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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 4706 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 〈𝐴, 𝐵〉 ∈ (N × N)) | |
| 2 | fvres 5599 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (N × N) → (( ·o ↾ (N × N))‘〈𝐴, 𝐵〉) = ( ·o ‘〈𝐴, 𝐵〉)) | |
| 3 | df-ov 5946 | . . . 4 ⊢ (𝐴 ·N 𝐵) = ( ·N ‘〈𝐴, 𝐵〉) | |
| 4 | df-mi 7418 | . . . . 5 ⊢ ·N = ( ·o ↾ (N × N)) | |
| 5 | 4 | fveq1i 5576 | . . . 4 ⊢ ( ·N ‘〈𝐴, 𝐵〉) = (( ·o ↾ (N × N))‘〈𝐴, 𝐵〉) |
| 6 | 3, 5 | eqtri 2225 | . . 3 ⊢ (𝐴 ·N 𝐵) = (( ·o ↾ (N × N))‘〈𝐴, 𝐵〉) |
| 7 | df-ov 5946 | . . 3 ⊢ (𝐴 ·o 𝐵) = ( ·o ‘〈𝐴, 𝐵〉) | |
| 8 | 2, 6, 7 | 3eqtr4g 2262 | . 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 104 = wceq 1372 ∈ wcel 2175 〈cop 3635 × cxp 4672 ↾ cres 4676 ‘cfv 5270 (class class class)co 5943 ·o comu 6499 Ncnpi 7384 ·N cmi 7386 |
| 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-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-nf 1483 df-sb 1785 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ral 2488 df-rex 2489 df-v 2773 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-br 4044 df-opab 4105 df-xp 4680 df-res 4686 df-iota 5231 df-fv 5278 df-ov 5946 df-mi 7418 |
| This theorem is referenced by: mulidpi 7430 mulclpi 7440 mulcompig 7443 mulasspig 7444 distrpig 7445 mulcanpig 7447 ltmpig 7451 archnqq 7529 enq0enq 7543 addcmpblnq0 7555 mulcmpblnq0 7556 mulcanenq0ec 7557 addclnq0 7563 mulclnq0 7564 nqpnq0nq 7565 nqnq0a 7566 nqnq0m 7567 nq0m0r 7568 distrnq0 7571 addassnq0lemcl 7573 |
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