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Theorem nnaordi 6197
Description: Ordering property of addition. Proposition 8.4 of [TakeutiZaring] p. 58, limited to natural numbers. (Contributed by NM, 3-Feb-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nnaordi ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))

Proof of Theorem nnaordi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5599 . . . . . . . . 9 (𝑥 = 𝐶 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝐶))
2 oveq2 5599 . . . . . . . . 9 (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
31, 2eleq12d 2153 . . . . . . . 8 (𝑥 = 𝐶 → ((𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝐶) ∈ (𝐵 +𝑜 𝐶)))
43imbi2d 228 . . . . . . 7 (𝑥 = 𝐶 → (((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥)) ↔ ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +𝑜 𝐶) ∈ (𝐵 +𝑜 𝐶))))
5 oveq2 5599 . . . . . . . . 9 (𝑥 = ∅ → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 ∅))
6 oveq2 5599 . . . . . . . . 9 (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
75, 6eleq12d 2153 . . . . . . . 8 (𝑥 = ∅ → ((𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 ∅) ∈ (𝐵 +𝑜 ∅)))
8 oveq2 5599 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦))
9 oveq2 5599 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
108, 9eleq12d 2153 . . . . . . . 8 (𝑥 = 𝑦 → ((𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑦)))
11 oveq2 5599 . . . . . . . . 9 (𝑥 = suc 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 suc 𝑦))
12 oveq2 5599 . . . . . . . . 9 (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
1311, 12eleq12d 2153 . . . . . . . 8 (𝑥 = suc 𝑦 → ((𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 suc 𝑦) ∈ (𝐵 +𝑜 suc 𝑦)))
14 simpr 108 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐴𝐵)
15 elnn 4383 . . . . . . . . . . 11 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)
1615ancoms 264 . . . . . . . . . 10 ((𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐴 ∈ ω)
17 nna0 6167 . . . . . . . . . 10 (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
1816, 17syl 14 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +𝑜 ∅) = 𝐴)
19 nna0 6167 . . . . . . . . . 10 (𝐵 ∈ ω → (𝐵 +𝑜 ∅) = 𝐵)
2019adantr 270 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐵 +𝑜 ∅) = 𝐵)
2114, 18, 203eltr4d 2166 . . . . . . . 8 ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +𝑜 ∅) ∈ (𝐵 +𝑜 ∅))
22 simprl 498 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐵 ∈ ω)
23 simpl 107 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑦 ∈ ω)
24 nnacl 6173 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 𝑦) ∈ ω)
2522, 23, 24syl2anc 403 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐵 +𝑜 𝑦) ∈ ω)
26 nnsucelsuc 6184 . . . . . . . . . . . 12 ((𝐵 +𝑜 𝑦) ∈ ω → ((𝐴 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑦) ↔ suc (𝐴 +𝑜 𝑦) ∈ suc (𝐵 +𝑜 𝑦)))
2725, 26syl 14 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → ((𝐴 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑦) ↔ suc (𝐴 +𝑜 𝑦) ∈ suc (𝐵 +𝑜 𝑦)))
2816adantl 271 . . . . . . . . . . . . . 14 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐴 ∈ ω)
29 nnon 4387 . . . . . . . . . . . . . 14 (𝐴 ∈ ω → 𝐴 ∈ On)
3028, 29syl 14 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐴 ∈ On)
31 nnon 4387 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → 𝑦 ∈ On)
3231adantr 270 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑦 ∈ On)
33 oasuc 6157 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
3430, 32, 33syl2anc 403 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
35 nnon 4387 . . . . . . . . . . . . . 14 (𝐵 ∈ ω → 𝐵 ∈ On)
3635ad2antrl 474 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐵 ∈ On)
37 oasuc 6157 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
3836, 32, 37syl2anc 403 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
3934, 38eleq12d 2153 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → ((𝐴 +𝑜 suc 𝑦) ∈ (𝐵 +𝑜 suc 𝑦) ↔ suc (𝐴 +𝑜 𝑦) ∈ suc (𝐵 +𝑜 𝑦)))
4027, 39bitr4d 189 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → ((𝐴 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑦) ↔ (𝐴 +𝑜 suc 𝑦) ∈ (𝐵 +𝑜 suc 𝑦)))
4140biimpd 142 . . . . . . . . 9 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → ((𝐴 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 suc 𝑦) ∈ (𝐵 +𝑜 suc 𝑦)))
4241ex 113 . . . . . . . 8 (𝑦 ∈ ω → ((𝐵 ∈ ω ∧ 𝐴𝐵) → ((𝐴 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 suc 𝑦) ∈ (𝐵 +𝑜 suc 𝑦))))
437, 10, 13, 21, 42finds2 4379 . . . . . . 7 (𝑥 ∈ ω → ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥)))
444, 43vtoclga 2675 . . . . . 6 (𝐶 ∈ ω → ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +𝑜 𝐶) ∈ (𝐵 +𝑜 𝐶)))
4544imp 122 . . . . 5 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐴 +𝑜 𝐶) ∈ (𝐵 +𝑜 𝐶))
4616adantl 271 . . . . . 6 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐴 ∈ ω)
47 simpl 107 . . . . . 6 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐶 ∈ ω)
48 nnacom 6177 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 +𝑜 𝐶) = (𝐶 +𝑜 𝐴))
4946, 47, 48syl2anc 403 . . . . 5 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐴 +𝑜 𝐶) = (𝐶 +𝑜 𝐴))
50 nnacom 6177 . . . . . . 7 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵 +𝑜 𝐶) = (𝐶 +𝑜 𝐵))
5150ancoms 264 . . . . . 6 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +𝑜 𝐶) = (𝐶 +𝑜 𝐵))
5251adantrr 463 . . . . 5 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐵 +𝑜 𝐶) = (𝐶 +𝑜 𝐵))
5345, 49, 523eltr3d 2165 . . . 4 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
54533impb 1135 . . 3 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
55543com12 1143 . 2 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐴𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
56553expia 1141 1 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  c0 3269  Oncon0 4154  suc csuc 4156  ωcom 4368  (class class class)co 5591   +𝑜 coa 6110
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 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-oadd 6117
This theorem is referenced by:  nnaord  6198  nnmordi  6205  addclpi  6789  addnidpig  6798  archnqq  6879  prarloclemarch2  6881  prarloclemlt  6955
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