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Mirrors > Home > ILE Home > Th. List > mulidpi | GIF version |
Description: 1 is an identity element for multiplication on positive integers. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
mulidpi | ⊢ (𝐴 ∈ N → (𝐴 ·N 1o) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1pi 7091 | . . 3 ⊢ 1o ∈ N | |
2 | mulpiord 7093 | . . 3 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N) → (𝐴 ·N 1o) = (𝐴 ·o 1o)) | |
3 | 1, 2 | mpan2 421 | . 2 ⊢ (𝐴 ∈ N → (𝐴 ·N 1o) = (𝐴 ·o 1o)) |
4 | pinn 7085 | . . 3 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
5 | nnm1 6388 | . . 3 ⊢ (𝐴 ∈ ω → (𝐴 ·o 1o) = 𝐴) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝐴 ∈ N → (𝐴 ·o 1o) = 𝐴) |
7 | 3, 6 | eqtrd 2150 | 1 ⊢ (𝐴 ∈ N → (𝐴 ·N 1o) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∈ wcel 1465 ωcom 4474 (class class class)co 5742 1oc1o 6274 ·o comu 6279 Ncnpi 7048 ·N cmi 7050 |
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 df-ni 7080 df-mi 7082 |
This theorem is referenced by: 1qec 7164 1lt2nq 7182 archnqq 7193 prarloclemarch2 7195 ltnnnq 7199 addpinq1 7240 prarloclemlt 7269 |
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