Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 6281 | . . 3 ⊢ 1o = suc ∅ | |
2 | 1 | oveq2i 5753 | . 2 ⊢ (𝐴 ·o 1o) = (𝐴 ·o suc ∅) |
3 | peano1 4478 | . . . 4 ⊢ ∅ ∈ ω | |
4 | nnmsuc 6341 | . . . 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 6339 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅) | |
7 | 6 | oveq1d 5757 | . . 3 ⊢ (𝐴 ∈ ω → ((𝐴 ·o ∅) +o 𝐴) = (∅ +o 𝐴)) |
8 | nna0r 6342 | . . 3 ⊢ (𝐴 ∈ ω → (∅ +o 𝐴) = 𝐴) | |
9 | 5, 7, 8 | 3eqtrd 2154 | . 2 ⊢ (𝐴 ∈ ω → (𝐴 ·o suc ∅) = 𝐴) |
10 | 2, 9 | syl5eq 2162 | 1 ⊢ (𝐴 ∈ ω → (𝐴 ·o 1o) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∈ wcel 1465 ∅c0 3333 suc csuc 4257 ωcom 4474 (class class class)co 5742 1oc1o 6274 +o coa 6278 ·o comu 6279 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-1o 6281 df-oadd 6285 df-omul 6286 |
This theorem is referenced by: nnm2 6389 mulidpi 7094 archnqq 7193 nq0a0 7233 nq02m 7241 |
Copyright terms: Public domain | W3C validator |