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Mirrors > Home > ILE Home > Th. List > sqr2irrlem | GIF version |
Description: Lemma concerning rationality of square root of 2. The core of the proof - if 𝐴 / 𝐵 = √(2), then 𝐴 and 𝐵 are even, so 𝐴 / 2 and 𝐵 / 2 are smaller representatives, which is absurd by the method of infinite descent (here implemented by strong induction). (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.) |
Ref | Expression |
---|---|
sqr2irrlem.1 | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
sqr2irrlem.2 | ⊢ (𝜑 → 𝐵 ∈ ℕ) |
sqrt2irrlem.3 | ⊢ (𝜑 → (√‘2) = (𝐴 / 𝐵)) |
Ref | Expression |
---|---|
sqr2irrlem | ⊢ (𝜑 → ((𝐴 / 2) ∈ ℤ ∧ (𝐵 / 2) ∈ ℕ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 7985 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ | |
2 | 0le2 8006 | . . . . . . . . . . . 12 ⊢ 0 ≤ 2 | |
3 | resqrtth 9629 | . . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ 0 ≤ 2) → ((√‘2)↑2) = 2) | |
4 | 1, 2, 3 | mp2an 402 | . . . . . . . . . . 11 ⊢ ((√‘2)↑2) = 2 |
5 | sqrt2irrlem.3 | . . . . . . . . . . . 12 ⊢ (𝜑 → (√‘2) = (𝐴 / 𝐵)) | |
6 | 5 | oveq1d 5527 | . . . . . . . . . . 11 ⊢ (𝜑 → ((√‘2)↑2) = ((𝐴 / 𝐵)↑2)) |
7 | 4, 6 | syl5eqr 2086 | . . . . . . . . . 10 ⊢ (𝜑 → 2 = ((𝐴 / 𝐵)↑2)) |
8 | sqr2irrlem.1 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
9 | 8 | zcnd 8361 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
10 | sqr2irrlem.2 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
11 | 10 | nncnd 7928 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
12 | 10 | nnap0d 7959 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 # 0) |
13 | 9, 11, 12 | sqdivapd 9394 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2))) |
14 | 7, 13 | eqtrd 2072 | . . . . . . . . 9 ⊢ (𝜑 → 2 = ((𝐴↑2) / (𝐵↑2))) |
15 | 14 | oveq1d 5527 | . . . . . . . 8 ⊢ (𝜑 → (2 · (𝐵↑2)) = (((𝐴↑2) / (𝐵↑2)) · (𝐵↑2))) |
16 | 9 | sqcld 9379 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴↑2) ∈ ℂ) |
17 | 10 | nnsqcld 9401 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐵↑2) ∈ ℕ) |
18 | 17 | nncnd 7928 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵↑2) ∈ ℂ) |
19 | 17 | nnap0d 7959 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵↑2) # 0) |
20 | 16, 18, 19 | divcanap1d 7766 | . . . . . . . 8 ⊢ (𝜑 → (((𝐴↑2) / (𝐵↑2)) · (𝐵↑2)) = (𝐴↑2)) |
21 | 15, 20 | eqtrd 2072 | . . . . . . 7 ⊢ (𝜑 → (2 · (𝐵↑2)) = (𝐴↑2)) |
22 | 21 | oveq1d 5527 | . . . . . 6 ⊢ (𝜑 → ((2 · (𝐵↑2)) / 2) = ((𝐴↑2) / 2)) |
23 | 2cnd 7988 | . . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ) | |
24 | 2ap0 8009 | . . . . . . . 8 ⊢ 2 # 0 | |
25 | 24 | a1i 9 | . . . . . . 7 ⊢ (𝜑 → 2 # 0) |
26 | 18, 23, 25 | divcanap3d 7770 | . . . . . 6 ⊢ (𝜑 → ((2 · (𝐵↑2)) / 2) = (𝐵↑2)) |
27 | 22, 26 | eqtr3d 2074 | . . . . 5 ⊢ (𝜑 → ((𝐴↑2) / 2) = (𝐵↑2)) |
28 | 27, 17 | eqeltrd 2114 | . . . 4 ⊢ (𝜑 → ((𝐴↑2) / 2) ∈ ℕ) |
29 | 28 | nnzd 8359 | . . 3 ⊢ (𝜑 → ((𝐴↑2) / 2) ∈ ℤ) |
30 | zesq 9367 | . . . 4 ⊢ (𝐴 ∈ ℤ → ((𝐴 / 2) ∈ ℤ ↔ ((𝐴↑2) / 2) ∈ ℤ)) | |
31 | 8, 30 | syl 14 | . . 3 ⊢ (𝜑 → ((𝐴 / 2) ∈ ℤ ↔ ((𝐴↑2) / 2) ∈ ℤ)) |
32 | 29, 31 | mpbird 156 | . 2 ⊢ (𝜑 → (𝐴 / 2) ∈ ℤ) |
33 | 2cn 7986 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
34 | 33 | sqvali 9333 | . . . . . . . 8 ⊢ (2↑2) = (2 · 2) |
35 | 34 | oveq2i 5523 | . . . . . . 7 ⊢ ((𝐴↑2) / (2↑2)) = ((𝐴↑2) / (2 · 2)) |
36 | 9, 23, 25 | sqdivapd 9394 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 / 2)↑2) = ((𝐴↑2) / (2↑2))) |
37 | 16, 23, 23, 25, 25 | divdivap1d 7796 | . . . . . . 7 ⊢ (𝜑 → (((𝐴↑2) / 2) / 2) = ((𝐴↑2) / (2 · 2))) |
38 | 35, 36, 37 | 3eqtr4a 2098 | . . . . . 6 ⊢ (𝜑 → ((𝐴 / 2)↑2) = (((𝐴↑2) / 2) / 2)) |
39 | 27 | oveq1d 5527 | . . . . . 6 ⊢ (𝜑 → (((𝐴↑2) / 2) / 2) = ((𝐵↑2) / 2)) |
40 | 38, 39 | eqtrd 2072 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 2)↑2) = ((𝐵↑2) / 2)) |
41 | zsqcl 9324 | . . . . . 6 ⊢ ((𝐴 / 2) ∈ ℤ → ((𝐴 / 2)↑2) ∈ ℤ) | |
42 | 32, 41 | syl 14 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 2)↑2) ∈ ℤ) |
43 | 40, 42 | eqeltrrd 2115 | . . . 4 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℤ) |
44 | 17 | nnrpd 8621 | . . . . . 6 ⊢ (𝜑 → (𝐵↑2) ∈ ℝ+) |
45 | 44 | rphalfcld 8635 | . . . . 5 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℝ+) |
46 | 45 | rpgt0d 8625 | . . . 4 ⊢ (𝜑 → 0 < ((𝐵↑2) / 2)) |
47 | elnnz 8255 | . . . 4 ⊢ (((𝐵↑2) / 2) ∈ ℕ ↔ (((𝐵↑2) / 2) ∈ ℤ ∧ 0 < ((𝐵↑2) / 2))) | |
48 | 43, 46, 47 | sylanbrc 394 | . . 3 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℕ) |
49 | nnesq 9368 | . . . 4 ⊢ (𝐵 ∈ ℕ → ((𝐵 / 2) ∈ ℕ ↔ ((𝐵↑2) / 2) ∈ ℕ)) | |
50 | 10, 49 | syl 14 | . . 3 ⊢ (𝜑 → ((𝐵 / 2) ∈ ℕ ↔ ((𝐵↑2) / 2) ∈ ℕ)) |
51 | 48, 50 | mpbird 156 | . 2 ⊢ (𝜑 → (𝐵 / 2) ∈ ℕ) |
52 | 32, 51 | jca 290 | 1 ⊢ (𝜑 → ((𝐴 / 2) ∈ ℤ ∧ (𝐵 / 2) ∈ ℕ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 class class class wbr 3764 ‘cfv 4902 (class class class)co 5512 ℝcr 6888 0cc0 6889 · cmul 6894 < clt 7060 ≤ cle 7061 # cap 7572 / cdiv 7651 ℕcn 7914 2c2 7964 ℤcz 8245 ↑cexp 9254 √csqrt 9594 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulrcl 6983 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-precex 6994 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-apti 6999 ax-pre-ltadd 7000 ax-pre-mulgt0 7001 ax-pre-mulext 7002 ax-arch 7003 ax-caucvg 7004 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-if 3332 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-frec 5978 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-reap 7566 df-ap 7573 df-div 7652 df-inn 7915 df-2 7973 df-3 7974 df-4 7975 df-n0 8182 df-z 8246 df-uz 8474 df-rp 8584 df-iseq 9212 df-iexp 9255 df-rsqrt 9596 |
This theorem is referenced by: sqrt2irr 9878 |
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