Theorem oevn0 8516

Description: Value of ordinal exponentiation at a nonzero base. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
oevn0	⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem oevn0

Step	Hyp	Ref	Expression
1		on0eln0 6419	. . . . 5 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
2		df-ne 2957	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
3	1, 2	bitrdi 290	. . . 4 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ ¬ 𝐴 = ∅))
4	3	adantr 486	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐴 ↔ ¬ 𝐴 = ∅))
5		oev 8515	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) = if(𝐴 = ∅, (1o ∖ 𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))
6		iffalse 4491	. . . . 5 ⊢ (¬ 𝐴 = ∅ → if(𝐴 = ∅, (1o ∖ 𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
7	5, 6	sylan9eq 2816	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ¬ 𝐴 = ∅) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
8	7	ex 418	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴 = ∅ → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))
9	4, 8	sylbid 243	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐴 → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))
10	9	imp 412	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  Vcvv 3451   ∖ cdif 3896  ∅c0 4279  ifcif 4482   ↦ cmpt 5186  Oncon0 6361  ‘cfv 6537  (class class class)co 7418  reccrdg 8410  1_oc1o 8462   ·_o comu 8467   ↑_o coe 8468

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-oexp 8475

This theorem is used by:  oe0  8523  oev2  8524  oesuclem  8526  oelim  8535