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Theorem naddoa 8740
Description: Natural addition of a natural is the same as regular addition. (Contributed by Scott Fenton, 20-Aug-2025.)
Assertion
Ref Expression
naddoa ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +no 𝐵) = (𝐴 +o 𝐵))

Proof of Theorem naddoa
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7439 . . . . 5 (𝑦 = ∅ → (𝐴 +no 𝑦) = (𝐴 +no ∅))
2 oveq2 7439 . . . . 5 (𝑦 = ∅ → (𝐴 +o 𝑦) = (𝐴 +o ∅))
31, 2eqeq12d 2753 . . . 4 (𝑦 = ∅ → ((𝐴 +no 𝑦) = (𝐴 +o 𝑦) ↔ (𝐴 +no ∅) = (𝐴 +o ∅)))
43imbi2d 340 . . 3 (𝑦 = ∅ → ((𝐴 ∈ On → (𝐴 +no 𝑦) = (𝐴 +o 𝑦)) ↔ (𝐴 ∈ On → (𝐴 +no ∅) = (𝐴 +o ∅))))
5 oveq2 7439 . . . . 5 (𝑦 = 𝑥 → (𝐴 +no 𝑦) = (𝐴 +no 𝑥))
6 oveq2 7439 . . . . 5 (𝑦 = 𝑥 → (𝐴 +o 𝑦) = (𝐴 +o 𝑥))
75, 6eqeq12d 2753 . . . 4 (𝑦 = 𝑥 → ((𝐴 +no 𝑦) = (𝐴 +o 𝑦) ↔ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)))
87imbi2d 340 . . 3 (𝑦 = 𝑥 → ((𝐴 ∈ On → (𝐴 +no 𝑦) = (𝐴 +o 𝑦)) ↔ (𝐴 ∈ On → (𝐴 +no 𝑥) = (𝐴 +o 𝑥))))
9 oveq2 7439 . . . . 5 (𝑦 = suc 𝑥 → (𝐴 +no 𝑦) = (𝐴 +no suc 𝑥))
10 oveq2 7439 . . . . 5 (𝑦 = suc 𝑥 → (𝐴 +o 𝑦) = (𝐴 +o suc 𝑥))
119, 10eqeq12d 2753 . . . 4 (𝑦 = suc 𝑥 → ((𝐴 +no 𝑦) = (𝐴 +o 𝑦) ↔ (𝐴 +no suc 𝑥) = (𝐴 +o suc 𝑥)))
1211imbi2d 340 . . 3 (𝑦 = suc 𝑥 → ((𝐴 ∈ On → (𝐴 +no 𝑦) = (𝐴 +o 𝑦)) ↔ (𝐴 ∈ On → (𝐴 +no suc 𝑥) = (𝐴 +o suc 𝑥))))
13 oveq2 7439 . . . . 5 (𝑦 = 𝐵 → (𝐴 +no 𝑦) = (𝐴 +no 𝐵))
14 oveq2 7439 . . . . 5 (𝑦 = 𝐵 → (𝐴 +o 𝑦) = (𝐴 +o 𝐵))
1513, 14eqeq12d 2753 . . . 4 (𝑦 = 𝐵 → ((𝐴 +no 𝑦) = (𝐴 +o 𝑦) ↔ (𝐴 +no 𝐵) = (𝐴 +o 𝐵)))
1615imbi2d 340 . . 3 (𝑦 = 𝐵 → ((𝐴 ∈ On → (𝐴 +no 𝑦) = (𝐴 +o 𝑦)) ↔ (𝐴 ∈ On → (𝐴 +no 𝐵) = (𝐴 +o 𝐵))))
17 naddrid 8721 . . . 4 (𝐴 ∈ On → (𝐴 +no ∅) = 𝐴)
18 oa0 8554 . . . 4 (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
1917, 18eqtr4d 2780 . . 3 (𝐴 ∈ On → (𝐴 +no ∅) = (𝐴 +o ∅))
20 suceq 6450 . . . . . . 7 ((𝐴 +no 𝑥) = (𝐴 +o 𝑥) → suc (𝐴 +no 𝑥) = suc (𝐴 +o 𝑥))
21203ad2ant3 1136 . . . . . 6 ((𝑥 ∈ ω ∧ 𝐴 ∈ On ∧ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → suc (𝐴 +no 𝑥) = suc (𝐴 +o 𝑥))
22 nnon 7893 . . . . . . . . 9 (𝑥 ∈ ω → 𝑥 ∈ On)
23 naddsuc2 8739 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +no suc 𝑥) = suc (𝐴 +no 𝑥))
2422, 23sylan2 593 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴 +no suc 𝑥) = suc (𝐴 +no 𝑥))
2524ancoms 458 . . . . . . 7 ((𝑥 ∈ ω ∧ 𝐴 ∈ On) → (𝐴 +no suc 𝑥) = suc (𝐴 +no 𝑥))
26253adant3 1133 . . . . . 6 ((𝑥 ∈ ω ∧ 𝐴 ∈ On ∧ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → (𝐴 +no suc 𝑥) = suc (𝐴 +no 𝑥))
27 onasuc 8566 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
2827ancoms 458 . . . . . . 7 ((𝑥 ∈ ω ∧ 𝐴 ∈ On) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
29283adant3 1133 . . . . . 6 ((𝑥 ∈ ω ∧ 𝐴 ∈ On ∧ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
3021, 26, 293eqtr4d 2787 . . . . 5 ((𝑥 ∈ ω ∧ 𝐴 ∈ On ∧ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → (𝐴 +no suc 𝑥) = (𝐴 +o suc 𝑥))
31303exp 1120 . . . 4 (𝑥 ∈ ω → (𝐴 ∈ On → ((𝐴 +no 𝑥) = (𝐴 +o 𝑥) → (𝐴 +no suc 𝑥) = (𝐴 +o suc 𝑥))))
3231a2d 29 . . 3 (𝑥 ∈ ω → ((𝐴 ∈ On → (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → (𝐴 ∈ On → (𝐴 +no suc 𝑥) = (𝐴 +o suc 𝑥))))
334, 8, 12, 16, 19, 32finds 7918 . 2 (𝐵 ∈ ω → (𝐴 ∈ On → (𝐴 +no 𝐵) = (𝐴 +o 𝐵)))
3433impcom 407 1 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +no 𝐵) = (𝐴 +o 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  c0 4333  Oncon0 6384  suc csuc 6386  (class class class)co 7431  ωcom 7887   +o coa 8503   +no cnadd 8703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-oadd 8510  df-nadd 8704
This theorem is referenced by:  omnaddcl  8741
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