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Mirrors > Home > ILE Home > Th. List > addpiord | GIF version |
Description: Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.) |
Ref | Expression |
---|---|
addpiord | ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐴 +𝑜 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 4394 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 〈𝐴, 𝐵〉 ∈ (N × N)) | |
2 | fvres 5219 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (N × N) → (( +𝑜 ↾ (N × N))‘〈𝐴, 𝐵〉) = ( +𝑜 ‘〈𝐴, 𝐵〉)) | |
3 | df-ov 5535 | . . . 4 ⊢ (𝐴 +N 𝐵) = ( +N ‘〈𝐴, 𝐵〉) | |
4 | df-pli 6495 | . . . . 5 ⊢ +N = ( +𝑜 ↾ (N × N)) | |
5 | 4 | fveq1i 5199 | . . . 4 ⊢ ( +N ‘〈𝐴, 𝐵〉) = (( +𝑜 ↾ (N × N))‘〈𝐴, 𝐵〉) |
6 | 3, 5 | eqtri 2101 | . . 3 ⊢ (𝐴 +N 𝐵) = (( +𝑜 ↾ (N × N))‘〈𝐴, 𝐵〉) |
7 | df-ov 5535 | . . 3 ⊢ (𝐴 +𝑜 𝐵) = ( +𝑜 ‘〈𝐴, 𝐵〉) | |
8 | 2, 6, 7 | 3eqtr4g 2138 | . 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 102 = wceq 1284 ∈ wcel 1433 〈cop 3401 × cxp 4361 ↾ cres 4365 ‘cfv 4922 (class class class)co 5532 +𝑜 coa 6021 Ncnpi 6462 +N cpli 6463 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-res 4375 df-iota 4887 df-fv 4930 df-ov 5535 df-pli 6495 |
This theorem is referenced by: addclpi 6517 addcompig 6519 addasspig 6520 distrpig 6523 addcanpig 6524 addnidpig 6526 ltexpi 6527 ltapig 6528 1lt2pi 6530 indpi 6532 archnqq 6607 prarloclemarch2 6609 nqnq0a 6644 |
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