Theorem naddonnn 43357

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 7456	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
2		oveq2 7456	. . . . 5 ⊢ (𝑥 = ∅ → (𝐴 +no 𝑥) = (𝐴 +no ∅))
3	1, 2	eqeq12d 2756	. . . 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 7456	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
6		oveq2 7456	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 +no 𝑥) = (𝐴 +no 𝑦))
7	5, 6	eqeq12d 2756	. . . 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 7456	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
10		oveq2 7456	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (𝐴 +no 𝑥) = (𝐴 +no suc 𝑦))
11	9, 10	eqeq12d 2756	. . . 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 7456	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵))
14		oveq2 7456	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 +no 𝑥) = (𝐴 +no 𝐵))
15	13, 14	eqeq12d 2756	. . . 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 8572	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
18		naddrid 8739	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 +no ∅) = 𝐴)
19	17, 18	eqtr4d 2783	. . 3 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = (𝐴 +no ∅))
20		nnon 7909	. . . . 5 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
21		suceq 6461	. . . . . . . . 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 8580	. . . . . . . . 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 8757	. . . . . . . . 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 2790	. . . . . . 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 7936	. 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 2108  ∅c0 4352  Oncon0 6395  suc csuc 6397  (class class class)co 7448  ωcom 7903   +_o coa 8519   +no cnadd 8721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-oadd 8526  df-nadd 8722

This theorem is referenced by:  naddwordnexlem3  43361  naddwordnexlem4  43363