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Theorem nnaword 6114
Description: Weak ordering property of addition. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nnaword ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))

Proof of Theorem nnaword
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 5546 . . . . . . 7 (𝑥 = 𝐶 → (𝑥 +𝑜 𝐴) = (𝐶 +𝑜 𝐴))
2 oveq1 5546 . . . . . . 7 (𝑥 = 𝐶 → (𝑥 +𝑜 𝐵) = (𝐶 +𝑜 𝐵))
31, 2sseq12d 3001 . . . . . 6 (𝑥 = 𝐶 → ((𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵) ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
43bibi2d 225 . . . . 5 (𝑥 = 𝐶 → ((𝐴𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵)) ↔ (𝐴𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵))))
54imbi2d 223 . . . 4 (𝑥 = 𝐶 → (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵))) ↔ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))))
6 oveq1 5546 . . . . . . 7 (𝑥 = ∅ → (𝑥 +𝑜 𝐴) = (∅ +𝑜 𝐴))
7 oveq1 5546 . . . . . . 7 (𝑥 = ∅ → (𝑥 +𝑜 𝐵) = (∅ +𝑜 𝐵))
86, 7sseq12d 3001 . . . . . 6 (𝑥 = ∅ → ((𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵) ↔ (∅ +𝑜 𝐴) ⊆ (∅ +𝑜 𝐵)))
98bibi2d 225 . . . . 5 (𝑥 = ∅ → ((𝐴𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵)) ↔ (𝐴𝐵 ↔ (∅ +𝑜 𝐴) ⊆ (∅ +𝑜 𝐵))))
10 oveq1 5546 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 +𝑜 𝐴) = (𝑦 +𝑜 𝐴))
11 oveq1 5546 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 +𝑜 𝐵) = (𝑦 +𝑜 𝐵))
1210, 11sseq12d 3001 . . . . . 6 (𝑥 = 𝑦 → ((𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵) ↔ (𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵)))
1312bibi2d 225 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵)) ↔ (𝐴𝐵 ↔ (𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵))))
14 oveq1 5546 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑥 +𝑜 𝐴) = (suc 𝑦 +𝑜 𝐴))
15 oveq1 5546 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑥 +𝑜 𝐵) = (suc 𝑦 +𝑜 𝐵))
1614, 15sseq12d 3001 . . . . . 6 (𝑥 = suc 𝑦 → ((𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵) ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵)))
1716bibi2d 225 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵)) ↔ (𝐴𝐵 ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵))))
18 nna0r 6087 . . . . . . . 8 (𝐴 ∈ ω → (∅ +𝑜 𝐴) = 𝐴)
1918eqcomd 2061 . . . . . . 7 (𝐴 ∈ ω → 𝐴 = (∅ +𝑜 𝐴))
2019adantr 265 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 = (∅ +𝑜 𝐴))
21 nna0r 6087 . . . . . . . 8 (𝐵 ∈ ω → (∅ +𝑜 𝐵) = 𝐵)
2221eqcomd 2061 . . . . . . 7 (𝐵 ∈ ω → 𝐵 = (∅ +𝑜 𝐵))
2322adantl 266 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐵 = (∅ +𝑜 𝐵))
2420, 23sseq12d 3001 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (∅ +𝑜 𝐴) ⊆ (∅ +𝑜 𝐵)))
25 nnacl 6089 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝑦 +𝑜 𝐴) ∈ ω)
26253adant3 935 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +𝑜 𝐴) ∈ ω)
27 nnacl 6089 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +𝑜 𝐵) ∈ ω)
28273adant2 934 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +𝑜 𝐵) ∈ ω)
29 nnsucsssuc 6101 . . . . . . . . . 10 (((𝑦 +𝑜 𝐴) ∈ ω ∧ (𝑦 +𝑜 𝐵) ∈ ω) → ((𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵) ↔ suc (𝑦 +𝑜 𝐴) ⊆ suc (𝑦 +𝑜 𝐵)))
3026, 28, 29syl2anc 397 . . . . . . . . 9 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵) ↔ suc (𝑦 +𝑜 𝐴) ⊆ suc (𝑦 +𝑜 𝐵)))
31 nnasuc 6085 . . . . . . . . . . . . 13 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
32 peano2 4345 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
33 nnacom 6093 . . . . . . . . . . . . . 14 ((𝐴 ∈ ω ∧ suc 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = (suc 𝑦 +𝑜 𝐴))
3432, 33sylan2 274 . . . . . . . . . . . . 13 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = (suc 𝑦 +𝑜 𝐴))
35 nnacom 6093 . . . . . . . . . . . . . 14 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 𝑦) = (𝑦 +𝑜 𝐴))
36 suceq 4166 . . . . . . . . . . . . . 14 ((𝐴 +𝑜 𝑦) = (𝑦 +𝑜 𝐴) → suc (𝐴 +𝑜 𝑦) = suc (𝑦 +𝑜 𝐴))
3735, 36syl 14 . . . . . . . . . . . . 13 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝐴 +𝑜 𝑦) = suc (𝑦 +𝑜 𝐴))
3831, 34, 373eqtr3rd 2097 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝑦 +𝑜 𝐴) = (suc 𝑦 +𝑜 𝐴))
3938ancoms 259 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → suc (𝑦 +𝑜 𝐴) = (suc 𝑦 +𝑜 𝐴))
40393adant3 935 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +𝑜 𝐴) = (suc 𝑦 +𝑜 𝐴))
41 nnasuc 6085 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
42 nnacom 6093 . . . . . . . . . . . . . 14 ((𝐵 ∈ ω ∧ suc 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = (suc 𝑦 +𝑜 𝐵))
4332, 42sylan2 274 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = (suc 𝑦 +𝑜 𝐵))
44 nnacom 6093 . . . . . . . . . . . . . 14 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 𝑦) = (𝑦 +𝑜 𝐵))
45 suceq 4166 . . . . . . . . . . . . . 14 ((𝐵 +𝑜 𝑦) = (𝑦 +𝑜 𝐵) → suc (𝐵 +𝑜 𝑦) = suc (𝑦 +𝑜 𝐵))
4644, 45syl 14 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝐵 +𝑜 𝑦) = suc (𝑦 +𝑜 𝐵))
4741, 43, 463eqtr3rd 2097 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝑦 +𝑜 𝐵) = (suc 𝑦 +𝑜 𝐵))
4847ancoms 259 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +𝑜 𝐵) = (suc 𝑦 +𝑜 𝐵))
49483adant2 934 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +𝑜 𝐵) = (suc 𝑦 +𝑜 𝐵))
5040, 49sseq12d 3001 . . . . . . . . 9 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc (𝑦 +𝑜 𝐴) ⊆ suc (𝑦 +𝑜 𝐵) ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵)))
5130, 50bitrd 181 . . . . . . . 8 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵) ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵)))
5251bibi2d 225 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴𝐵 ↔ (𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵)) ↔ (𝐴𝐵 ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵))))
5352biimpd 136 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴𝐵 ↔ (𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵)) → (𝐴𝐵 ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵))))
54533expib 1118 . . . . 5 (𝑦 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴𝐵 ↔ (𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵)) → (𝐴𝐵 ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵)))))
559, 13, 17, 24, 54finds2 4351 . . . 4 (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵))))
565, 55vtoclga 2636 . . 3 (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵))))
5756impcom 120 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐶 ∈ ω) → (𝐴𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
58573impa 1110 1 ((𝐴 ∈ ω ∧ 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  w3a 896   = wceq 1259  wcel 1409  wss 2944  c0 3251  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:  nnacan  6115  nnawordi  6118
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