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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 6321 | . . 3 ⊢ 1o = suc ∅ | |
2 | 1 | oveq2i 5793 | . 2 ⊢ (𝐴 ·o 1o) = (𝐴 ·o suc ∅) |
3 | peano1 4516 | . . . 4 ⊢ ∅ ∈ ω | |
4 | nnmsuc 6381 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ ∅ ∈ ω) → (𝐴 ·o suc ∅) = ((𝐴 ·o ∅) +o 𝐴)) | |
5 | 3, 4 | mpan2 422 | . . 3 ⊢ (𝐴 ∈ ω → (𝐴 ·o suc ∅) = ((𝐴 ·o ∅) +o 𝐴)) |
6 | nnm0 6379 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅) | |
7 | 6 | oveq1d 5797 | . . 3 ⊢ (𝐴 ∈ ω → ((𝐴 ·o ∅) +o 𝐴) = (∅ +o 𝐴)) |
8 | nna0r 6382 | . . 3 ⊢ (𝐴 ∈ ω → (∅ +o 𝐴) = 𝐴) | |
9 | 5, 7, 8 | 3eqtrd 2177 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 ·o suc ∅) = 𝐴) |
10 | 2, 9 | syl5eq 2185 | 1 ⊢ (𝐴 ∈ ω → (𝐴 ·o 1o) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 ∅c0 3368 suc csuc 4295 ωcom 4512 (class class class)co 5782 1oc1o 6314 +o coa 6318 ·o comu 6319 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-oadd 6325 df-omul 6326 |
This theorem is referenced by: nnm2 6429 mulidpi 7150 archnqq 7249 nq0a0 7289 nq02m 7297 |
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