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Theorem naddonnn 43413
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.
StepHypRef Expression
1 oveq2 7440 . . . . 5 (𝑥 = ∅ → (𝐴 +o 𝑥) = (𝐴 +o ∅))
2 oveq2 7440 . . . . 5 (𝑥 = ∅ → (𝐴 +no 𝑥) = (𝐴 +no ∅))
31, 2eqeq12d 2752 . . . 4 (𝑥 = ∅ → ((𝐴 +o 𝑥) = (𝐴 +no 𝑥) ↔ (𝐴 +o ∅) = (𝐴 +no ∅)))
43imbi2d 340 . . 3 (𝑥 = ∅ → ((𝐴 ∈ On → (𝐴 +o 𝑥) = (𝐴 +no 𝑥)) ↔ (𝐴 ∈ On → (𝐴 +o ∅) = (𝐴 +no ∅))))
5 oveq2 7440 . . . . 5 (𝑥 = 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o 𝑦))
6 oveq2 7440 . . . . 5 (𝑥 = 𝑦 → (𝐴 +no 𝑥) = (𝐴 +no 𝑦))
75, 6eqeq12d 2752 . . . 4 (𝑥 = 𝑦 → ((𝐴 +o 𝑥) = (𝐴 +no 𝑥) ↔ (𝐴 +o 𝑦) = (𝐴 +no 𝑦)))
87imbi2d 340 . . 3 (𝑥 = 𝑦 → ((𝐴 ∈ On → (𝐴 +o 𝑥) = (𝐴 +no 𝑥)) ↔ (𝐴 ∈ On → (𝐴 +o 𝑦) = (𝐴 +no 𝑦))))
9 oveq2 7440 . . . . 5 (𝑥 = suc 𝑦 → (𝐴 +o 𝑥) = (𝐴 +o suc 𝑦))
10 oveq2 7440 . . . . 5 (𝑥 = suc 𝑦 → (𝐴 +no 𝑥) = (𝐴 +no suc 𝑦))
119, 10eqeq12d 2752 . . . 4 (𝑥 = suc 𝑦 → ((𝐴 +o 𝑥) = (𝐴 +no 𝑥) ↔ (𝐴 +o suc 𝑦) = (𝐴 +no suc 𝑦)))
1211imbi2d 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 7440 . . . . 5 (𝑥 = 𝐵 → (𝐴 +o 𝑥) = (𝐴 +o 𝐵))
14 oveq2 7440 . . . . 5 (𝑥 = 𝐵 → (𝐴 +no 𝑥) = (𝐴 +no 𝐵))
1513, 14eqeq12d 2752 . . . 4 (𝑥 = 𝐵 → ((𝐴 +o 𝑥) = (𝐴 +no 𝑥) ↔ (𝐴 +o 𝐵) = (𝐴 +no 𝐵)))
1615imbi2d 340 . . 3 (𝑥 = 𝐵 → ((𝐴 ∈ On → (𝐴 +o 𝑥) = (𝐴 +no 𝑥)) ↔ (𝐴 ∈ On → (𝐴 +o 𝐵) = (𝐴 +no 𝐵))))
17 oa0 8555 . . . 4 (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
18 naddrid 8722 . . . 4 (𝐴 ∈ On → (𝐴 +no ∅) = 𝐴)
1917, 18eqtr4d 2779 . . 3 (𝐴 ∈ On → (𝐴 +o ∅) = (𝐴 +no ∅))
20 nnon 7894 . . . . 5 (𝑦 ∈ ω → 𝑦 ∈ On)
21 suceq 6449 . . . . . . . . 9 ((𝐴 +o 𝑦) = (𝐴 +no 𝑦) → suc (𝐴 +o 𝑦) = suc (𝐴 +no 𝑦))
2221adantl 481 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 +o 𝑦) = (𝐴 +no 𝑦)) → suc (𝐴 +o 𝑦) = suc (𝐴 +no 𝑦))
23 oasuc 8563 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
2423adantr 480 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 +o 𝑦) = (𝐴 +no 𝑦)) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
25 naddsuc2 8740 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +no suc 𝑦) = suc (𝐴 +no 𝑦))
2625adantr 480 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 +o 𝑦) = (𝐴 +no 𝑦)) → (𝐴 +no suc 𝑦) = suc (𝐴 +no 𝑦))
2722, 24, 263eqtr4d 2786 . . . . . . 7 (((𝐴 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 +o 𝑦) = (𝐴 +no 𝑦)) → (𝐴 +o suc 𝑦) = (𝐴 +no suc 𝑦))
2827ex 412 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝑦) = (𝐴 +no 𝑦) → (𝐴 +o suc 𝑦) = (𝐴 +no suc 𝑦)))
2928expcom 413 . . . . 5 (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 +o 𝑦) = (𝐴 +no 𝑦) → (𝐴 +o suc 𝑦) = (𝐴 +no suc 𝑦))))
3020, 29syl 17 . . . 4 (𝑦 ∈ ω → (𝐴 ∈ On → ((𝐴 +o 𝑦) = (𝐴 +no 𝑦) → (𝐴 +o suc 𝑦) = (𝐴 +no suc 𝑦))))
3130a2d 29 . . 3 (𝑦 ∈ ω → ((𝐴 ∈ On → (𝐴 +o 𝑦) = (𝐴 +no 𝑦)) → (𝐴 ∈ On → (𝐴 +o suc 𝑦) = (𝐴 +no suc 𝑦))))
324, 8, 12, 16, 19, 31finds 7919 . 2 (𝐵 ∈ ω → (𝐴 ∈ On → (𝐴 +o 𝐵) = (𝐴 +no 𝐵)))
3332impcom 407 1 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) = (𝐴 +no 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  c0 4332  Oncon0 6383  suc csuc 6385  (class class class)co 7432  ωcom 7888   +o coa 8504   +no cnadd 8704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-oadd 8511  df-nadd 8705
This theorem is referenced by:  naddwordnexlem3  43417  naddwordnexlem4  43419
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