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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 6417 | . . 3 ⊢ 2o = suc 1o | |
2 | 1 | oveq2i 5885 | . 2 ⊢ (𝐴 ·o 2o) = (𝐴 ·o suc 1o) |
3 | 1onn 6520 | . . . 4 ⊢ 1o ∈ ω | |
4 | nnmsuc 6477 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 1o ∈ ω) → (𝐴 ·o suc 1o) = ((𝐴 ·o 1o) +o 𝐴)) | |
5 | 3, 4 | mpan2 425 | . . 3 ⊢ (𝐴 ∈ ω → (𝐴 ·o suc 1o) = ((𝐴 ·o 1o) +o 𝐴)) |
6 | nnm1 6525 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ·o 1o) = 𝐴) | |
7 | 6 | oveq1d 5889 | . . 3 ⊢ (𝐴 ∈ ω → ((𝐴 ·o 1o) +o 𝐴) = (𝐴 +o 𝐴)) |
8 | 5, 7 | eqtrd 2210 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 ·o suc 1o) = (𝐴 +o 𝐴)) |
9 | 2, 8 | eqtrid 2222 | 1 ⊢ (𝐴 ∈ ω → (𝐴 ·o 2o) = (𝐴 +o 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 suc csuc 4365 ωcom 4589 (class class class)co 5874 1oc1o 6409 2oc2o 6410 +o coa 6413 ·o comu 6414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-iord 4366 df-on 4368 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-irdg 6370 df-1o 6416 df-2o 6417 df-oadd 6420 df-omul 6421 |
This theorem is referenced by: nn2m 6527 |
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