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Theorem nnaordi 6532
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 5903 . . . . . . . . 9 (𝑥 = 𝐶 → (𝐴 +o 𝑥) = (𝐴 +o 𝐶))
2 oveq2 5903 . . . . . . . . 9 (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
31, 2eleq12d 2260 . . . . . . . 8 (𝑥 = 𝐶 → ((𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥) ↔ (𝐴 +o 𝐶) ∈ (𝐵 +o 𝐶)))
43imbi2d 230 . . . . . . 7 (𝑥 = 𝐶 → (((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥)) ↔ ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +o 𝐶) ∈ (𝐵 +o 𝐶))))
5 oveq2 5903 . . . . . . . . 9 (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
6 oveq2 5903 . . . . . . . . 9 (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
75, 6eleq12d 2260 . . . . . . . 8 (𝑥 = ∅ → ((𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥) ↔ (𝐴 +o ∅) ∈ (𝐵 +o ∅)))
8 oveq2 5903 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
9 oveq2 5903 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
108, 9eleq12d 2260 . . . . . . . 8 (𝑥 = 𝑦 → ((𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥) ↔ (𝐴 +o 𝑦) ∈ (𝐵 +o 𝑦)))
11 oveq2 5903 . . . . . . . . 9 (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
12 oveq2 5903 . . . . . . . . 9 (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
1311, 12eleq12d 2260 . . . . . . . 8 (𝑥 = suc 𝑦 → ((𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥) ↔ (𝐴 +o suc 𝑦) ∈ (𝐵 +o suc 𝑦)))
14 simpr 110 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐴𝐵)
15 elnn 4623 . . . . . . . . . . 11 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)
1615ancoms 268 . . . . . . . . . 10 ((𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐴 ∈ ω)
17 nna0 6498 . . . . . . . . . 10 (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
1816, 17syl 14 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +o ∅) = 𝐴)
19 nna0 6498 . . . . . . . . . 10 (𝐵 ∈ ω → (𝐵 +o ∅) = 𝐵)
2019adantr 276 . . . . . . . . 9 ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐵 +o ∅) = 𝐵)
2114, 18, 203eltr4d 2273 . . . . . . . 8 ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +o ∅) ∈ (𝐵 +o ∅))
22 simprl 529 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐵 ∈ ω)
23 simpl 109 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑦 ∈ ω)
24 nnacl 6504 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o 𝑦) ∈ ω)
2522, 23, 24syl2anc 411 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐵 +o 𝑦) ∈ ω)
26 nnsucelsuc 6515 . . . . . . . . . . . 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 277 . . . . . . . . . . . . . 14 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐴 ∈ ω)
29 nnon 4627 . . . . . . . . . . . . . 14 (𝐴 ∈ ω → 𝐴 ∈ On)
3028, 29syl 14 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐴 ∈ On)
31 nnon 4627 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → 𝑦 ∈ On)
3231adantr 276 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑦 ∈ On)
33 oasuc 6488 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
3430, 32, 33syl2anc 411 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
35 nnon 4627 . . . . . . . . . . . . . 14 (𝐵 ∈ ω → 𝐵 ∈ On)
3635ad2antrl 490 . . . . . . . . . . . . 13 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐵 ∈ On)
37 oasuc 6488 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
3836, 32, 37syl2anc 411 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
3934, 38eleq12d 2260 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → ((𝐴 +o suc 𝑦) ∈ (𝐵 +o suc 𝑦) ↔ suc (𝐴 +o 𝑦) ∈ suc (𝐵 +o 𝑦)))
4027, 39bitr4d 191 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → ((𝐴 +o 𝑦) ∈ (𝐵 +o 𝑦) ↔ (𝐴 +o suc 𝑦) ∈ (𝐵 +o suc 𝑦)))
4140biimpd 144 . . . . . . . . 9 ((𝑦 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → ((𝐴 +o 𝑦) ∈ (𝐵 +o 𝑦) → (𝐴 +o suc 𝑦) ∈ (𝐵 +o suc 𝑦)))
4241ex 115 . . . . . . . 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 4618 . . . . . . 7 (𝑥 ∈ ω → ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +o 𝑥) ∈ (𝐵 +o 𝑥)))
444, 43vtoclga 2818 . . . . . 6 (𝐶 ∈ ω → ((𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐴 +o 𝐶) ∈ (𝐵 +o 𝐶)))
4544imp 124 . . . . 5 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐴 +o 𝐶) ∈ (𝐵 +o 𝐶))
4616adantl 277 . . . . . 6 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐴 ∈ ω)
47 simpl 109 . . . . . 6 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝐶 ∈ ω)
48 nnacom 6508 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 +o 𝐶) = (𝐶 +o 𝐴))
4946, 47, 48syl2anc 411 . . . . 5 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐴 +o 𝐶) = (𝐶 +o 𝐴))
50 nnacom 6508 . . . . . . 7 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵 +o 𝐶) = (𝐶 +o 𝐵))
5150ancoms 268 . . . . . 6 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 +o 𝐶) = (𝐶 +o 𝐵))
5251adantrr 479 . . . . 5 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐵 +o 𝐶) = (𝐶 +o 𝐵))
5345, 49, 523eltr3d 2272 . . . 4 ((𝐶 ∈ ω ∧ (𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
54533impb 1201 . . 3 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
55543com12 1209 . 2 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω ∧ 𝐴𝐵) → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))
56553expia 1207 1 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160  c0 3437  Oncon0 4381  suc csuc 4383  ωcom 4607  (class class class)co 5895   +o coa 6437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-irdg 6394  df-oadd 6444
This theorem is referenced by:  nnaord  6533  nnmordi  6540  addclpi  7355  addnidpig  7364  archnqq  7445  prarloclemarch2  7447  prarloclemlt  7521
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