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Theorem nnm2 8263
 Description: Multiply an element of ω by 2o. (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
nnm2 (𝐴 ∈ ω → (𝐴 ·o 2o) = (𝐴 +o 𝐴))

Proof of Theorem nnm2
StepHypRef Expression
1 df-2o 8090 . . 3 2o = suc 1o
21oveq2i 7150 . 2 (𝐴 ·o 2o) = (𝐴 ·o suc 1o)
3 1onn 8252 . . . 4 1o ∈ ω
4 nnmsuc 8220 . . . 4 ((𝐴 ∈ ω ∧ 1o ∈ ω) → (𝐴 ·o suc 1o) = ((𝐴 ·o 1o) +o 𝐴))
53, 4mpan2 690 . . 3 (𝐴 ∈ ω → (𝐴 ·o suc 1o) = ((𝐴 ·o 1o) +o 𝐴))
6 nnm1 8262 . . . 4 (𝐴 ∈ ω → (𝐴 ·o 1o) = 𝐴)
76oveq1d 7154 . . 3 (𝐴 ∈ ω → ((𝐴 ·o 1o) +o 𝐴) = (𝐴 +o 𝐴))
85, 7eqtrd 2836 . 2 (𝐴 ∈ ω → (𝐴 ·o suc 1o) = (𝐴 +o 𝐴))
92, 8syl5eq 2848 1 (𝐴 ∈ ω → (𝐴 ·o 2o) = (𝐴 +o 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2112  suc csuc 6165  (class class class)co 7139  ωcom 7564  1oc1o 8082  2oc2o 8083   +o coa 8086   ·o comu 8087 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-omul 8094 This theorem is referenced by:  nn2m  8264  omopthlem1  8269  omopthlem2  8270
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