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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 5072 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 〈𝐴, 𝐵〉 ∈ (N × N)) | |
2 | fvres 6117 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (N × N) → (( ·𝑜 ↾ (N × N))‘〈𝐴, 𝐵〉) = ( ·𝑜 ‘〈𝐴, 𝐵〉)) | |
3 | df-ov 6552 | . . . 4 ⊢ (𝐴 ·N 𝐵) = ( ·N ‘〈𝐴, 𝐵〉) | |
4 | df-mi 9575 | . . . . 5 ⊢ ·N = ( ·𝑜 ↾ (N × N)) | |
5 | 4 | fveq1i 6104 | . . . 4 ⊢ ( ·N ‘〈𝐴, 𝐵〉) = (( ·𝑜 ↾ (N × N))‘〈𝐴, 𝐵〉) |
6 | 3, 5 | eqtri 2632 | . . 3 ⊢ (𝐴 ·N 𝐵) = (( ·𝑜 ↾ (N × N))‘〈𝐴, 𝐵〉) |
7 | df-ov 6552 | . . 3 ⊢ (𝐴 ·𝑜 𝐵) = ( ·𝑜 ‘〈𝐴, 𝐵〉) | |
8 | 2, 6, 7 | 3eqtr4g 2669 | . 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 383 = wceq 1475 ∈ wcel 1977 〈cop 4131 × cxp 5036 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 ·𝑜 comu 7445 Ncnpi 9545 ·N cmi 9547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-res 5050 df-iota 5768 df-fv 5812 df-ov 6552 df-mi 9575 |
This theorem is referenced by: mulidpi 9587 mulclpi 9594 mulcompi 9597 mulasspi 9598 distrpi 9599 mulcanpi 9601 ltmpi 9605 |
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