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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 4571 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 〈𝐴, 𝐵〉 ∈ (N × N)) | |
2 | fvres 5445 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (N × N) → (( ·o ↾ (N × N))‘〈𝐴, 𝐵〉) = ( ·o ‘〈𝐴, 𝐵〉)) | |
3 | df-ov 5777 | . . . 4 ⊢ (𝐴 ·N 𝐵) = ( ·N ‘〈𝐴, 𝐵〉) | |
4 | df-mi 7114 | . . . . 5 ⊢ ·N = ( ·o ↾ (N × N)) | |
5 | 4 | fveq1i 5422 | . . . 4 ⊢ ( ·N ‘〈𝐴, 𝐵〉) = (( ·o ↾ (N × N))‘〈𝐴, 𝐵〉) |
6 | 3, 5 | eqtri 2160 | . . 3 ⊢ (𝐴 ·N 𝐵) = (( ·o ↾ (N × N))‘〈𝐴, 𝐵〉) |
7 | df-ov 5777 | . . 3 ⊢ (𝐴 ·o 𝐵) = ( ·o ‘〈𝐴, 𝐵〉) | |
8 | 2, 6, 7 | 3eqtr4g 2197 | . 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 1331 ∈ wcel 1480 〈cop 3530 × cxp 4537 ↾ cres 4541 ‘cfv 5123 (class class class)co 5774 ·o comu 6311 Ncnpi 7080 ·N cmi 7082 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-res 4551 df-iota 5088 df-fv 5131 df-ov 5777 df-mi 7114 |
This theorem is referenced by: mulidpi 7126 mulclpi 7136 mulcompig 7139 mulasspig 7140 distrpig 7141 mulcanpig 7143 ltmpig 7147 archnqq 7225 enq0enq 7239 addcmpblnq0 7251 mulcmpblnq0 7252 mulcanenq0ec 7253 addclnq0 7259 mulclnq0 7260 nqpnq0nq 7261 nqnq0a 7262 nqnq0m 7263 nq0m0r 7264 distrnq0 7267 addassnq0lemcl 7269 |
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