![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > mulcanpig | GIF version |
Description: Multiplication cancellation law for positive integers. (Contributed by Jim Kingdon, 29-Aug-2019.) |
Ref | Expression |
---|---|
mulcanpig | ⊢ ((A ∈ N ∧ B ∈ N ∧ 𝐶 ∈ N) → ((A ·N B) = (A ·N 𝐶) ↔ B = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulpiord 6301 | . . . . . 6 ⊢ ((A ∈ N ∧ B ∈ N) → (A ·N B) = (A ·𝑜 B)) | |
2 | 1 | adantr 261 | . . . . 5 ⊢ (((A ∈ N ∧ B ∈ N) ∧ 𝐶 ∈ N) → (A ·N B) = (A ·𝑜 B)) |
3 | mulpiord 6301 | . . . . . 6 ⊢ ((A ∈ N ∧ 𝐶 ∈ N) → (A ·N 𝐶) = (A ·𝑜 𝐶)) | |
4 | 3 | adantlr 446 | . . . . 5 ⊢ (((A ∈ N ∧ B ∈ N) ∧ 𝐶 ∈ N) → (A ·N 𝐶) = (A ·𝑜 𝐶)) |
5 | 2, 4 | eqeq12d 2051 | . . . 4 ⊢ (((A ∈ N ∧ B ∈ N) ∧ 𝐶 ∈ N) → ((A ·N B) = (A ·N 𝐶) ↔ (A ·𝑜 B) = (A ·𝑜 𝐶))) |
6 | pinn 6293 | . . . . . . . . 9 ⊢ (A ∈ N → A ∈ 𝜔) | |
7 | pinn 6293 | . . . . . . . . 9 ⊢ (B ∈ N → B ∈ 𝜔) | |
8 | pinn 6293 | . . . . . . . . 9 ⊢ (𝐶 ∈ N → 𝐶 ∈ 𝜔) | |
9 | elni2 6298 | . . . . . . . . . . . 12 ⊢ (A ∈ N ↔ (A ∈ 𝜔 ∧ ∅ ∈ A)) | |
10 | 9 | simprbi 260 | . . . . . . . . . . 11 ⊢ (A ∈ N → ∅ ∈ A) |
11 | nnmcan 6028 | . . . . . . . . . . . 12 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ A) → ((A ·𝑜 B) = (A ·𝑜 𝐶) ↔ B = 𝐶)) | |
12 | 11 | biimpd 132 | . . . . . . . . . . 11 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ ∅ ∈ A) → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶)) |
13 | 10, 12 | sylan2 270 | . . . . . . . . . 10 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) ∧ A ∈ N) → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶)) |
14 | 13 | ex 108 | . . . . . . . . 9 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔 ∧ 𝐶 ∈ 𝜔) → (A ∈ N → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶))) |
15 | 6, 7, 8, 14 | syl3an 1176 | . . . . . . . 8 ⊢ ((A ∈ N ∧ B ∈ N ∧ 𝐶 ∈ N) → (A ∈ N → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶))) |
16 | 15 | 3exp 1102 | . . . . . . 7 ⊢ (A ∈ N → (B ∈ N → (𝐶 ∈ N → (A ∈ N → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶))))) |
17 | 16 | com4r 80 | . . . . . 6 ⊢ (A ∈ N → (A ∈ N → (B ∈ N → (𝐶 ∈ N → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶))))) |
18 | 17 | pm2.43i 43 | . . . . 5 ⊢ (A ∈ N → (B ∈ N → (𝐶 ∈ N → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶)))) |
19 | 18 | imp31 243 | . . . 4 ⊢ (((A ∈ N ∧ B ∈ N) ∧ 𝐶 ∈ N) → ((A ·𝑜 B) = (A ·𝑜 𝐶) → B = 𝐶)) |
20 | 5, 19 | sylbid 139 | . . 3 ⊢ (((A ∈ N ∧ B ∈ N) ∧ 𝐶 ∈ N) → ((A ·N B) = (A ·N 𝐶) → B = 𝐶)) |
21 | 20 | 3impa 1098 | . 2 ⊢ ((A ∈ N ∧ B ∈ N ∧ 𝐶 ∈ N) → ((A ·N B) = (A ·N 𝐶) → B = 𝐶)) |
22 | oveq2 5463 | . 2 ⊢ (B = 𝐶 → (A ·N B) = (A ·N 𝐶)) | |
23 | 21, 22 | impbid1 130 | 1 ⊢ ((A ∈ N ∧ B ∈ N ∧ 𝐶 ∈ N) → ((A ·N B) = (A ·N 𝐶) ↔ B = 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∧ w3a 884 = wceq 1242 ∈ wcel 1390 ∅c0 3218 𝜔com 4256 (class class class)co 5455 ·𝑜 comu 5938 Ncnpi 6256 ·N cmi 6258 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254 |
This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-id 4021 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-oadd 5944 df-omul 5945 df-ni 6288 df-mi 6290 |
This theorem is referenced by: enqer 6342 |
Copyright terms: Public domain | W3C validator |