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Theorem nnaword 6506
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 5876 . . . . . . 7 (𝑥 = 𝐶 → (𝑥 +o 𝐴) = (𝐶 +o 𝐴))
2 oveq1 5876 . . . . . . 7 (𝑥 = 𝐶 → (𝑥 +o 𝐵) = (𝐶 +o 𝐵))
31, 2sseq12d 3186 . . . . . 6 (𝑥 = 𝐶 → ((𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵) ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
43bibi2d 232 . . . . 5 (𝑥 = 𝐶 → ((𝐴𝐵 ↔ (𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵)) ↔ (𝐴𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵))))
54imbi2d 230 . . . 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 5876 . . . . . . 7 (𝑥 = ∅ → (𝑥 +o 𝐴) = (∅ +o 𝐴))
7 oveq1 5876 . . . . . . 7 (𝑥 = ∅ → (𝑥 +o 𝐵) = (∅ +o 𝐵))
86, 7sseq12d 3186 . . . . . 6 (𝑥 = ∅ → ((𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵) ↔ (∅ +o 𝐴) ⊆ (∅ +o 𝐵)))
98bibi2d 232 . . . . 5 (𝑥 = ∅ → ((𝐴𝐵 ↔ (𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵)) ↔ (𝐴𝐵 ↔ (∅ +o 𝐴) ⊆ (∅ +o 𝐵))))
10 oveq1 5876 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 +o 𝐴) = (𝑦 +o 𝐴))
11 oveq1 5876 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 +o 𝐵) = (𝑦 +o 𝐵))
1210, 11sseq12d 3186 . . . . . 6 (𝑥 = 𝑦 → ((𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵) ↔ (𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵)))
1312bibi2d 232 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝐵 ↔ (𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵)) ↔ (𝐴𝐵 ↔ (𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵))))
14 oveq1 5876 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑥 +o 𝐴) = (suc 𝑦 +o 𝐴))
15 oveq1 5876 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑥 +o 𝐵) = (suc 𝑦 +o 𝐵))
1614, 15sseq12d 3186 . . . . . 6 (𝑥 = suc 𝑦 → ((𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵) ↔ (suc 𝑦 +o 𝐴) ⊆ (suc 𝑦 +o 𝐵)))
1716bibi2d 232 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴𝐵 ↔ (𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵)) ↔ (𝐴𝐵 ↔ (suc 𝑦 +o 𝐴) ⊆ (suc 𝑦 +o 𝐵))))
18 nna0r 6473 . . . . . . . 8 (𝐴 ∈ ω → (∅ +o 𝐴) = 𝐴)
1918eqcomd 2183 . . . . . . 7 (𝐴 ∈ ω → 𝐴 = (∅ +o 𝐴))
2019adantr 276 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 = (∅ +o 𝐴))
21 nna0r 6473 . . . . . . . 8 (𝐵 ∈ ω → (∅ +o 𝐵) = 𝐵)
2221eqcomd 2183 . . . . . . 7 (𝐵 ∈ ω → 𝐵 = (∅ +o 𝐵))
2322adantl 277 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐵 = (∅ +o 𝐵))
2420, 23sseq12d 3186 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (∅ +o 𝐴) ⊆ (∅ +o 𝐵)))
25 nnacl 6475 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝑦 +o 𝐴) ∈ ω)
26253adant3 1017 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +o 𝐴) ∈ ω)
27 nnacl 6475 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +o 𝐵) ∈ ω)
28273adant2 1016 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝑦 +o 𝐵) ∈ ω)
29 nnsucsssuc 6487 . . . . . . . . . 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 411 . . . . . . . . 9 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵) ↔ suc (𝑦 +o 𝐴) ⊆ suc (𝑦 +o 𝐵)))
31 nnasuc 6471 . . . . . . . . . . . . 13 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
32 peano2 4591 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
33 nnacom 6479 . . . . . . . . . . . . . 14 ((𝐴 ∈ ω ∧ suc 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = (suc 𝑦 +o 𝐴))
3432, 33sylan2 286 . . . . . . . . . . . . 13 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = (suc 𝑦 +o 𝐴))
35 nnacom 6479 . . . . . . . . . . . . . 14 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) = (𝑦 +o 𝐴))
36 suceq 4399 . . . . . . . . . . . . . 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 2219 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝑦 +o 𝐴) = (suc 𝑦 +o 𝐴))
3938ancoms 268 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → suc (𝑦 +o 𝐴) = (suc 𝑦 +o 𝐴))
40393adant3 1017 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +o 𝐴) = (suc 𝑦 +o 𝐴))
41 nnasuc 6471 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
42 nnacom 6479 . . . . . . . . . . . . . 14 ((𝐵 ∈ ω ∧ suc 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = (suc 𝑦 +o 𝐵))
4332, 42sylan2 286 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o suc 𝑦) = (suc 𝑦 +o 𝐵))
44 nnacom 6479 . . . . . . . . . . . . . 14 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → (𝐵 +o 𝑦) = (𝑦 +o 𝐵))
45 suceq 4399 . . . . . . . . . . . . . 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 2219 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝑦 +o 𝐵) = (suc 𝑦 +o 𝐵))
4847ancoms 268 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +o 𝐵) = (suc 𝑦 +o 𝐵))
49483adant2 1016 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → suc (𝑦 +o 𝐵) = (suc 𝑦 +o 𝐵))
5040, 49sseq12d 3186 . . . . . . . . 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 188 . . . . . . . 8 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵) ↔ (suc 𝑦 +o 𝐴) ⊆ (suc 𝑦 +o 𝐵)))
5251bibi2d 232 . . . . . . 7 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴𝐵 ↔ (𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵)) ↔ (𝐴𝐵 ↔ (suc 𝑦 +o 𝐴) ⊆ (suc 𝑦 +o 𝐵))))
5352biimpd 144 . . . . . 6 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴𝐵 ↔ (𝑦 +o 𝐴) ⊆ (𝑦 +o 𝐵)) → (𝐴𝐵 ↔ (suc 𝑦 +o 𝐴) ⊆ (suc 𝑦 +o 𝐵))))
54533expib 1206 . . . . 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 4597 . . . 4 (𝑥 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (𝑥 +o 𝐴) ⊆ (𝑥 +o 𝐵))))
565, 55vtoclga 2803 . . 3 (𝐶 ∈ ω → ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵))))
5756impcom 125 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐶 ∈ ω) → (𝐴𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵)))
58573impa 1194 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 104  wb 105  w3a 978   = wceq 1353  wcel 2148  wss 3129  c0 3422  suc csuc 4362  ωcom 4586  (class class class)co 5869   +o coa 6408
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-oadd 6415
This theorem is referenced by:  nnacan  6507  nnawordi  6510
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