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Theorem om1 8160
Description: Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. (Contributed by NM, 29-Oct-1995.)
Assertion
Ref Expression
om1 (𝐴 ∈ On → (𝐴 ·o 1o) = 𝐴)

Proof of Theorem om1
StepHypRef Expression
1 df-1o 8094 . . . 4 1o = suc ∅
21oveq2i 7159 . . 3 (𝐴 ·o 1o) = (𝐴 ·o suc ∅)
3 peano1 7593 . . . 4 ∅ ∈ ω
4 onmsuc 8146 . . . 4 ((𝐴 ∈ On ∧ ∅ ∈ ω) → (𝐴 ·o suc ∅) = ((𝐴 ·o ∅) +o 𝐴))
53, 4mpan2 689 . . 3 (𝐴 ∈ On → (𝐴 ·o suc ∅) = ((𝐴 ·o ∅) +o 𝐴))
62, 5syl5eq 2866 . 2 (𝐴 ∈ On → (𝐴 ·o 1o) = ((𝐴 ·o ∅) +o 𝐴))
7 om0 8134 . . 3 (𝐴 ∈ On → (𝐴 ·o ∅) = ∅)
87oveq1d 7163 . 2 (𝐴 ∈ On → ((𝐴 ·o ∅) +o 𝐴) = (∅ +o 𝐴))
9 oa0r 8155 . 2 (𝐴 ∈ On → (∅ +o 𝐴) = 𝐴)
106, 8, 93eqtrd 2858 1 (𝐴 ∈ On → (𝐴 ·o 1o) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1531  wcel 2108  c0 4289  Oncon0 6184  suc csuc 6186  (class class class)co 7148  ωcom 7572  1oc1o 8087   +o coa 8091   ·o comu 8092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-omul 8099
This theorem is referenced by:  oe1m  8163  omword1  8191  oeordi  8205  oeoalem  8214  oeoa  8215  oeeui  8220  oaabs2  8264  infxpenc  9436
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