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Theorem oe0m 7767
Description: Ordinal exponentiation with zero mantissa. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
oe0m (𝐴 ∈ On → (∅ ↑𝑜 𝐴) = (1𝑜𝐴))

Proof of Theorem oe0m
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 0elon 5939 . . 3 ∅ ∈ On
2 oev 7763 . . 3 ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ↑𝑜 𝐴) = if(∅ = ∅, (1𝑜𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 ∅)), 1𝑜)‘𝐴)))
31, 2mpan 708 . 2 (𝐴 ∈ On → (∅ ↑𝑜 𝐴) = if(∅ = ∅, (1𝑜𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 ∅)), 1𝑜)‘𝐴)))
4 eqid 2760 . . 3 ∅ = ∅
54iftruei 4237 . 2 if(∅ = ∅, (1𝑜𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 ∅)), 1𝑜)‘𝐴)) = (1𝑜𝐴)
63, 5syl6eq 2810 1 (𝐴 ∈ On → (∅ ↑𝑜 𝐴) = (1𝑜𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  cdif 3712  c0 4058  ifcif 4230  cmpt 4881  Oncon0 5884  cfv 6049  (class class class)co 6813  reccrdg 7674  1𝑜c1o 7722   ·𝑜 comu 7727  𝑜 coe 7728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-suc 5890  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oexp 7735
This theorem is referenced by:  oe0m0  7769  oe0m1  7770  cantnflem2  8760
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