| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > addpiord | Structured version Visualization version GIF version | ||
| Description: Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| addpiord | ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opelxpi 5722 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 〈𝐴, 𝐵〉 ∈ (N × N)) | |
| 2 | fvres 6925 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (N × N) → (( +o ↾ (N × N))‘〈𝐴, 𝐵〉) = ( +o ‘〈𝐴, 𝐵〉)) | |
| 3 | df-ov 7434 | . . . 4 ⊢ (𝐴 +N 𝐵) = ( +N ‘〈𝐴, 𝐵〉) | |
| 4 | df-pli 10913 | . . . . 5 ⊢ +N = ( +o ↾ (N × N)) | |
| 5 | 4 | fveq1i 6907 | . . . 4 ⊢ ( +N ‘〈𝐴, 𝐵〉) = (( +o ↾ (N × N))‘〈𝐴, 𝐵〉) |
| 6 | 3, 5 | eqtri 2765 | . . 3 ⊢ (𝐴 +N 𝐵) = (( +o ↾ (N × N))‘〈𝐴, 𝐵〉) |
| 7 | df-ov 7434 | . . 3 ⊢ (𝐴 +o 𝐵) = ( +o ‘〈𝐴, 𝐵〉) | |
| 8 | 2, 6, 7 | 3eqtr4g 2802 | . 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 395 = wceq 1540 ∈ wcel 2108 〈cop 4632 × cxp 5683 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 +o coa 8503 Ncnpi 10884 +N cpli 10885 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-res 5697 df-iota 6514 df-fv 6569 df-ov 7434 df-pli 10913 |
| This theorem is referenced by: addclpi 10932 addcompi 10934 addasspi 10935 distrpi 10938 addcanpi 10939 addnidpi 10941 ltexpi 10942 ltapi 10943 1lt2pi 10945 indpi 10947 |
| Copyright terms: Public domain | W3C validator |