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Theorem nnaordi 6111
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 5547 . . . . . . . . 9 (𝑥 = 𝐶 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝐶))
2 oveq2 5547 . . . . . . . . 9 (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
31, 2eleq12d 2124 . . . . . . . 8 (𝑥 = 𝐶 → ((𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝐶) ∈ (𝐵 +𝑜 𝐶)))
43imbi2d 223 . . . . . . 7 (𝑥 = 𝐶 → (((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥)) ↔ ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +𝑜 𝐶) ∈ (𝐵 +𝑜 𝐶))))
5 oveq2 5547 . . . . . . . . 9 (𝑥 = ∅ → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 ∅))
6 oveq2 5547 . . . . . . . . 9 (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
75, 6eleq12d 2124 . . . . . . . 8 (𝑥 = ∅ → ((𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 ∅) ∈ (𝐵 +𝑜 ∅)))
8 oveq2 5547 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦))
9 oveq2 5547 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
108, 9eleq12d 2124 . . . . . . . 8 (𝑥 = 𝑦 → ((𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑦)))
11 oveq2 5547 . . . . . . . . 9 (𝑥 = suc 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 suc 𝑦))
12 oveq2 5547 . . . . . . . . 9 (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
1311, 12eleq12d 2124 . . . . . . . 8 (𝑥 = suc 𝑦 → ((𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥) ↔ (𝐴 +𝑜 suc 𝑦) ∈ (𝐵 +𝑜 suc 𝑦)))
14 simpr 107 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐴𝐵)
15 elnn 4355 . . . . . . . . . . 11 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)
1615ancoms 259 . . . . . . . . . 10 ((𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐴 ∈ ω)
17 nna0 6083 . . . . . . . . . 10 (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
1816, 17syl 14 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +𝑜 ∅) = 𝐴)
19 nna0 6083 . . . . . . . . . 10 (𝐵 ∈ ω → (𝐵 +𝑜 ∅) = 𝐵)
2019adantr 265 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐵 +𝑜 ∅) = 𝐵)
2114, 18, 203eltr4d 2137 . . . . . . . 8 ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +𝑜 ∅) ∈ (𝐵 +𝑜 ∅))
22 simprl 491 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐵 ∈ ω)
23 simpl 106 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑦 ∈ ω)
24 nnacl 6089 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 𝑦) ∈ ω)
2522, 23, 24syl2anc 397 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐵 +𝑜 𝑦) ∈ ω)
26 nnsucelsuc 6100 . . . . . . . . . . . 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 266 . . . . . . . . . . . . . 14 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐴 ∈ ω)
29 nnon 4359 . . . . . . . . . . . . . 14 (𝐴 ∈ ω → 𝐴 ∈ On)
3028, 29syl 14 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐴 ∈ On)
31 nnon 4359 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → 𝑦 ∈ On)
3231adantr 265 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑦 ∈ On)
33 oasuc 6074 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
3430, 32, 33syl2anc 397 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
35 nnon 4359 . . . . . . . . . . . . . 14 (𝐵 ∈ ω → 𝐵 ∈ On)
3635ad2antrl 467 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐵 ∈ On)
37 oasuc 6074 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
3836, 32, 37syl2anc 397 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
3934, 38eleq12d 2124 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → ((𝐴 +𝑜 suc 𝑦) ∈ (𝐵 +𝑜 suc 𝑦) ↔ suc (𝐴 +𝑜 𝑦) ∈ suc (𝐵 +𝑜 𝑦)))
4027, 39bitr4d 184 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → ((𝐴 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑦) ↔ (𝐴 +𝑜 suc 𝑦) ∈ (𝐵 +𝑜 suc 𝑦)))
4140biimpd 136 . . . . . . . . 9 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → ((𝐴 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 suc 𝑦) ∈ (𝐵 +𝑜 suc 𝑦)))
4241ex 112 . . . . . . . 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 4351 . . . . . . 7 (𝑥 ∈ ω → ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +𝑜 𝑥) ∈ (𝐵 +𝑜 𝑥)))
444, 43vtoclga 2636 . . . . . 6 (𝐶 ∈ ω → ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +𝑜 𝐶) ∈ (𝐵 +𝑜 𝐶)))
4544imp 119 . . . . 5 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐴 +𝑜 𝐶) ∈ (𝐵 +𝑜 𝐶))
4616adantl 266 . . . . . 6 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐴 ∈ ω)
47 simpl 106 . . . . . 6 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐶 ∈ ω)
48 nnacom 6093 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 +𝑜 𝐶) = (𝐶 +𝑜 𝐴))
4946, 47, 48syl2anc 397 . . . . 5 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐴 +𝑜 𝐶) = (𝐶 +𝑜 𝐴))
50 nnacom 6093 . . . . . . 7 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵 +𝑜 𝐶) = (𝐶 +𝑜 𝐵))
5150ancoms 259 . . . . . 6 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +𝑜 𝐶) = (𝐶 +𝑜 𝐵))
5251adantrr 456 . . . . 5 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐵 +𝑜 𝐶) = (𝐶 +𝑜 𝐵))
5345, 49, 523eltr3d 2136 . . . 4 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
54533impb 1111 . . 3 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
55543com12 1119 . 2 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐴𝐵) → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵))
56553expia 1117 1 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 +𝑜 𝐴) ∈ (𝐶 +𝑜 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  c0 3251  Oncon0 4127  suc csuc 4129  ωcom 4340  (class class class)co 5539   +𝑜 coa 6028
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 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-oadd 6035
This theorem is referenced by:  nnaord  6112  nnmordi  6119  addclpi  6482  addnidpig  6491  archnqq  6572  prarloclemarch2  6574  prarloclemlt  6648
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