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Theorem oe0m 7543
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 5737 . . 3 ∅ ∈ On
2 oev 7539 . . 3 ((∅ ∈ On ∧ 𝐴 ∈ On) → (∅ ↑𝑜 𝐴) = if(∅ = ∅, (1𝑜𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 ∅)), 1𝑜)‘𝐴)))
31, 2mpan 705 . 2 (𝐴 ∈ On → (∅ ↑𝑜 𝐴) = if(∅ = ∅, (1𝑜𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 ∅)), 1𝑜)‘𝐴)))
4 eqid 2621 . . 3 ∅ = ∅
54iftruei 4065 . 2 if(∅ = ∅, (1𝑜𝐴), (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 ∅)), 1𝑜)‘𝐴)) = (1𝑜𝐴)
63, 5syl6eq 2671 1 (𝐴 ∈ On → (∅ ↑𝑜 𝐴) = (1𝑜𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3186  cdif 3552  c0 3891  ifcif 4058  cmpt 4673  Oncon0 5682  cfv 5847  (class class class)co 6604  reccrdg 7450  1𝑜c1o 7498   ·𝑜 comu 7503  𝑜 coe 7504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-suc 5688  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oexp 7511
This theorem is referenced by:  oe0m0  7545  oe0m1  7546  cantnflem2  8531
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