![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 4579 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 〈𝐴, 𝐵〉 ∈ (N × N)) | |
2 | fvres 5453 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (N × N) → (( +o ↾ (N × N))‘〈𝐴, 𝐵〉) = ( +o ‘〈𝐴, 𝐵〉)) | |
3 | df-ov 5785 | . . . 4 ⊢ (𝐴 +N 𝐵) = ( +N ‘〈𝐴, 𝐵〉) | |
4 | df-pli 7137 | . . . . 5 ⊢ +N = ( +o ↾ (N × N)) | |
5 | 4 | fveq1i 5430 | . . . 4 ⊢ ( +N ‘〈𝐴, 𝐵〉) = (( +o ↾ (N × N))‘〈𝐴, 𝐵〉) |
6 | 3, 5 | eqtri 2161 | . . 3 ⊢ (𝐴 +N 𝐵) = (( +o ↾ (N × N))‘〈𝐴, 𝐵〉) |
7 | df-ov 5785 | . . 3 ⊢ (𝐴 +o 𝐵) = ( +o ‘〈𝐴, 𝐵〉) | |
8 | 2, 6, 7 | 3eqtr4g 2198 | . 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 103 = wceq 1332 ∈ wcel 1481 〈cop 3535 × cxp 4545 ↾ cres 4549 ‘cfv 5131 (class class class)co 5782 +o coa 6318 Ncnpi 7104 +N cpli 7105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-res 4559 df-iota 5096 df-fv 5139 df-ov 5785 df-pli 7137 |
This theorem is referenced by: addclpi 7159 addcompig 7161 addasspig 7162 distrpig 7165 addcanpig 7166 addnidpig 7168 ltexpi 7169 ltapig 7170 1lt2pi 7172 indpi 7174 archnqq 7249 prarloclemarch2 7251 nqnq0a 7286 |
Copyright terms: Public domain | W3C validator |