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Mirrors > Home > ILE Home > Th. List > addasspig | GIF version |
Description: Addition of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
Ref | Expression |
---|---|
addasspig | ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 +N 𝐵) +N 𝐶) = (𝐴 +N (𝐵 +N 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 7310 | . . 3 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | pinn 7310 | . . 3 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
3 | pinn 7310 | . . 3 ⊢ (𝐶 ∈ N → 𝐶 ∈ ω) | |
4 | nnaass 6488 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +o 𝐵) +o 𝐶) = (𝐴 +o (𝐵 +o 𝐶))) | |
5 | 1, 2, 3, 4 | syl3an 1280 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 +o 𝐵) +o 𝐶) = (𝐴 +o (𝐵 +o 𝐶))) |
6 | addclpi 7328 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) ∈ N) | |
7 | addpiord 7317 | . . . . 5 ⊢ (((𝐴 +N 𝐵) ∈ N ∧ 𝐶 ∈ N) → ((𝐴 +N 𝐵) +N 𝐶) = ((𝐴 +N 𝐵) +o 𝐶)) | |
8 | 6, 7 | sylan 283 | . . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → ((𝐴 +N 𝐵) +N 𝐶) = ((𝐴 +N 𝐵) +o 𝐶)) |
9 | addpiord 7317 | . . . . . 6 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +N 𝐵) = (𝐴 +o 𝐵)) | |
10 | 9 | oveq1d 5892 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 +N 𝐵) +o 𝐶) = ((𝐴 +o 𝐵) +o 𝐶)) |
11 | 10 | adantr 276 | . . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → ((𝐴 +N 𝐵) +o 𝐶) = ((𝐴 +o 𝐵) +o 𝐶)) |
12 | 8, 11 | eqtrd 2210 | . . 3 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ 𝐶 ∈ N) → ((𝐴 +N 𝐵) +N 𝐶) = ((𝐴 +o 𝐵) +o 𝐶)) |
13 | 12 | 3impa 1194 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 +N 𝐵) +N 𝐶) = ((𝐴 +o 𝐵) +o 𝐶)) |
14 | addclpi 7328 | . . . . 5 ⊢ ((𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐵 +N 𝐶) ∈ N) | |
15 | addpiord 7317 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ (𝐵 +N 𝐶) ∈ N) → (𝐴 +N (𝐵 +N 𝐶)) = (𝐴 +o (𝐵 +N 𝐶))) | |
16 | 14, 15 | sylan2 286 | . . . 4 ⊢ ((𝐴 ∈ N ∧ (𝐵 ∈ N ∧ 𝐶 ∈ N)) → (𝐴 +N (𝐵 +N 𝐶)) = (𝐴 +o (𝐵 +N 𝐶))) |
17 | addpiord 7317 | . . . . . 6 ⊢ ((𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐵 +N 𝐶) = (𝐵 +o 𝐶)) | |
18 | 17 | oveq2d 5893 | . . . . 5 ⊢ ((𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 +o (𝐵 +N 𝐶)) = (𝐴 +o (𝐵 +o 𝐶))) |
19 | 18 | adantl 277 | . . . 4 ⊢ ((𝐴 ∈ N ∧ (𝐵 ∈ N ∧ 𝐶 ∈ N)) → (𝐴 +o (𝐵 +N 𝐶)) = (𝐴 +o (𝐵 +o 𝐶))) |
20 | 16, 19 | eqtrd 2210 | . . 3 ⊢ ((𝐴 ∈ N ∧ (𝐵 ∈ N ∧ 𝐶 ∈ N)) → (𝐴 +N (𝐵 +N 𝐶)) = (𝐴 +o (𝐵 +o 𝐶))) |
21 | 20 | 3impb 1199 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 +N (𝐵 +N 𝐶)) = (𝐴 +o (𝐵 +o 𝐶))) |
22 | 5, 13, 21 | 3eqtr4d 2220 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 +N 𝐵) +N 𝐶) = (𝐴 +N (𝐵 +N 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ωcom 4591 (class class class)co 5877 +o coa 6416 Ncnpi 7273 +N cpli 7274 |
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-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-iord 4368 df-on 4370 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-oadd 6423 df-ni 7305 df-pli 7306 |
This theorem is referenced by: addassnqg 7383 |
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