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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 4654 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 〈𝐴, 𝐵〉 ∈ (N × N)) | |
2 | fvres 5534 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (N × N) → (( ·o ↾ (N × N))‘〈𝐴, 𝐵〉) = ( ·o ‘〈𝐴, 𝐵〉)) | |
3 | df-ov 5871 | . . . 4 ⊢ (𝐴 ·N 𝐵) = ( ·N ‘〈𝐴, 𝐵〉) | |
4 | df-mi 7283 | . . . . 5 ⊢ ·N = ( ·o ↾ (N × N)) | |
5 | 4 | fveq1i 5511 | . . . 4 ⊢ ( ·N ‘〈𝐴, 𝐵〉) = (( ·o ↾ (N × N))‘〈𝐴, 𝐵〉) |
6 | 3, 5 | eqtri 2198 | . . 3 ⊢ (𝐴 ·N 𝐵) = (( ·o ↾ (N × N))‘〈𝐴, 𝐵〉) |
7 | df-ov 5871 | . . 3 ⊢ (𝐴 ·o 𝐵) = ( ·o ‘〈𝐴, 𝐵〉) | |
8 | 2, 6, 7 | 3eqtr4g 2235 | . 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 1353 ∈ wcel 2148 〈cop 3594 × cxp 4620 ↾ cres 4624 ‘cfv 5211 (class class class)co 5868 ·o comu 6408 Ncnpi 7249 ·N cmi 7251 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-xp 4628 df-res 4634 df-iota 5173 df-fv 5219 df-ov 5871 df-mi 7283 |
This theorem is referenced by: mulidpi 7295 mulclpi 7305 mulcompig 7308 mulasspig 7309 distrpig 7310 mulcanpig 7312 ltmpig 7316 archnqq 7394 enq0enq 7408 addcmpblnq0 7420 mulcmpblnq0 7421 mulcanenq0ec 7422 addclnq0 7428 mulclnq0 7429 nqpnq0nq 7430 nqnq0a 7431 nqnq0m 7432 nq0m0r 7433 distrnq0 7436 addassnq0lemcl 7438 |
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