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Theorem nnaword 6172
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 5571 . . . . . . 7 (𝑥 = 𝐶 → (𝑥 +𝑜 𝐴) = (𝐶 +𝑜 𝐴))
2 oveq1 5571 . . . . . . 7 (𝑥 = 𝐶 → (𝑥 +𝑜 𝐵) = (𝐶 +𝑜 𝐵))
31, 2sseq12d 3038 . . . . . 6 (𝑥 = 𝐶 → ((𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵) ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
43bibi2d 230 . . . . 5 (𝑥 = 𝐶 → ((𝐴𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵)) ↔ (𝐴𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵))))
54imbi2d 228 . . . 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 5571 . . . . . . 7 (𝑥 = ∅ → (𝑥 +𝑜 𝐴) = (∅ +𝑜 𝐴))
7 oveq1 5571 . . . . . . 7 (𝑥 = ∅ → (𝑥 +𝑜 𝐵) = (∅ +𝑜 𝐵))
86, 7sseq12d 3038 . . . . . 6 (𝑥 = ∅ → ((𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵) ↔ (∅ +𝑜 𝐴) ⊆ (∅ +𝑜 𝐵)))
98bibi2d 230 . . . . 5 (𝑥 = ∅ → ((𝐴𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵)) ↔ (𝐴𝐵 ↔ (∅ +𝑜 𝐴) ⊆ (∅ +𝑜 𝐵))))
10 oveq1 5571 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 +𝑜 𝐴) = (𝑦 +𝑜 𝐴))
11 oveq1 5571 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 +𝑜 𝐵) = (𝑦 +𝑜 𝐵))
1210, 11sseq12d 3038 . . . . . 6 (𝑥 = 𝑦 → ((𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵) ↔ (𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵)))
1312bibi2d 230 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵)) ↔ (𝐴𝐵 ↔ (𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵))))
14 oveq1 5571 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑥 +𝑜 𝐴) = (suc 𝑦 +𝑜 𝐴))
15 oveq1 5571 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑥 +𝑜 𝐵) = (suc 𝑦 +𝑜 𝐵))
1614, 15sseq12d 3038 . . . . . 6 (𝑥 = suc 𝑦 → ((𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵) ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵)))
1716bibi2d 230 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵)) ↔ (𝐴𝐵 ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵))))
18 nna0r 6143 . . . . . . . 8 (𝐴 ∈ ω → (∅ +𝑜 𝐴) = 𝐴)
1918eqcomd 2088 . . . . . . 7 (𝐴 ∈ ω → 𝐴 = (∅ +𝑜 𝐴))
2019adantr 270 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 = (∅ +𝑜 𝐴))
21 nna0r 6143 . . . . . . . 8 (𝐵 ∈ ω → (∅ +𝑜 𝐵) = 𝐵)
2221eqcomd 2088 . . . . . . 7 (𝐵 ∈ ω → 𝐵 = (∅ +𝑜 𝐵))
2322adantl 271 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐵 = (∅ +𝑜 𝐵))
2420, 23sseq12d 3038 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (∅ +𝑜 𝐴) ⊆ (∅ +𝑜 𝐵)))
25 nnacl 6145 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝑦 +𝑜 𝐴) ∈ ω)
26253adant3 959 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +𝑜 𝐴) ∈ ω)
27 nnacl 6145 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +𝑜 𝐵) ∈ ω)
28273adant2 958 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +𝑜 𝐵) ∈ ω)
29 nnsucsssuc 6157 . . . . . . . . . 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 403 . . . . . . . . 9 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵) ↔ suc (𝑦 +𝑜 𝐴) ⊆ suc (𝑦 +𝑜 𝐵)))
31 nnasuc 6141 . . . . . . . . . . . . 13 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
32 peano2 4365 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
33 nnacom 6149 . . . . . . . . . . . . . 14 ((𝐴 ∈ ω ∧ suc 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = (suc 𝑦 +𝑜 𝐴))
3432, 33sylan2 280 . . . . . . . . . . . . 13 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = (suc 𝑦 +𝑜 𝐴))
35 nnacom 6149 . . . . . . . . . . . . . 14 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 𝑦) = (𝑦 +𝑜 𝐴))
36 suceq 4186 . . . . . . . . . . . . . 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 2124 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝑦 +𝑜 𝐴) = (suc 𝑦 +𝑜 𝐴))
3938ancoms 264 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → suc (𝑦 +𝑜 𝐴) = (suc 𝑦 +𝑜 𝐴))
40393adant3 959 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +𝑜 𝐴) = (suc 𝑦 +𝑜 𝐴))
41 nnasuc 6141 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
42 nnacom 6149 . . . . . . . . . . . . . 14 ((𝐵 ∈ ω ∧ suc 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = (suc 𝑦 +𝑜 𝐵))
4332, 42sylan2 280 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = (suc 𝑦 +𝑜 𝐵))
44 nnacom 6149 . . . . . . . . . . . . . 14 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 𝑦) = (𝑦 +𝑜 𝐵))
45 suceq 4186 . . . . . . . . . . . . . 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 2124 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝑦 +𝑜 𝐵) = (suc 𝑦 +𝑜 𝐵))
4847ancoms 264 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +𝑜 𝐵) = (suc 𝑦 +𝑜 𝐵))
49483adant2 958 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +𝑜 𝐵) = (suc 𝑦 +𝑜 𝐵))
5040, 49sseq12d 3038 . . . . . . . . 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 186 . . . . . . . 8 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵) ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵)))
5251bibi2d 230 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴𝐵 ↔ (𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵)) ↔ (𝐴𝐵 ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵))))
5352biimpd 142 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴𝐵 ↔ (𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵)) → (𝐴𝐵 ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵))))
54533expib 1142 . . . . 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 4371 . . . 4 (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵))))
565, 55vtoclga 2673 . . 3 (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵))))
5756impcom 123 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐶 ∈ ω) → (𝐴𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
58573impa 1134 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 102  wb 103  w3a 920   = wceq 1285  wcel 1434  wss 2983  c0 3268  suc csuc 4149  ωcom 4360  (class class class)co 5564   +𝑜 coa 6083
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 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358
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 2612  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-id 4077  df-iord 4150  df-on 4152  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-irdg 6040  df-oadd 6090
This theorem is referenced by:  nnacan  6173  nnawordi  6176
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