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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 5714 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ (N × N)) | |
2 | fvres 6911 | . . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (N × N) → (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩) = ( +o ‘⟨𝐴, 𝐵⟩)) | |
3 | df-ov 7412 | . . . 4 ⊢ (𝐴 +N 𝐵) = ( +N ‘⟨𝐴, 𝐵⟩) | |
4 | df-pli 10868 | . . . . 5 ⊢ +N = ( +o ↾ (N × N)) | |
5 | 4 | fveq1i 6893 | . . . 4 ⊢ ( +N ‘⟨𝐴, 𝐵⟩) = (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩) |
6 | 3, 5 | eqtri 2761 | . . 3 ⊢ (𝐴 +N 𝐵) = (( +o ↾ (N × N))‘⟨𝐴, 𝐵⟩) |
7 | df-ov 7412 | . . 3 ⊢ (𝐴 +o 𝐵) = ( +o ‘⟨𝐴, 𝐵⟩) | |
8 | 2, 6, 7 | 3eqtr4g 2798 | . 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 397 = wceq 1542 ∈ wcel 2107 ⟨cop 4635 × cxp 5675 ↾ cres 5679 ‘cfv 6544 (class class class)co 7409 +o coa 8463 Ncnpi 10839 +N cpli 10840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5683 df-res 5689 df-iota 6496 df-fv 6552 df-ov 7412 df-pli 10868 |
This theorem is referenced by: addclpi 10887 addcompi 10889 addasspi 10890 distrpi 10893 addcanpi 10894 addnidpi 10896 ltexpi 10897 ltapi 10898 1lt2pi 10900 indpi 10902 |
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