Theorem naddoa 8712

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 7411	. . . . 5 ⊢ (𝑦 = ∅ → (𝐴 +no 𝑦) = (𝐴 +no ∅))
2		oveq2 7411	. . . . 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 7411	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝐴 +no 𝑦) = (𝐴 +no 𝑥))
6		oveq2 7411	. . . . 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 7411	. . . . 5 ⊢ (𝑦 = suc 𝑥 → (𝐴 +no 𝑦) = (𝐴 +no suc 𝑥))
10		oveq2 7411	. . . . 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 7411	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 +no 𝑦) = (𝐴 +no 𝐵))
14		oveq2 7411	. . . . 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 8693	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 +no ∅) = 𝐴)
18		oa0 8526	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
19	17, 18	eqtr4d 2773	. . 3 ⊢ (𝐴 ∈ On → (𝐴 +no ∅) = (𝐴 +o ∅))
20		suceq 6419	. . . . . . 7 ⊢ ((𝐴 +no 𝑥) = (𝐴 +o 𝑥) → suc (𝐴 +no 𝑥) = suc (𝐴 +o 𝑥))
21	20	3ad2ant3 1135	. . . . . 6 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ On ∧ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → suc (𝐴 +no 𝑥) = suc (𝐴 +o 𝑥))
22		nnon 7865	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On)
23		naddsuc2 8711	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +no suc 𝑥) = suc (𝐴 +no 𝑥))
24	22, 23	sylan2 593	. . . . . . . 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 1132	. . . . . 6 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ On ∧ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → (𝐴 +no suc 𝑥) = suc (𝐴 +no 𝑥))
27		onasuc 8538	. . . . . . . 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 1132	. . . . . 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 1119	. . . 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 7890	. 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 1086   = wceq 1540   ∈ wcel 2108  ∅c0 4308  Oncon0 6352  suc csuc 6354  (class class class)co 7403  ωcom 7859   +_o coa 8475   +no cnadd 8675

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-oadd 8482  df-nadd 8676

This theorem is referenced by:  omnaddcl  8713