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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 4676 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 〈𝐴, 𝐵〉 ∈ (N × N)) | |
2 | fvres 5558 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (N × N) → (( +o ↾ (N × N))‘〈𝐴, 𝐵〉) = ( +o ‘〈𝐴, 𝐵〉)) | |
3 | df-ov 5899 | . . . 4 ⊢ (𝐴 +N 𝐵) = ( +N ‘〈𝐴, 𝐵〉) | |
4 | df-pli 7334 | . . . . 5 ⊢ +N = ( +o ↾ (N × N)) | |
5 | 4 | fveq1i 5535 | . . . 4 ⊢ ( +N ‘〈𝐴, 𝐵〉) = (( +o ↾ (N × N))‘〈𝐴, 𝐵〉) |
6 | 3, 5 | eqtri 2210 | . . 3 ⊢ (𝐴 +N 𝐵) = (( +o ↾ (N × N))‘〈𝐴, 𝐵〉) |
7 | df-ov 5899 | . . 3 ⊢ (𝐴 +o 𝐵) = ( +o ‘〈𝐴, 𝐵〉) | |
8 | 2, 6, 7 | 3eqtr4g 2247 | . 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 104 = wceq 1364 ∈ wcel 2160 〈cop 3610 × cxp 4642 ↾ cres 4646 ‘cfv 5235 (class class class)co 5896 +o coa 6438 Ncnpi 7301 +N cpli 7302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-xp 4650 df-res 4656 df-iota 5196 df-fv 5243 df-ov 5899 df-pli 7334 |
This theorem is referenced by: addclpi 7356 addcompig 7358 addasspig 7359 distrpig 7362 addcanpig 7363 addnidpig 7365 ltexpi 7366 ltapig 7367 1lt2pi 7369 indpi 7371 archnqq 7446 prarloclemarch2 7448 nqnq0a 7483 |
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