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Mirrors > Home > ILE Home > Th. List > mulpiord | GIF version |
Description: Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.) |
Ref | Expression |
---|---|
mulpiord | ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·𝑜 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 4469 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 〈𝐴, 𝐵〉 ∈ (N × N)) | |
2 | fvres 5329 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (N × N) → (( ·𝑜 ↾ (N × N))‘〈𝐴, 𝐵〉) = ( ·𝑜 ‘〈𝐴, 𝐵〉)) | |
3 | df-ov 5655 | . . . 4 ⊢ (𝐴 ·N 𝐵) = ( ·N ‘〈𝐴, 𝐵〉) | |
4 | df-mi 6865 | . . . . 5 ⊢ ·N = ( ·𝑜 ↾ (N × N)) | |
5 | 4 | fveq1i 5306 | . . . 4 ⊢ ( ·N ‘〈𝐴, 𝐵〉) = (( ·𝑜 ↾ (N × N))‘〈𝐴, 𝐵〉) |
6 | 3, 5 | eqtri 2108 | . . 3 ⊢ (𝐴 ·N 𝐵) = (( ·𝑜 ↾ (N × N))‘〈𝐴, 𝐵〉) |
7 | df-ov 5655 | . . 3 ⊢ (𝐴 ·𝑜 𝐵) = ( ·𝑜 ‘〈𝐴, 𝐵〉) | |
8 | 2, 6, 7 | 3eqtr4g 2145 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (N × N) → (𝐴 ·N 𝐵) = (𝐴 ·𝑜 𝐵)) |
9 | 1, 8 | syl 14 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·𝑜 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1289 ∈ wcel 1438 〈cop 3449 × cxp 4436 ↾ cres 4440 ‘cfv 5015 (class class class)co 5652 ·𝑜 comu 6179 Ncnpi 6831 ·N cmi 6833 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-xp 4444 df-res 4450 df-iota 4980 df-fv 5023 df-ov 5655 df-mi 6865 |
This theorem is referenced by: mulidpi 6877 mulclpi 6887 mulcompig 6890 mulasspig 6891 distrpig 6892 mulcanpig 6894 ltmpig 6898 archnqq 6976 enq0enq 6990 addcmpblnq0 7002 mulcmpblnq0 7003 mulcanenq0ec 7004 addclnq0 7010 mulclnq0 7011 nqpnq0nq 7012 nqnq0a 7013 nqnq0m 7014 nq0m0r 7015 distrnq0 7018 addassnq0lemcl 7020 |
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