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Mirrors > Home > MPE Home > Th. List > mulpiord | Structured version Visualization version GIF version |
Description: Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mulpiord | ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 5585 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 〈𝐴, 𝐵〉 ∈ (N × N)) | |
2 | fvres 6682 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (N × N) → (( ·o ↾ (N × N))‘〈𝐴, 𝐵〉) = ( ·o ‘〈𝐴, 𝐵〉)) | |
3 | df-ov 7148 | . . . 4 ⊢ (𝐴 ·N 𝐵) = ( ·N ‘〈𝐴, 𝐵〉) | |
4 | df-mi 10284 | . . . . 5 ⊢ ·N = ( ·o ↾ (N × N)) | |
5 | 4 | fveq1i 6664 | . . . 4 ⊢ ( ·N ‘〈𝐴, 𝐵〉) = (( ·o ↾ (N × N))‘〈𝐴, 𝐵〉) |
6 | 3, 5 | eqtri 2841 | . . 3 ⊢ (𝐴 ·N 𝐵) = (( ·o ↾ (N × N))‘〈𝐴, 𝐵〉) |
7 | df-ov 7148 | . . 3 ⊢ (𝐴 ·o 𝐵) = ( ·o ‘〈𝐴, 𝐵〉) | |
8 | 2, 6, 7 | 3eqtr4g 2878 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (N × N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵)) |
9 | 1, 8 | syl 17 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 〈cop 4563 × cxp 5546 ↾ cres 5550 ‘cfv 6348 (class class class)co 7145 ·o comu 8089 Ncnpi 10254 ·N cmi 10256 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-res 5560 df-iota 6307 df-fv 6356 df-ov 7148 df-mi 10284 |
This theorem is referenced by: mulidpi 10296 mulclpi 10303 mulcompi 10306 mulasspi 10307 distrpi 10308 mulcanpi 10310 ltmpi 10314 |
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