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Theorem nnaword 6310
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 5697 . . . . . . 7 (𝑥 = 𝐶 → (𝑥 +o 𝐴) = (𝐶 +o 𝐴))
2 oveq1 5697 . . . . . . 7 (𝑥 = 𝐶 → (𝑥 +o 𝐵) = (𝐶 +o 𝐵))
31, 2sseq12d 3070 . . . . . 6 (𝑥 = 𝐶 → ((𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵) ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
43bibi2d 231 . . . . 5 (𝑥 = 𝐶 → ((𝐴𝐵 ↔ (𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵)) ↔ (𝐴𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵))))
54imbi2d 229 . . . 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 5697 . . . . . . 7 (𝑥 = ∅ → (𝑥 +o 𝐴) = (∅ +o 𝐴))
7 oveq1 5697 . . . . . . 7 (𝑥 = ∅ → (𝑥 +o 𝐵) = (∅ +o 𝐵))
86, 7sseq12d 3070 . . . . . 6 (𝑥 = ∅ → ((𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵) ↔ (∅ +o 𝐴) ⊆ (∅ +o 𝐵)))
98bibi2d 231 . . . . 5 (𝑥 = ∅ → ((𝐴𝐵 ↔ (𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵)) ↔ (𝐴𝐵 ↔ (∅ +o 𝐴) ⊆ (∅ +o 𝐵))))
10 oveq1 5697 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 +o 𝐴) = (𝑦 +o 𝐴))
11 oveq1 5697 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 +o 𝐵) = (𝑦 +o 𝐵))
1210, 11sseq12d 3070 . . . . . 6 (𝑥 = 𝑦 → ((𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵) ↔ (𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵)))
1312bibi2d 231 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝐵 ↔ (𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵)) ↔ (𝐴𝐵 ↔ (𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵))))
14 oveq1 5697 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑥 +o 𝐴) = (suc 𝑦 +o 𝐴))
15 oveq1 5697 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑥 +o 𝐵) = (suc 𝑦 +o 𝐵))
1614, 15sseq12d 3070 . . . . . 6 (𝑥 = suc 𝑦 → ((𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵) ↔ (suc 𝑦 +o 𝐴) ⊆ (suc 𝑦 +o 𝐵)))
1716bibi2d 231 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴𝐵 ↔ (𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵)) ↔ (𝐴𝐵 ↔ (suc 𝑦 +o 𝐴) ⊆ (suc 𝑦 +o 𝐵))))
18 nna0r 6279 . . . . . . . 8 (𝐴 ∈ ω → (∅ +o 𝐴) = 𝐴)
1918eqcomd 2100 . . . . . . 7 (𝐴 ∈ ω → 𝐴 = (∅ +o 𝐴))
2019adantr 271 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 = (∅ +o 𝐴))
21 nna0r 6279 . . . . . . . 8 (𝐵 ∈ ω → (∅ +o 𝐵) = 𝐵)
2221eqcomd 2100 . . . . . . 7 (𝐵 ∈ ω → 𝐵 = (∅ +o 𝐵))
2322adantl 272 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐵 = (∅ +o 𝐵))
2420, 23sseq12d 3070 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (∅ +o 𝐴) ⊆ (∅ +o 𝐵)))
25 nnacl 6281 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝑦 +o 𝐴) ∈ ω)
26253adant3 966 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +o 𝐴) ∈ ω)
27 nnacl 6281 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +o 𝐵) ∈ ω)
28273adant2 965 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +o 𝐵) ∈ ω)
29 nnsucsssuc 6293 . . . . . . . . . 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 404 . . . . . . . . 9 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵) ↔ suc (𝑦 +o 𝐴) ⊆ suc (𝑦 +o 𝐵)))
31 nnasuc 6277 . . . . . . . . . . . . 13 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
32 peano2 4438 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
33 nnacom 6285 . . . . . . . . . . . . . 14 ((𝐴 ∈ ω ∧ suc 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = (suc 𝑦 +o 𝐴))
3432, 33sylan2 281 . . . . . . . . . . . . 13 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = (suc 𝑦 +o 𝐴))
35 nnacom 6285 . . . . . . . . . . . . . 14 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) = (𝑦 +o 𝐴))
36 suceq 4253 . . . . . . . . . . . . . 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 2136 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝑦 +o 𝐴) = (suc 𝑦 +o 𝐴))
3938ancoms 265 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → suc (𝑦 +o 𝐴) = (suc 𝑦 +o 𝐴))
40393adant3 966 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +o 𝐴) = (suc 𝑦 +o 𝐴))
41 nnasuc 6277 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
42 nnacom 6285 . . . . . . . . . . . . . 14 ((𝐵 ∈ ω ∧ suc 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = (suc 𝑦 +o 𝐵))
4332, 42sylan2 281 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = (suc 𝑦 +o 𝐵))
44 nnacom 6285 . . . . . . . . . . . . . 14 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o 𝑦) = (𝑦 +o 𝐵))
45 suceq 4253 . . . . . . . . . . . . . 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 2136 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝑦 +o 𝐵) = (suc 𝑦 +o 𝐵))
4847ancoms 265 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +o 𝐵) = (suc 𝑦 +o 𝐵))
49483adant2 965 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +o 𝐵) = (suc 𝑦 +o 𝐵))
5040, 49sseq12d 3070 . . . . . . . . 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 187 . . . . . . . 8 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵) ↔ (suc 𝑦 +o 𝐴) ⊆ (suc 𝑦 +o 𝐵)))
5251bibi2d 231 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴𝐵 ↔ (𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵)) ↔ (𝐴𝐵 ↔ (suc 𝑦 +o 𝐴) ⊆ (suc 𝑦 +o 𝐵))))
5352biimpd 143 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴𝐵 ↔ (𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵)) → (𝐴𝐵 ↔ (suc 𝑦 +o 𝐴) ⊆ (suc 𝑦 +o 𝐵))))
54533expib 1149 . . . . 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 4444 . . . 4 (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵))))
565, 55vtoclga 2699 . . 3 (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵))))
5756impcom 124 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐶 ∈ ω) → (𝐴𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
58573impa 1141 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 103  wb 104  w3a 927   = wceq 1296  wcel 1445  wss 3013  c0 3302  suc csuc 4216  ωcom 4433  (class class class)co 5690   +o coa 6216
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-oadd 6223
This theorem is referenced by:  nnacan  6311  nnawordi  6314
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