Theorem naddonnn 42722

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 7413	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
2		oveq2 7413	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 +no 𝑥) = (𝐴 +no ∅))
3	1, 2	eqeq12d 2742	. . . 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 7413	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
6		oveq2 7413	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 +no 𝑥) = (𝐴 +no 𝑦))
7	5, 6	eqeq12d 2742	. . . 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 7413	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
10		oveq2 7413	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 +no 𝑥) = (𝐴 +no suc 𝑦))
11	9, 10	eqeq12d 2742	. . . 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 7413	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵))
14		oveq2 7413	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 +no 𝑥) = (𝐴 +no 𝐵))
15	13, 14	eqeq12d 2742	. . . 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 8517	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
18		naddrid 8684	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 +no ∅) = 𝐴)
19	17, 18	eqtr4d 2769	. . 3 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = (𝐴 +no ∅))
20		nnon 7858	. . . . 5 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
21		suceq 6424	. . . . . . . . 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 8525	. . . . . . . . 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 42719	. . . . . . . . 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 7886	. 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 1533   ∈ wcel 2098  ∅c0 4317  Oncon0 6358  suc csuc 6360  (class class class)co 7405  ωcom 7852   +_o coa 8464   +no cnadd 8666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-oadd 8471  df-nadd 8667

This theorem is referenced by:  naddwordnexlem3  42726  naddwordnexlem4  42728