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Theorem nnaword 6084
 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 5519 . . . . . . 7 (𝑥 = 𝐶 → (𝑥 +𝑜 𝐴) = (𝐶 +𝑜 𝐴))
2 oveq1 5519 . . . . . . 7 (𝑥 = 𝐶 → (𝑥 +𝑜 𝐵) = (𝐶 +𝑜 𝐵))
31, 2sseq12d 2974 . . . . . 6 (𝑥 = 𝐶 → ((𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵) ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
43bibi2d 221 . . . . 5 (𝑥 = 𝐶 → ((𝐴𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵)) ↔ (𝐴𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵))))
54imbi2d 219 . . . 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 5519 . . . . . . 7 (𝑥 = ∅ → (𝑥 +𝑜 𝐴) = (∅ +𝑜 𝐴))
7 oveq1 5519 . . . . . . 7 (𝑥 = ∅ → (𝑥 +𝑜 𝐵) = (∅ +𝑜 𝐵))
86, 7sseq12d 2974 . . . . . 6 (𝑥 = ∅ → ((𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵) ↔ (∅ +𝑜 𝐴) ⊆ (∅ +𝑜 𝐵)))
98bibi2d 221 . . . . 5 (𝑥 = ∅ → ((𝐴𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵)) ↔ (𝐴𝐵 ↔ (∅ +𝑜 𝐴) ⊆ (∅ +𝑜 𝐵))))
10 oveq1 5519 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 +𝑜 𝐴) = (𝑦 +𝑜 𝐴))
11 oveq1 5519 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 +𝑜 𝐵) = (𝑦 +𝑜 𝐵))
1210, 11sseq12d 2974 . . . . . 6 (𝑥 = 𝑦 → ((𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵) ↔ (𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵)))
1312bibi2d 221 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵)) ↔ (𝐴𝐵 ↔ (𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵))))
14 oveq1 5519 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑥 +𝑜 𝐴) = (suc 𝑦 +𝑜 𝐴))
15 oveq1 5519 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑥 +𝑜 𝐵) = (suc 𝑦 +𝑜 𝐵))
1614, 15sseq12d 2974 . . . . . 6 (𝑥 = suc 𝑦 → ((𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵) ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵)))
1716bibi2d 221 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵)) ↔ (𝐴𝐵 ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵))))
18 nna0r 6057 . . . . . . . 8 (𝐴 ∈ ω → (∅ +𝑜 𝐴) = 𝐴)
1918eqcomd 2045 . . . . . . 7 (𝐴 ∈ ω → 𝐴 = (∅ +𝑜 𝐴))
2019adantr 261 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 = (∅ +𝑜 𝐴))
21 nna0r 6057 . . . . . . . 8 (𝐵 ∈ ω → (∅ +𝑜 𝐵) = 𝐵)
2221eqcomd 2045 . . . . . . 7 (𝐵 ∈ ω → 𝐵 = (∅ +𝑜 𝐵))
2322adantl 262 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐵 = (∅ +𝑜 𝐵))
2420, 23sseq12d 2974 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (∅ +𝑜 𝐴) ⊆ (∅ +𝑜 𝐵)))
25 nnacl 6059 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝑦 +𝑜 𝐴) ∈ ω)
26253adant3 924 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +𝑜 𝐴) ∈ ω)
27 nnacl 6059 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +𝑜 𝐵) ∈ ω)
28273adant2 923 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +𝑜 𝐵) ∈ ω)
29 nnsucsssuc 6071 . . . . . . . . . 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 391 . . . . . . . . 9 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵) ↔ suc (𝑦 +𝑜 𝐴) ⊆ suc (𝑦 +𝑜 𝐵)))
31 nnasuc 6055 . . . . . . . . . . . . 13 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
32 peano2 4318 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
33 nnacom 6063 . . . . . . . . . . . . . 14 ((𝐴 ∈ ω ∧ suc 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = (suc 𝑦 +𝑜 𝐴))
3432, 33sylan2 270 . . . . . . . . . . . . 13 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = (suc 𝑦 +𝑜 𝐴))
35 nnacom 6063 . . . . . . . . . . . . . 14 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 𝑦) = (𝑦 +𝑜 𝐴))
36 suceq 4139 . . . . . . . . . . . . . 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 2081 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝑦 +𝑜 𝐴) = (suc 𝑦 +𝑜 𝐴))
3938ancoms 255 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → suc (𝑦 +𝑜 𝐴) = (suc 𝑦 +𝑜 𝐴))
40393adant3 924 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +𝑜 𝐴) = (suc 𝑦 +𝑜 𝐴))
41 nnasuc 6055 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
42 nnacom 6063 . . . . . . . . . . . . . 14 ((𝐵 ∈ ω ∧ suc 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = (suc 𝑦 +𝑜 𝐵))
4332, 42sylan2 270 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 suc 𝑦) = (suc 𝑦 +𝑜 𝐵))
44 nnacom 6063 . . . . . . . . . . . . . 14 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +𝑜 𝑦) = (𝑦 +𝑜 𝐵))
45 suceq 4139 . . . . . . . . . . . . . 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 2081 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝑦 +𝑜 𝐵) = (suc 𝑦 +𝑜 𝐵))
4847ancoms 255 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +𝑜 𝐵) = (suc 𝑦 +𝑜 𝐵))
49483adant2 923 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +𝑜 𝐵) = (suc 𝑦 +𝑜 𝐵))
5040, 49sseq12d 2974 . . . . . . . . 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 177 . . . . . . . 8 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵) ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵)))
5251bibi2d 221 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴𝐵 ↔ (𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵)) ↔ (𝐴𝐵 ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵))))
5352biimpd 132 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴𝐵 ↔ (𝑦 +𝑜 𝐴) ⊆ (𝑦 +𝑜 𝐵)) → (𝐴𝐵 ↔ (suc 𝑦 +𝑜 𝐴) ⊆ (suc 𝑦 +𝑜 𝐵))))
54533expib 1107 . . . . 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 4324 . . . 4 (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (𝑥 +𝑜 𝐴) ⊆ (𝑥 +𝑜 𝐵))))
565, 55vtoclga 2619 . . 3 (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵))))
5756impcom 116 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐶 ∈ ω) → (𝐴𝐵 ↔ (𝐶 +𝑜 𝐴) ⊆ (𝐶 +𝑜 𝐵)))
58573impa 1099 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 97   ↔ wb 98   ∧ w3a 885   = wceq 1243   ∈ wcel 1393   ⊆ wss 2917  ∅c0 3224  suc csuc 4102  ωcom 4313  (class class class)co 5512   +𝑜 coa 5998 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005 This theorem is referenced by:  nnacan  6085  nnawordi  6088
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