Theorem oa0suclim 43894

Description: Closed form expression of the value of ordinal addition for the cases when the second ordinal is zero, a successor ordinal, or a limit ordinal. Definition 2.3 of [Schloeder] p. 4. See oa0 8501, oasuc 8509, and oalim 8517. (Contributed by RP, 18-Jan-2025.)

Assertion

Ref	Expression
oa0suclim	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 +o 𝐵) = 𝐴) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 +o 𝐵) = suc (𝐴 +o 𝐶)) ∧ (Lim 𝐵 → (𝐴 +o 𝐵) = ∪ 𝑐 ∈ 𝐵 (𝐴 +o 𝑐))))

Distinct variable groups:   𝐴,𝑐   𝐵,𝑐

Allowed substitution hint:   𝐶(𝑐)

Proof of Theorem oa0suclim

Step	Hyp	Ref	Expression
1		oa0 8501	. 2 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
2		oasuc 8509	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 +o suc 𝐶) = suc (𝐴 +o 𝐶))
3		oalim 8517	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (𝐴 +o 𝐵) = ∪ 𝑐 ∈ 𝐵 (𝐴 +o 𝑐))
4	3	anassrs 472	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → (𝐴 +o 𝐵) = ∪ 𝑐 ∈ 𝐵 (𝐴 +o 𝑐))
5	1, 2, 4	onov0suclim 43893	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 +o 𝐵) = 𝐴) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 +o 𝐵) = suc (𝐴 +o 𝐶)) ∧ (Lim 𝐵 → (𝐴 +o 𝐵) = ∪ 𝑐 ∈ 𝐵 (𝐴 +o 𝑐))))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∅c0 4294  ∪ ciun 4960  Oncon0 6361  Lim wlim 6362  suc csuc 6363  (class class class)co 7411   +_o coa 8450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-oadd 8457

This theorem is referenced by: (None)