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Mirrors > Home > ILE Home > Th. List > nnm1 | GIF version |
Description: Multiply an element of ω by 1o. (Contributed by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
nnm1 | ⊢ (𝐴 ∈ ω → (𝐴 ·o 1o) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-1o 6313 | . . 3 ⊢ 1o = suc ∅ | |
2 | 1 | oveq2i 5785 | . 2 ⊢ (𝐴 ·o 1o) = (𝐴 ·o suc ∅) |
3 | peano1 4508 | . . . 4 ⊢ ∅ ∈ ω | |
4 | nnmsuc 6373 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ ∅ ∈ ω) → (𝐴 ·o suc ∅) = ((𝐴 ·o ∅) +o 𝐴)) | |
5 | 3, 4 | mpan2 421 | . . 3 ⊢ (𝐴 ∈ ω → (𝐴 ·o suc ∅) = ((𝐴 ·o ∅) +o 𝐴)) |
6 | nnm0 6371 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅) | |
7 | 6 | oveq1d 5789 | . . 3 ⊢ (𝐴 ∈ ω → ((𝐴 ·o ∅) +o 𝐴) = (∅ +o 𝐴)) |
8 | nna0r 6374 | . . 3 ⊢ (𝐴 ∈ ω → (∅ +o 𝐴) = 𝐴) | |
9 | 5, 7, 8 | 3eqtrd 2176 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 ·o suc ∅) = 𝐴) |
10 | 2, 9 | syl5eq 2184 | 1 ⊢ (𝐴 ∈ ω → (𝐴 ·o 1o) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ∅c0 3363 suc csuc 4287 ωcom 4504 (class class class)co 5774 1oc1o 6306 +o coa 6310 ·o comu 6311 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-oadd 6317 df-omul 6318 |
This theorem is referenced by: nnm2 6421 mulidpi 7126 archnqq 7225 nq0a0 7265 nq02m 7273 |
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