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Mirrors > Home > ILE Home > Th. List > nnm2 | GIF version |
Description: Multiply an element of ω by 2o. (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
nnm2 | ⊢ (𝐴 ∈ ω → (𝐴 ·o 2o) = (𝐴 +o 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2o 6377 | . . 3 ⊢ 2o = suc 1o | |
2 | 1 | oveq2i 5848 | . 2 ⊢ (𝐴 ·o 2o) = (𝐴 ·o suc 1o) |
3 | 1onn 6480 | . . . 4 ⊢ 1o ∈ ω | |
4 | nnmsuc 6437 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ ω) → (𝐴 ·o suc 1o) = ((𝐴 ·o 1o) +o 𝐴)) | |
5 | 3, 4 | mpan2 422 | . . 3 ⊢ (𝐴 ∈ ω → (𝐴 ·o suc 1o) = ((𝐴 ·o 1o) +o 𝐴)) |
6 | nnm1 6484 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ·o 1o) = 𝐴) | |
7 | 6 | oveq1d 5852 | . . 3 ⊢ (𝐴 ∈ ω → ((𝐴 ·o 1o) +o 𝐴) = (𝐴 +o 𝐴)) |
8 | 5, 7 | eqtrd 2197 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 ·o suc 1o) = (𝐴 +o 𝐴)) |
9 | 2, 8 | syl5eq 2209 | 1 ⊢ (𝐴 ∈ ω → (𝐴 ·o 2o) = (𝐴 +o 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 suc csuc 4338 ωcom 4562 (class class class)co 5837 1oc1o 6369 2oc2o 6370 +o coa 6373 ·o comu 6374 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4092 ax-sep 4095 ax-nul 4103 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-iinf 4560 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-iun 3863 df-br 3978 df-opab 4039 df-mpt 4040 df-tr 4076 df-id 4266 df-iord 4339 df-on 4341 df-suc 4344 df-iom 4563 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-ov 5840 df-oprab 5841 df-mpo 5842 df-1st 6101 df-2nd 6102 df-recs 6265 df-irdg 6330 df-1o 6376 df-2o 6377 df-oadd 6380 df-omul 6381 |
This theorem is referenced by: nn2m 6486 |
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