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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 5719 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 〈𝐴, 𝐵〉 ∈ (N × N)) | |
2 | fvres 6920 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (N × N) → (( +o ↾ (N × N))‘〈𝐴, 𝐵〉) = ( +o ‘〈𝐴, 𝐵〉)) | |
3 | df-ov 7427 | . . . 4 ⊢ (𝐴 +N 𝐵) = ( +N ‘〈𝐴, 𝐵〉) | |
4 | df-pli 10916 | . . . . 5 ⊢ +N = ( +o ↾ (N × N)) | |
5 | 4 | fveq1i 6902 | . . . 4 ⊢ ( +N ‘〈𝐴, 𝐵〉) = (( +o ↾ (N × N))‘〈𝐴, 𝐵〉) |
6 | 3, 5 | eqtri 2754 | . . 3 ⊢ (𝐴 +N 𝐵) = (( +o ↾ (N × N))‘〈𝐴, 𝐵〉) |
7 | df-ov 7427 | . . 3 ⊢ (𝐴 +o 𝐵) = ( +o ‘〈𝐴, 𝐵〉) | |
8 | 2, 6, 7 | 3eqtr4g 2791 | . 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 394 = wceq 1534 ∈ wcel 2099 〈cop 4639 × cxp 5680 ↾ cres 5684 ‘cfv 6554 (class class class)co 7424 +o coa 8493 Ncnpi 10887 +N cpli 10888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-xp 5688 df-res 5694 df-iota 6506 df-fv 6562 df-ov 7427 df-pli 10916 |
This theorem is referenced by: addclpi 10935 addcompi 10937 addasspi 10938 distrpi 10941 addcanpi 10942 addnidpi 10944 ltexpi 10945 ltapi 10946 1lt2pi 10948 indpi 10950 |
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