Theorem naddonnn 43385

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 7439	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
2		oveq2 7439	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 +no 𝑥) = (𝐴 +no ∅))
3	1, 2	eqeq12d 2751	. . . 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 7439	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
6		oveq2 7439	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 +no 𝑥) = (𝐴 +no 𝑦))
7	5, 6	eqeq12d 2751	. . . 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 7439	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
10		oveq2 7439	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 +no 𝑥) = (𝐴 +no suc 𝑦))
11	9, 10	eqeq12d 2751	. . . 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 7439	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵))
14		oveq2 7439	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 +no 𝑥) = (𝐴 +no 𝐵))
15	13, 14	eqeq12d 2751	. . . 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 8553	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
18		naddrid 8720	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 +no ∅) = 𝐴)
19	17, 18	eqtr4d 2778	. . 3 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = (𝐴 +no ∅))
20		nnon 7893	. . . . 5 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
21		suceq 6452	. . . . . . . . 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 8561	. . . . . . . . 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 8738	. . . . . . . . 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 2785	. . . . . . 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 7919	. 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 1537   ∈ wcel 2106  ∅c0 4339  Oncon0 6386  suc csuc 6388  (class class class)co 7431  ωcom 7887   +_o coa 8502   +no cnadd 8702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-oadd 8509  df-nadd 8703

This theorem is referenced by:  naddwordnexlem3  43389  naddwordnexlem4  43391