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Theorem nnaord 6110
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 6109 . . 3 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
213adant1 931 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
3 oveq2 5545 . . . . . 6 (𝐴 = 𝐵 → (𝐶 +𝑜 𝐴) = (𝐶 +𝑜 𝐵))
43a1i 9 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 = 𝐵 → (𝐶 +𝑜 𝐴) = (𝐶 +𝑜 𝐵)))
5 nnaordi 6109 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵𝐴 → (𝐶 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐴)))
653adant2 932 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵𝐴 → (𝐶 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐴)))
74, 6orim12d 708 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 = 𝐵𝐵𝐴) → ((𝐶 +𝑜 𝐴) = (𝐶 +𝑜 𝐵) ∨ (𝐶 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐴))))
87con3d 569 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (¬ ((𝐶 +𝑜 𝐴) = (𝐶 +𝑜 𝐵) ∨ (𝐶 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐴)) → ¬ (𝐴 = 𝐵𝐵𝐴)))
9 df-3an 896 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐶 ∈ ω))
10 ancom 257 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐶 ∈ ω) ↔ (𝐶 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)))
11 anandi 532 . . . . . 6 ((𝐶 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) ↔ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐶 ∈ ω ∧ 𝐵 ∈ ω)))
129, 10, 113bitri 199 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ↔ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐶 ∈ ω ∧ 𝐵 ∈ ω)))
13 nnacl 6087 . . . . . 6 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 +𝑜 𝐴) ∈ ω)
14 nnacl 6087 . . . . . 6 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 +𝑜 𝐵) ∈ ω)
1513, 14anim12i 325 . . . . 5 (((𝐶 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐶 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐶 +𝑜 𝐴) ∈ ω ∧ (𝐶 +𝑜 𝐵) ∈ ω))
1612, 15sylbi 118 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 +𝑜 𝐴) ∈ ω ∧ (𝐶 +𝑜 𝐵) ∈ ω))
17 nntri2 6101 . . . 4 (((𝐶 +𝑜 𝐴) ∈ ω ∧ (𝐶 +𝑜 𝐵) ∈ ω) → ((𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵) ↔ ¬ ((𝐶 +𝑜 𝐴) = (𝐶 +𝑜 𝐵) ∨ (𝐶 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐴))))
1816, 17syl 14 . . 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))))
19 nntri2 6101 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
20193adant3 933 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
218, 18, 203imtr4d 196 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵) → 𝐴𝐵))
222, 21impbid 124 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 ↔ (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wo 637  w3a 894   = wceq 1257  wcel 1407  ωcom 4338  (class class class)co 5537   +𝑜 coa 6026
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-coll 3897  ax-sep 3900  ax-nul 3908  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-setind 4287  ax-iinf 4336
This theorem depends on definitions:  df-bi 114  df-3or 895  df-3an 896  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-ral 2326  df-rex 2327  df-reu 2328  df-rab 2330  df-v 2574  df-sbc 2785  df-csb 2878  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-nul 3250  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-int 3641  df-iun 3684  df-br 3790  df-opab 3844  df-mpt 3845  df-tr 3880  df-id 4055  df-iord 4128  df-on 4130  df-suc 4133  df-iom 4339  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934  df-fv 4935  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 5985  df-oadd 6033
This theorem is referenced by:  nnaordr  6111  nnaordex  6128  ltapig  6464  1lt2pi  6466
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