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| 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 5703 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 〈𝐴, 𝐵〉 ∈ (N × N)) | |
| 2 | fvres 6907 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (N × N) → (( +o ↾ (N × N))‘〈𝐴, 𝐵〉) = ( +o ‘〈𝐴, 𝐵〉)) | |
| 3 | df-ov 7426 | . . . 4 ⊢ (𝐴 +N 𝐵) = ( +N ‘〈𝐴, 𝐵〉) | |
| 4 | df-pli 10876 | . . . . 5 ⊢ +N = ( +o ↾ (N × N)) | |
| 5 | 4 | fveq1i 6889 | . . . 4 ⊢ ( +N ‘〈𝐴, 𝐵〉) = (( +o ↾ (N × N))‘〈𝐴, 𝐵〉) |
| 6 | 3, 5 | eqtri 2789 | . . 3 ⊢ (𝐴 +N 𝐵) = (( +o ↾ (N × N))‘〈𝐴, 𝐵〉) |
| 7 | df-ov 7426 | . . 3 ⊢ (𝐴 +o 𝐵) = ( +o ‘〈𝐴, 𝐵〉) | |
| 8 | 2, 6, 7 | 3eqtr4g 2826 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (N × N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵)) |
| 9 | 1, 8 | syl 18 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵)) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 〈cop 4600 × cxp 5664 ↾ cres 5668 ‘cfv 6543 (class class class)co 7423 +o coa 8459 Ncnpi 10847 +N cpli 10848 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-ext 2738 ax-sep 5262 ax-pr 5409 |
| This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-sb 2100 df-clab 2745 df-cleq 2758 df-clel 2841 df-ral 3083 df-rex 3093 df-rab 3420 df-v 3460 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4493 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-br 5115 df-opab 5179 df-xp 5672 df-res 5678 df-iota 6499 df-fv 6551 df-ov 7426 df-pli 10876 |
| This theorem is used by: addclpi 10895 addcompi 10897 addasspi 10898 distrpi 10901 addcanpi 10902 addnidpi 10904 ltexpi 10905 ltapi 10906 1lt2pi 10908 indpi 10910 |
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