Theorem naddonnn 43427

Description: Natural addition with a natural number on the right results in a value equal to that of ordinal addition. (Contributed by RP, 1-Jan-2025.)

Assertion

Ref	Expression
naddonnn	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) = (𝐴 +no 𝐵))

Proof of Theorem naddonnn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7354	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
2		oveq2 7354	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 +no 𝑥) = (𝐴 +no ∅))
3	1, 2	eqeq12d 2747	. . . 4 ⊢ (𝑥 = ∅ → ((𝐴 +o 𝑥) = (𝐴 +no 𝑥) ↔ (𝐴 +o ∅) = (𝐴 +no ∅)))
4	3	imbi2d 340	. . 3 ⊢ (𝑥 = ∅ → ((𝐴 ∈ On → (𝐴 +o 𝑥) = (𝐴 +no 𝑥)) ↔ (𝐴 ∈ On → (𝐴 +o ∅) = (𝐴 +no ∅))))
5		oveq2 7354	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
6		oveq2 7354	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 +no 𝑥) = (𝐴 +no 𝑦))
7	5, 6	eqeq12d 2747	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 +o 𝑥) = (𝐴 +no 𝑥) ↔ (𝐴 +o 𝑦) = (𝐴 +no 𝑦)))
8	7	imbi2d 340	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ On → (𝐴 +o 𝑥) = (𝐴 +no 𝑥)) ↔ (𝐴 ∈ On → (𝐴 +o 𝑦) = (𝐴 +no 𝑦))))
9		oveq2 7354	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
10		oveq2 7354	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 +no 𝑥) = (𝐴 +no suc 𝑦))
11	9, 10	eqeq12d 2747	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((𝐴 +o 𝑥) = (𝐴 +no 𝑥) ↔ (𝐴 +o suc 𝑦) = (𝐴 +no suc 𝑦)))
12	11	imbi2d 340	. . 3 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ∈ On → (𝐴 +o 𝑥) = (𝐴 +no 𝑥)) ↔ (𝐴 ∈ On → (𝐴 +o suc 𝑦) = (𝐴 +no suc 𝑦))))
13		oveq2 7354	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵))
14		oveq2 7354	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 +no 𝑥) = (𝐴 +no 𝐵))
15	13, 14	eqeq12d 2747	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 +o 𝑥) = (𝐴 +no 𝑥) ↔ (𝐴 +o 𝐵) = (𝐴 +no 𝐵)))
16	15	imbi2d 340	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ On → (𝐴 +o 𝑥) = (𝐴 +no 𝑥)) ↔ (𝐴 ∈ On → (𝐴 +o 𝐵) = (𝐴 +no 𝐵))))
17		oa0 8431	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
18		naddrid 8598	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 +no ∅) = 𝐴)
19	17, 18	eqtr4d 2769	. . 3 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = (𝐴 +no ∅))
20		nnon 7802	. . . . 5 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
21		suceq 6374	. . . . . . . . 9 ⊢ ((𝐴 +o 𝑦) = (𝐴 +no 𝑦) → suc (𝐴 +o 𝑦) = suc (𝐴 +no 𝑦))
22	21	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 +o 𝑦) = (𝐴 +no 𝑦)) → suc (𝐴 +o 𝑦) = suc (𝐴 +no 𝑦))
23		oasuc 8439	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
24	23	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 +o 𝑦) = (𝐴 +no 𝑦)) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
25		naddsuc2 8616	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +no suc 𝑦) = suc (𝐴 +no 𝑦))
26	25	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 +o 𝑦) = (𝐴 +no 𝑦)) → (𝐴 +no suc 𝑦) = suc (𝐴 +no 𝑦))
27	22, 24, 26	3eqtr4d 2776	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 +o 𝑦) = (𝐴 +no 𝑦)) → (𝐴 +o suc 𝑦) = (𝐴 +no suc 𝑦))
28	27	ex 412	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑦) = (𝐴 +no 𝑦) → (𝐴 +o suc 𝑦) = (𝐴 +no suc 𝑦)))
29	28	expcom 413	. . . . 5 ⊢ (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 +o 𝑦) = (𝐴 +no 𝑦) → (𝐴 +o suc 𝑦) = (𝐴 +no suc 𝑦))))
30	20, 29	syl 17	. . . 4 ⊢ (𝑦 ∈ ω → (𝐴 ∈ On → ((𝐴 +o 𝑦) = (𝐴 +no 𝑦) → (𝐴 +o suc 𝑦) = (𝐴 +no suc 𝑦))))
31	30	a2d 29	. . 3 ⊢ (𝑦 ∈ ω → ((𝐴 ∈ On → (𝐴 +o 𝑦) = (𝐴 +no 𝑦)) → (𝐴 ∈ On → (𝐴 +o suc 𝑦) = (𝐴 +no suc 𝑦))))
32	4, 8, 12, 16, 19, 31	finds 7826	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ On → (𝐴 +o 𝐵) = (𝐴 +no 𝐵)))
33	32	impcom 407	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) = (𝐴 +no 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∅c0 4283  Oncon0 6306  suc csuc 6308  (class class class)co 7346  ωcom 7796   +_o coa 8382   +no cnadd 8580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-se 5570  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-oadd 8389  df-nadd 8581

This theorem is referenced by:  naddwordnexlem3  43431  naddwordnexlem4  43433