Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4541 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 〈𝐴, 𝐵〉 ∈ (N × N)) | |
2 | fvres 5413 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (N × N) → (( ·o ↾ (N × N))‘〈𝐴, 𝐵〉) = ( ·o ‘〈𝐴, 𝐵〉)) | |
3 | df-ov 5745 | . . . 4 ⊢ (𝐴 ·N 𝐵) = ( ·N ‘〈𝐴, 𝐵〉) | |
4 | df-mi 7082 | . . . . 5 ⊢ ·N = ( ·o ↾ (N × N)) | |
5 | 4 | fveq1i 5390 | . . . 4 ⊢ ( ·N ‘〈𝐴, 𝐵〉) = (( ·o ↾ (N × N))‘〈𝐴, 𝐵〉) |
6 | 3, 5 | eqtri 2138 | . . 3 ⊢ (𝐴 ·N 𝐵) = (( ·o ↾ (N × N))‘〈𝐴, 𝐵〉) |
7 | df-ov 5745 | . . 3 ⊢ (𝐴 ·o 𝐵) = ( ·o ‘〈𝐴, 𝐵〉) | |
8 | 2, 6, 7 | 3eqtr4g 2175 | . 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 103 = wceq 1316 ∈ wcel 1465 〈cop 3500 × cxp 4507 ↾ cres 4511 ‘cfv 5093 (class class class)co 5742 ·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-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-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-res 4521 df-iota 5058 df-fv 5101 df-ov 5745 df-mi 7082 |
This theorem is referenced by: mulidpi 7094 mulclpi 7104 mulcompig 7107 mulasspig 7108 distrpig 7109 mulcanpig 7111 ltmpig 7115 archnqq 7193 enq0enq 7207 addcmpblnq0 7219 mulcmpblnq0 7220 mulcanenq0ec 7221 addclnq0 7227 mulclnq0 7228 nqpnq0nq 7229 nqnq0a 7230 nqnq0m 7231 nq0m0r 7232 distrnq0 7235 addassnq0lemcl 7237 |
Copyright terms: Public domain | W3C validator |