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Theorem nnaord 7645
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 7644 . . 3 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
213adant1 1077 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
3 oveq2 6613 . . . . . 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 7644 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵𝐴 → (𝐶 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐴)))
653adant2 1078 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵𝐴 → (𝐶 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐴)))
74, 6orim12d 882 . . . 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 1038 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐶 ∈ ω))
10 ancom 466 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐶 ∈ ω) ↔ (𝐶 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)))
11 anandi 870 . . . . . 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 7637 . . . . . . 7 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 +𝑜 𝐴) ∈ ω)
14 nnord 7021 . . . . . . 7 ((𝐶 +𝑜 𝐴) ∈ ω → Ord (𝐶 +𝑜 𝐴))
1513, 14syl 17 . . . . . 6 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → Ord (𝐶 +𝑜 𝐴))
16 nnacl 7637 . . . . . . 7 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 +𝑜 𝐵) ∈ ω)
17 nnord 7021 . . . . . . 7 ((𝐶 +𝑜 𝐵) ∈ ω → Ord (𝐶 +𝑜 𝐵))
1816, 17syl 17 . . . . . 6 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → Ord (𝐶 +𝑜 𝐵))
1915, 18anim12i 589 . . . . 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 5720 . . . 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 7021 . . . . . 6 (𝐴 ∈ ω → Ord 𝐴)
24 nnord 7021 . . . . . 6 (𝐵 ∈ ω → Ord 𝐵)
2523, 24anim12i 589 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (Ord 𝐴 ∧ Ord 𝐵))
26253adant3 1079 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (Ord 𝐴 ∧ Ord 𝐵))
27 ordtri2 5720 . . . 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 383  wa 384  w3a 1036   = wceq 1480  wcel 1992  Ord word 5684  (class class class)co 6605  ωcom 7013   +𝑜 coa 7503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-oadd 7510
This theorem is referenced by:  nnaordr  7646  nnaword  7653  nnaordex  7664  nnneo  7677  unfilem1  8169  ltapi  9670  1lt2pi  9672
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