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Theorem naddoa 8738
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 7438 . . . . 5 (𝑦 = ∅ → (𝐴 +no 𝑦) = (𝐴 +no ∅))
2 oveq2 7438 . . . . 5 (𝑦 = ∅ → (𝐴 +o 𝑦) = (𝐴 +o ∅))
31, 2eqeq12d 2750 . . . 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 7438 . . . . 5 (𝑦 = 𝑥 → (𝐴 +no 𝑦) = (𝐴 +no 𝑥))
6 oveq2 7438 . . . . 5 (𝑦 = 𝑥 → (𝐴 +o 𝑦) = (𝐴 +o 𝑥))
75, 6eqeq12d 2750 . . . 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 7438 . . . . 5 (𝑦 = suc 𝑥 → (𝐴 +no 𝑦) = (𝐴 +no suc 𝑥))
10 oveq2 7438 . . . . 5 (𝑦 = suc 𝑥 → (𝐴 +o 𝑦) = (𝐴 +o suc 𝑥))
119, 10eqeq12d 2750 . . . 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 7438 . . . . 5 (𝑦 = 𝐵 → (𝐴 +no 𝑦) = (𝐴 +no 𝐵))
14 oveq2 7438 . . . . 5 (𝑦 = 𝐵 → (𝐴 +o 𝑦) = (𝐴 +o 𝐵))
1513, 14eqeq12d 2750 . . . 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 8719 . . . 4 (𝐴 ∈ On → (𝐴 +no ∅) = 𝐴)
18 oa0 8552 . . . 4 (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
1917, 18eqtr4d 2777 . . 3 (𝐴 ∈ On → (𝐴 +no ∅) = (𝐴 +o ∅))
20 suceq 6451 . . . . . . 7 ((𝐴 +no 𝑥) = (𝐴 +o 𝑥) → suc (𝐴 +no 𝑥) = suc (𝐴 +o 𝑥))
21203ad2ant3 1134 . . . . . 6 ((𝑥 ∈ ω ∧ 𝐴 ∈ On ∧ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → suc (𝐴 +no 𝑥) = suc (𝐴 +o 𝑥))
22 nnon 7892 . . . . . . . . 9 (𝑥 ∈ ω → 𝑥 ∈ On)
23 naddsuc2 8737 . . . . . . . . 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 1131 . . . . . 6 ((𝑥 ∈ ω ∧ 𝐴 ∈ On ∧ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → (𝐴 +no suc 𝑥) = suc (𝐴 +no 𝑥))
27 onasuc 8564 . . . . . . . 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 1131 . . . . . 6 ((𝑥 ∈ ω ∧ 𝐴 ∈ On ∧ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
3021, 26, 293eqtr4d 2784 . . . . 5 ((𝑥 ∈ ω ∧ 𝐴 ∈ On ∧ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → (𝐴 +no suc 𝑥) = (𝐴 +o suc 𝑥))
31303exp 1118 . . . 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 1086   = wceq 1536  wcel 2105  c0 4338  Oncon0 6385  suc csuc 6387  (class class class)co 7430  ωcom 7886   +o coa 8501   +no cnadd 8701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-oadd 8508  df-nadd 8702
This theorem is referenced by:  omnaddcl  8739
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