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Mirrors > Home > ILE Home > Th. List > sqrt2irrlem | GIF version |
Description: Lemma for sqrt2irr 10921. This is 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 |
---|---|
sqrt2irrlem.1 | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
sqrt2irrlem.2 | ⊢ (𝜑 → 𝐵 ∈ ℕ) |
sqrt2irrlem.3 | ⊢ (𝜑 → (√‘2) = (𝐴 / 𝐵)) |
Ref | Expression |
---|---|
sqrt2irrlem | ⊢ (𝜑 → ((𝐴 / 2) ∈ ℤ ∧ (𝐵 / 2) ∈ ℕ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 8386 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ | |
2 | 0le2 8406 | . . . . . . . . . . . 12 ⊢ 0 ≤ 2 | |
3 | resqrtth 10291 | . . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ 0 ≤ 2) → ((√‘2)↑2) = 2) | |
4 | 1, 2, 3 | mp2an 417 | . . . . . . . . . . 11 ⊢ ((√‘2)↑2) = 2 |
5 | sqrt2irrlem.3 | . . . . . . . . . . . 12 ⊢ (𝜑 → (√‘2) = (𝐴 / 𝐵)) | |
6 | 5 | oveq1d 5606 | . . . . . . . . . . 11 ⊢ (𝜑 → ((√‘2)↑2) = ((𝐴 / 𝐵)↑2)) |
7 | 4, 6 | syl5eqr 2129 | . . . . . . . . . 10 ⊢ (𝜑 → 2 = ((𝐴 / 𝐵)↑2)) |
8 | sqrt2irrlem.1 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
9 | 8 | zcnd 8765 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
10 | sqrt2irrlem.2 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
11 | 10 | nncnd 8330 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
12 | 10 | nnap0d 8361 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 # 0) |
13 | 9, 11, 12 | sqdivapd 9934 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2))) |
14 | 7, 13 | eqtrd 2115 | . . . . . . . . 9 ⊢ (𝜑 → 2 = ((𝐴↑2) / (𝐵↑2))) |
15 | 14 | oveq1d 5606 | . . . . . . . 8 ⊢ (𝜑 → (2 · (𝐵↑2)) = (((𝐴↑2) / (𝐵↑2)) · (𝐵↑2))) |
16 | 9 | sqcld 9919 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴↑2) ∈ ℂ) |
17 | 10 | nnsqcld 9942 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐵↑2) ∈ ℕ) |
18 | 17 | nncnd 8330 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵↑2) ∈ ℂ) |
19 | 17 | nnap0d 8361 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵↑2) # 0) |
20 | 16, 18, 19 | divcanap1d 8155 | . . . . . . . 8 ⊢ (𝜑 → (((𝐴↑2) / (𝐵↑2)) · (𝐵↑2)) = (𝐴↑2)) |
21 | 15, 20 | eqtrd 2115 | . . . . . . 7 ⊢ (𝜑 → (2 · (𝐵↑2)) = (𝐴↑2)) |
22 | 21 | oveq1d 5606 | . . . . . 6 ⊢ (𝜑 → ((2 · (𝐵↑2)) / 2) = ((𝐴↑2) / 2)) |
23 | 2cnd 8389 | . . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ) | |
24 | 2ap0 8409 | . . . . . . . 8 ⊢ 2 # 0 | |
25 | 24 | a1i 9 | . . . . . . 7 ⊢ (𝜑 → 2 # 0) |
26 | 18, 23, 25 | divcanap3d 8159 | . . . . . 6 ⊢ (𝜑 → ((2 · (𝐵↑2)) / 2) = (𝐵↑2)) |
27 | 22, 26 | eqtr3d 2117 | . . . . 5 ⊢ (𝜑 → ((𝐴↑2) / 2) = (𝐵↑2)) |
28 | 27, 17 | eqeltrd 2159 | . . . 4 ⊢ (𝜑 → ((𝐴↑2) / 2) ∈ ℕ) |
29 | 28 | nnzd 8763 | . . 3 ⊢ (𝜑 → ((𝐴↑2) / 2) ∈ ℤ) |
30 | zesq 9907 | . . . 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 165 | . 2 ⊢ (𝜑 → (𝐴 / 2) ∈ ℤ) |
33 | 2cn 8387 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
34 | 33 | sqvali 9871 | . . . . . . . 8 ⊢ (2↑2) = (2 · 2) |
35 | 34 | oveq2i 5602 | . . . . . . 7 ⊢ ((𝐴↑2) / (2↑2)) = ((𝐴↑2) / (2 · 2)) |
36 | 9, 23, 25 | sqdivapd 9934 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 / 2)↑2) = ((𝐴↑2) / (2↑2))) |
37 | 16, 23, 23, 25, 25 | divdivap1d 8185 | . . . . . . 7 ⊢ (𝜑 → (((𝐴↑2) / 2) / 2) = ((𝐴↑2) / (2 · 2))) |
38 | 35, 36, 37 | 3eqtr4a 2141 | . . . . . 6 ⊢ (𝜑 → ((𝐴 / 2)↑2) = (((𝐴↑2) / 2) / 2)) |
39 | 27 | oveq1d 5606 | . . . . . 6 ⊢ (𝜑 → (((𝐴↑2) / 2) / 2) = ((𝐵↑2) / 2)) |
40 | 38, 39 | eqtrd 2115 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 2)↑2) = ((𝐵↑2) / 2)) |
41 | zsqcl 9862 | . . . . . 6 ⊢ ((𝐴 / 2) ∈ ℤ → ((𝐴 / 2)↑2) ∈ ℤ) | |
42 | 32, 41 | syl 14 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 2)↑2) ∈ ℤ) |
43 | 40, 42 | eqeltrrd 2160 | . . . 4 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℤ) |
44 | 17 | nnrpd 9067 | . . . . . 6 ⊢ (𝜑 → (𝐵↑2) ∈ ℝ+) |
45 | 44 | rphalfcld 9081 | . . . . 5 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℝ+) |
46 | 45 | rpgt0d 9071 | . . . 4 ⊢ (𝜑 → 0 < ((𝐵↑2) / 2)) |
47 | elnnz 8656 | . . . 4 ⊢ (((𝐵↑2) / 2) ∈ ℕ ↔ (((𝐵↑2) / 2) ∈ ℤ ∧ 0 < ((𝐵↑2) / 2))) | |
48 | 43, 46, 47 | sylanbrc 408 | . . 3 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℕ) |
49 | nnesq 9908 | . . . 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 165 | . 2 ⊢ (𝜑 → (𝐵 / 2) ∈ ℕ) |
52 | 32, 51 | jca 300 | 1 ⊢ (𝜑 → ((𝐴 / 2) ∈ ℤ ∧ (𝐵 / 2) ∈ ℕ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1285 ∈ wcel 1434 class class class wbr 3811 ‘cfv 4969 (class class class)co 5591 ℝcr 7252 0cc0 7253 · cmul 7258 < clt 7425 ≤ cle 7426 # cap 7958 / cdiv 8037 ℕcn 8316 2c2 8366 ℤcz 8646 ↑cexp 9791 √csqrt 10256 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-iinf 4366 ax-cnex 7339 ax-resscn 7340 ax-1cn 7341 ax-1re 7342 ax-icn 7343 ax-addcl 7344 ax-addrcl 7345 ax-mulcl 7346 ax-mulrcl 7347 ax-addcom 7348 ax-mulcom 7349 ax-addass 7350 ax-mulass 7351 ax-distr 7352 ax-i2m1 7353 ax-0lt1 7354 ax-1rid 7355 ax-0id 7356 ax-rnegex 7357 ax-precex 7358 ax-cnre 7359 ax-pre-ltirr 7360 ax-pre-ltwlin 7361 ax-pre-lttrn 7362 ax-pre-apti 7363 ax-pre-ltadd 7364 ax-pre-mulgt0 7365 ax-pre-mulext 7366 ax-arch 7367 ax-caucvg 7368 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-if 3374 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4084 df-po 4087 df-iso 4088 df-iord 4157 df-on 4159 df-ilim 4160 df-suc 4162 df-iom 4369 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-res 4413 df-ima 4414 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-f1 4974 df-fo 4975 df-f1o 4976 df-fv 4977 df-riota 5547 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-1st 5846 df-2nd 5847 df-recs 6002 df-frec 6088 df-pnf 7427 df-mnf 7428 df-xr 7429 df-ltxr 7430 df-le 7431 df-sub 7558 df-neg 7559 df-reap 7952 df-ap 7959 df-div 8038 df-inn 8317 df-2 8375 df-3 8376 df-4 8377 df-n0 8566 df-z 8647 df-uz 8915 df-rp 9030 df-iseq 9741 df-iexp 9792 df-rsqrt 10258 |
This theorem is referenced by: sqrt2irr 10921 |
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