-rdg 7671  df-oadd 7729
This theorem is referenced by:  nnaordr  7865  nnaword  7872  nnaordex  7883  nnneo  7896  unfilem1  8385  ltapi  9913  1lt2pi  9915
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