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Theorem nnaord 6141
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 6140 . . 3 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
213adant1 957 . 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 6140 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵𝐴 → (𝐶 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐴)))
653adant2 958 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵𝐴 → (𝐶 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐴)))
74, 6orim12d 733 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 = 𝐵𝐵𝐴) → ((𝐶 +𝑜 𝐴) = (𝐶 +𝑜 𝐵) ∨ (𝐶 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐴))))
87con3d 594 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (¬ ((𝐶 +𝑜 𝐴) = (𝐶 +𝑜 𝐵) ∨ (𝐶 +𝑜 𝐵) ∈ (𝐶 +𝑜 𝐴)) → ¬ (𝐴 = 𝐵𝐵𝐴)))
9 df-3an 922 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐶 ∈ ω))
10 ancom 262 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐶 ∈ ω) ↔ (𝐶 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)))
11 anandi 555 . . . . . 6 ((𝐶 ∈ ω ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) ↔ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐶 ∈ ω ∧ 𝐵 ∈ ω)))
129, 10, 113bitri 204 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ↔ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐶 ∈ ω ∧ 𝐵 ∈ ω)))
13 nnacl 6117 . . . . . 6 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 +𝑜 𝐴) ∈ ω)
14 nnacl 6117 . . . . . 6 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 +𝑜 𝐵) ∈ ω)
1513, 14anim12i 331 . . . . 5 (((𝐶 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐶 ∈ ω ∧ 𝐵 ∈ ω)) → ((𝐶 +𝑜 𝐴) ∈ ω ∧ (𝐶 +𝑜 𝐵) ∈ ω))
1612, 15sylbi 119 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 +𝑜 𝐴) ∈ ω ∧ (𝐶 +𝑜 𝐵) ∈ ω))
17 nntri2 6131 . . . 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 6131 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
20193adant3 959 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
218, 18, 203imtr4d 201 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵) → 𝐴𝐵))
222, 21impbid 127 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 102  wb 103  wo 662  w3a 920   = wceq 1285  wcel 1434  ωcom 4333  (class class class)co 5537   +𝑜 coa 6056
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-id 4050  df-iord 4123  df-on 4125  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 6013  df-oadd 6063
This theorem is referenced by:  nnaordr  6142  nnaordex  6159  ltapig  6579  1lt2pi  6581
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