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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 7344 | . . 3 ⊢ 1o ∈ N | |
2 | mulpiord 7346 | . . 3 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N) → (𝐴 ·N 1o) = (𝐴 ·o 1o)) | |
3 | 1, 2 | mpan2 425 | . 2 ⊢ (𝐴 ∈ N → (𝐴 ·N 1o) = (𝐴 ·o 1o)) |
4 | pinn 7338 | . . 3 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
5 | nnm1 6550 | . . 3 ⊢ (𝐴 ∈ ω → (𝐴 ·o 1o) = 𝐴) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝐴 ∈ N → (𝐴 ·o 1o) = 𝐴) |
7 | 3, 6 | eqtrd 2222 | 1 ⊢ (𝐴 ∈ N → (𝐴 ·N 1o) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 ωcom 4607 (class class class)co 5896 1oc1o 6434 ·o comu 6439 Ncnpi 7301 ·N cmi 7303 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-iord 4384 df-on 4386 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5899 df-oprab 5900 df-mpo 5901 df-1st 6165 df-2nd 6166 df-recs 6330 df-irdg 6395 df-1o 6441 df-oadd 6445 df-omul 6446 df-ni 7333 df-mi 7335 |
This theorem is referenced by: 1qec 7417 1lt2nq 7435 archnqq 7446 prarloclemarch2 7448 ltnnnq 7452 addpinq1 7493 prarloclemlt 7522 |
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