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 7428	. . . . 5 ⊢ (𝑦 = ∅ → (𝐴 +no 𝑦) = (𝐴 +no ∅))
2		oveq2 7428	. . . . 5 ⊢ (𝑦 = ∅ → (𝐴 +o 𝑦) = (𝐴 +o ∅))
3	1, 2	eqeq12d 2776	. . . 4 ⊢ (𝑦 = ∅ → ((𝐴 +no 𝑦) = (𝐴 +o 𝑦) ↔ (𝐴 +no ∅) = (𝐴 +o ∅)))
4	3	imbi2d 343	. . 3 ⊢ (𝑦 = ∅ → ((𝐴 ∈ On → (𝐴 +no 𝑦) = (𝐴 +o 𝑦)) ↔ (𝐴 ∈ On → (𝐴 +no ∅) = (𝐴 +o ∅))))
5		oveq2 7428	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝐴 +no 𝑦) = (𝐴 +no 𝑥))
6		oveq2 7428	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝐴 +o 𝑦) = (𝐴 +o 𝑥))
7	5, 6	eqeq12d 2776	. . . 4 ⊢ (𝑦 = 𝑥 → ((𝐴 +no 𝑦) = (𝐴 +o 𝑦) ↔ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)))
8	7	imbi2d 343	. . 3 ⊢ (𝑦 = 𝑥 → ((𝐴 ∈ On → (𝐴 +no 𝑦) = (𝐴 +o 𝑦)) ↔ (𝐴 ∈ On → (𝐴 +no 𝑥) = (𝐴 +o 𝑥))))
9		oveq2 7428	. . . . 5 ⊢ (𝑦 = suc 𝑥 → (𝐴 +no 𝑦) = (𝐴 +no suc 𝑥))
10		oveq2 7428	. . . . 5 ⊢ (𝑦 = suc 𝑥 → (𝐴 +o 𝑦) = (𝐴 +o suc 𝑥))
11	9, 10	eqeq12d 2776	. . . 4 ⊢ (𝑦 = suc 𝑥 → ((𝐴 +no 𝑦) = (𝐴 +o 𝑦) ↔ (𝐴 +no suc 𝑥) = (𝐴 +o suc 𝑥)))
12	11	imbi2d 343	. . 3 ⊢ (𝑦 = suc 𝑥 → ((𝐴 ∈ On → (𝐴 +no 𝑦) = (𝐴 +o 𝑦)) ↔ (𝐴 ∈ On → (𝐴 +no suc 𝑥) = (𝐴 +o suc 𝑥))))
13		oveq2 7428	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 +no 𝑦) = (𝐴 +no 𝐵))
14		oveq2 7428	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 +o 𝑦) = (𝐴 +o 𝐵))
15	13, 14	eqeq12d 2776	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 +no 𝑦) = (𝐴 +o 𝑦) ↔ (𝐴 +no 𝐵) = (𝐴 +o 𝐵)))
16	15	imbi2d 343	. . 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 8524	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
19	17, 18	eqtr4d 2798	. . 3 ⊢ (𝐴 ∈ On → (𝐴 +no ∅) = (𝐴 +o ∅))
20		suceq 6431	. . . . . . 7 ⊢ ((𝐴 +no 𝑥) = (𝐴 +o 𝑥) → suc (𝐴 +no 𝑥) = suc (𝐴 +o 𝑥))
21	20	3ad2ant3 1153	. . . . . 6 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ On ∧ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → suc (𝐴 +no 𝑥) = suc (𝐴 +o 𝑥))
22		nnon 7883	. . . . . . . . 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 605	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴 +no suc 𝑥) = suc (𝐴 +no 𝑥))
25	24	ancoms 464	. . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ On) → (𝐴 +no suc 𝑥) = suc (𝐴 +no 𝑥))
26	25	3adant3 1150	. . . . . 6 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ On ∧ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → (𝐴 +no suc 𝑥) = suc (𝐴 +no 𝑥))
27		onasuc 8536	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
28	27	ancoms 464	. . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ On) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
29	28	3adant3 1150	. . . . . 6 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ On ∧ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
30	21, 26, 29	3eqtr4d 2805	. . . . 5 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ On ∧ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → (𝐴 +no suc 𝑥) = (𝐴 +o suc 𝑥))
31	30	3exp 1137	. . . 4 ⊢ (𝑥 ∈ ω → (𝐴 ∈ On → ((𝐴 +no 𝑥) = (𝐴 +o 𝑥) → (𝐴 +no suc 𝑥) = (𝐴 +o suc 𝑥))))
32	31	a2d 30	. . 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 7908	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ On → (𝐴 +no 𝐵) = (𝐴 +o 𝐵)))
34	33	impcom 413	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +no 𝐵) = (𝐴 +o 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∅c0 4278  Oncon0 6362  suc csuc 6364  (class class class)co 7420  ωcom 7877   +_o coa 8473   +no cnadd 8674

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-oadd 8480  df-nadd 8675

This theorem is used by:  omnaddcl  8713