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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 5713 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ (N × N)) | |
2 | fvres 6910 | . . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (N × N) → (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩) = ( +o ‘⟨𝐴, 𝐵⟩)) | |
3 | df-ov 7414 | . . . 4 ⊢ (𝐴 +N 𝐵) = ( +N ‘⟨𝐴, 𝐵⟩) | |
4 | df-pli 10870 | . . . . 5 ⊢ +N = ( +o ↾ (N × N)) | |
5 | 4 | fveq1i 6892 | . . . 4 ⊢ ( +N ‘⟨𝐴, 𝐵⟩) = (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩) |
6 | 3, 5 | eqtri 2760 | . . 3 ⊢ (𝐴 +N 𝐵) = (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩) |
7 | df-ov 7414 | . . 3 ⊢ (𝐴 +o 𝐵) = ( +o ‘⟨𝐴, 𝐵⟩) | |
8 | 2, 6, 7 | 3eqtr4g 2797 | . 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 396 = wceq 1541 ∈ wcel 2106 ⟨cop 4634 × cxp 5674 ↾ cres 5678 ‘cfv 6543 (class class class)co 7411 +o coa 8465 Ncnpi 10841 +N cpli 10842 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-xp 5682 df-res 5688 df-iota 6495 df-fv 6551 df-ov 7414 df-pli 10870 |
This theorem is referenced by: addclpi 10889 addcompi 10891 addasspi 10892 distrpi 10895 addcanpi 10896 addnidpi 10898 ltexpi 10899 ltapi 10900 1lt2pi 10902 indpi 10904 |
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