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Mirrors > Home > ILE Home > Th. List > distrpig | GIF version |
Description: Multiplication of positive integers is distributive. (Contributed by Jim Kingdon, 26-Aug-2019.) |
Ref | Expression |
---|---|
distrpig | ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N (𝐵 +N 𝐶)) = ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 7337 | . . 3 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | |
2 | pinn 7337 | . . 3 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω) | |
3 | pinn 7337 | . . 3 ⊢ (𝐶 ∈ N → 𝐶 ∈ ω) | |
4 | nndi 6510 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶))) | |
5 | 1, 2, 3, 4 | syl3an 1291 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶))) |
6 | addclpi 7355 | . . . . 5 ⊢ ((𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐵 +N 𝐶) ∈ N) | |
7 | mulpiord 7345 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ (𝐵 +N 𝐶) ∈ N) → (𝐴 ·N (𝐵 +N 𝐶)) = (𝐴 ·o (𝐵 +N 𝐶))) | |
8 | 6, 7 | sylan2 286 | . . . 4 ⊢ ((𝐴 ∈ N ∧ (𝐵 ∈ N ∧ 𝐶 ∈ N)) → (𝐴 ·N (𝐵 +N 𝐶)) = (𝐴 ·o (𝐵 +N 𝐶))) |
9 | addpiord 7344 | . . . . . 6 ⊢ ((𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐵 +N 𝐶) = (𝐵 +o 𝐶)) | |
10 | 9 | oveq2d 5911 | . . . . 5 ⊢ ((𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·o (𝐵 +N 𝐶)) = (𝐴 ·o (𝐵 +o 𝐶))) |
11 | 10 | adantl 277 | . . . 4 ⊢ ((𝐴 ∈ N ∧ (𝐵 ∈ N ∧ 𝐶 ∈ N)) → (𝐴 ·o (𝐵 +N 𝐶)) = (𝐴 ·o (𝐵 +o 𝐶))) |
12 | 8, 11 | eqtrd 2222 | . . 3 ⊢ ((𝐴 ∈ N ∧ (𝐵 ∈ N ∧ 𝐶 ∈ N)) → (𝐴 ·N (𝐵 +N 𝐶)) = (𝐴 ·o (𝐵 +o 𝐶))) |
13 | 12 | 3impb 1201 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N (𝐵 +N 𝐶)) = (𝐴 ·o (𝐵 +o 𝐶))) |
14 | mulclpi 7356 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) ∈ N) | |
15 | mulclpi 7356 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) ∈ N) | |
16 | addpiord 7344 | . . . . 5 ⊢ (((𝐴 ·N 𝐵) ∈ N ∧ (𝐴 ·N 𝐶) ∈ N) → ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)) = ((𝐴 ·N 𝐵) +o (𝐴 ·N 𝐶))) | |
17 | 14, 15, 16 | syl2an 289 | . . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐴 ∈ N ∧ 𝐶 ∈ N)) → ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)) = ((𝐴 ·N 𝐵) +o (𝐴 ·N 𝐶))) |
18 | mulpiord 7345 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 𝐵) = (𝐴 ·o 𝐵)) | |
19 | mulpiord 7345 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N 𝐶) = (𝐴 ·o 𝐶)) | |
20 | 18, 19 | oveqan12d 5914 | . . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐴 ∈ N ∧ 𝐶 ∈ N)) → ((𝐴 ·N 𝐵) +o (𝐴 ·N 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶))) |
21 | 17, 20 | eqtrd 2222 | . . 3 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐴 ∈ N ∧ 𝐶 ∈ N)) → ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶))) |
22 | 21 | 3impdi 1304 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶))) |
23 | 5, 13, 22 | 3eqtr4d 2232 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·N (𝐵 +N 𝐶)) = ((𝐴 ·N 𝐵) +N (𝐴 ·N 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 ωcom 4607 (class class class)co 5895 +o coa 6437 ·o comu 6438 Ncnpi 7300 +N cpli 7301 ·N cmi 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-in1 615 ax-in2 616 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-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-iord 4384 df-on 4386 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5898 df-oprab 5899 df-mpo 5900 df-1st 6164 df-2nd 6165 df-recs 6329 df-irdg 6394 df-oadd 6444 df-omul 6445 df-ni 7332 df-pli 7333 df-mi 7334 |
This theorem is referenced by: addcmpblnq 7395 addassnqg 7410 distrnqg 7415 ltanqg 7428 ltexnqq 7436 |
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