Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  oevn0 Structured version   Visualization version   GIF version

Theorem oevn0 7748
 Description: Value of ordinal exponentiation at a nonzero mantissa. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
oevn0 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem oevn0
StepHypRef Expression
1 on0eln0 5923 . . . . 5 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
2 df-ne 2943 . . . . 5 (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
31, 2syl6bb 276 . . . 4 (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ ¬ 𝐴 = ∅))
43adantr 466 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐴 ↔ ¬ 𝐴 = ∅))
5 oev 7747 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) = if(𝐴 = ∅, (1𝑜𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)))
6 iffalse 4232 . . . . 5 𝐴 = ∅ → if(𝐴 = ∅, (1𝑜𝐵), (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
75, 6sylan9eq 2824 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ¬ 𝐴 = ∅) → (𝐴𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
87ex 397 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐴 = ∅ → (𝐴𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)))
94, 8sylbid 230 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐴 → (𝐴𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)))
109imp 393 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1630   ∈ wcel 2144   ≠ wne 2942  Vcvv 3349   ∖ cdif 3718  ∅c0 4061  ifcif 4223   ↦ cmpt 4861  Oncon0 5866  ‘cfv 6031  (class class class)co 6792  reccrdg 7657  1𝑜c1o 7705   ·𝑜 comu 7710   ↑𝑜 coe 7711 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oexp 7718 This theorem is referenced by:  oe0  7755  oev2  7756  oesuclem  7758  oelim  7767
 Copyright terms: Public domain W3C validator