Theorem naddoa 8627

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.

Step	Hyp	Ref	Expression
1		oveq2 7364	. . . . 5 ⊢ (𝑦 = ∅ → (𝐴 +no 𝑦) = (𝐴 +no ∅))
2		oveq2 7364	. . . . 5 ⊢ (𝑦 = ∅ → (𝐴 +o 𝑦) = (𝐴 +o ∅))
3	1, 2	eqeq12d 2751	. . . 4 ⊢ (𝑦 = ∅ → ((𝐴 +no 𝑦) = (𝐴 +o 𝑦) ↔ (𝐴 +no ∅) = (𝐴 +o ∅)))
4	3	imbi2d 340	. . 3 ⊢ (𝑦 = ∅ → ((𝐴 ∈ On → (𝐴 +no 𝑦) = (𝐴 +o 𝑦)) ↔ (𝐴 ∈ On → (𝐴 +no ∅) = (𝐴 +o ∅))))
5		oveq2 7364	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝐴 +no 𝑦) = (𝐴 +no 𝑥))
6		oveq2 7364	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝐴 +o 𝑦) = (𝐴 +o 𝑥))
7	5, 6	eqeq12d 2751	. . . 4 ⊢ (𝑦 = 𝑥 → ((𝐴 +no 𝑦) = (𝐴 +o 𝑦) ↔ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)))
8	7	imbi2d 340	. . 3 ⊢ (𝑦 = 𝑥 → ((𝐴 ∈ On → (𝐴 +no 𝑦) = (𝐴 +o 𝑦)) ↔ (𝐴 ∈ On → (𝐴 +no 𝑥) = (𝐴 +o 𝑥))))
9		oveq2 7364	. . . . 5 ⊢ (𝑦 = suc 𝑥 → (𝐴 +no 𝑦) = (𝐴 +no suc 𝑥))
10		oveq2 7364	. . . . 5 ⊢ (𝑦 = suc 𝑥 → (𝐴 +o 𝑦) = (𝐴 +o suc 𝑥))
11	9, 10	eqeq12d 2751	. . . 4 ⊢ (𝑦 = suc 𝑥 → ((𝐴 +no 𝑦) = (𝐴 +o 𝑦) ↔ (𝐴 +no suc 𝑥) = (𝐴 +o suc 𝑥)))
12	11	imbi2d 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 7364	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 +no 𝑦) = (𝐴 +no 𝐵))
14		oveq2 7364	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 +o 𝑦) = (𝐴 +o 𝐵))
15	13, 14	eqeq12d 2751	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 +no 𝑦) = (𝐴 +o 𝑦) ↔ (𝐴 +no 𝐵) = (𝐴 +o 𝐵)))
16	15	imbi2d 340	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ On → (𝐴 +no 𝑦) = (𝐴 +o 𝑦)) ↔ (𝐴 ∈ On → (𝐴 +no 𝐵) = (𝐴 +o 𝐵))))
17		naddrid 8608	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 +no ∅) = 𝐴)
18		oa0 8440	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
19	17, 18	eqtr4d 2773	. . 3 ⊢ (𝐴 ∈ On → (𝐴 +no ∅) = (𝐴 +o ∅))
20		suceq 6380	. . . . . . 7 ⊢ ((𝐴 +no 𝑥) = (𝐴 +o 𝑥) → suc (𝐴 +no 𝑥) = suc (𝐴 +o 𝑥))
21	20	3ad2ant3 1136	. . . . . 6 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ On ∧ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → suc (𝐴 +no 𝑥) = suc (𝐴 +o 𝑥))
22		nnon 7812	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On)
23		naddsuc2 8626	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +no suc 𝑥) = suc (𝐴 +no 𝑥))
24	22, 23	sylan2 594	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴 +no suc 𝑥) = suc (𝐴 +no 𝑥))
25	24	ancoms 458	. . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ On) → (𝐴 +no suc 𝑥) = suc (𝐴 +no 𝑥))
26	25	3adant3 1133	. . . . . 6 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ On ∧ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → (𝐴 +no suc 𝑥) = suc (𝐴 +no 𝑥))
27		onasuc 8452	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
28	27	ancoms 458	. . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ On) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
29	28	3adant3 1133	. . . . . 6 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ On ∧ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
30	21, 26, 29	3eqtr4d 2780	. . . . 5 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ On ∧ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → (𝐴 +no suc 𝑥) = (𝐴 +o suc 𝑥))
31	30	3exp 1120	. . . 4 ⊢ (𝑥 ∈ ω → (𝐴 ∈ On → ((𝐴 +no 𝑥) = (𝐴 +o 𝑥) → (𝐴 +no suc 𝑥) = (𝐴 +o suc 𝑥))))
32	31	a2d 29	. . 3 ⊢ (𝑥 ∈ ω → ((𝐴 ∈ On → (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → (𝐴 ∈ On → (𝐴 +no suc 𝑥) = (𝐴 +o suc 𝑥))))
33	4, 8, 12, 16, 19, 32	finds 7836	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ On → (𝐴 +no 𝐵) = (𝐴 +o 𝐵)))
34	33	impcom 407	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +no 𝐵) = (𝐴 +o 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∅c0 4263  Oncon0 6312  suc csuc 6314  (class class class)co 7356  ωcom 7806   +_o coa 8391   +no cnadd 8590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-se 5574  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-oadd 8398  df-nadd 8591

This theorem is referenced by:  omnaddcl  8628