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Mirrors > Home > ILE Home > Th. List > sqrt2irrlem | GIF version |
Description: Lemma for sqrt2irr 12197. 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 9020 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ | |
2 | 0le2 9040 | . . . . . . . . . . . 12 ⊢ 0 ≤ 2 | |
3 | resqrtth 11075 | . . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ 0 ≤ 2) → ((√‘2)↑2) = 2) | |
4 | 1, 2, 3 | mp2an 426 | . . . . . . . . . . 11 ⊢ ((√‘2)↑2) = 2 |
5 | sqrt2irrlem.3 | . . . . . . . . . . . 12 ⊢ (𝜑 → (√‘2) = (𝐴 / 𝐵)) | |
6 | 5 | oveq1d 5912 | . . . . . . . . . . 11 ⊢ (𝜑 → ((√‘2)↑2) = ((𝐴 / 𝐵)↑2)) |
7 | 4, 6 | eqtr3id 2236 | . . . . . . . . . 10 ⊢ (𝜑 → 2 = ((𝐴 / 𝐵)↑2)) |
8 | sqrt2irrlem.1 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
9 | 8 | zcnd 9407 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
10 | sqrt2irrlem.2 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
11 | 10 | nncnd 8964 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
12 | 10 | nnap0d 8996 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 # 0) |
13 | 9, 11, 12 | sqdivapd 10701 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2))) |
14 | 7, 13 | eqtrd 2222 | . . . . . . . . 9 ⊢ (𝜑 → 2 = ((𝐴↑2) / (𝐵↑2))) |
15 | 14 | oveq1d 5912 | . . . . . . . 8 ⊢ (𝜑 → (2 · (𝐵↑2)) = (((𝐴↑2) / (𝐵↑2)) · (𝐵↑2))) |
16 | 9 | sqcld 10686 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴↑2) ∈ ℂ) |
17 | 10 | nnsqcld 10709 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐵↑2) ∈ ℕ) |
18 | 17 | nncnd 8964 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵↑2) ∈ ℂ) |
19 | 17 | nnap0d 8996 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵↑2) # 0) |
20 | 16, 18, 19 | divcanap1d 8779 | . . . . . . . 8 ⊢ (𝜑 → (((𝐴↑2) / (𝐵↑2)) · (𝐵↑2)) = (𝐴↑2)) |
21 | 15, 20 | eqtrd 2222 | . . . . . . 7 ⊢ (𝜑 → (2 · (𝐵↑2)) = (𝐴↑2)) |
22 | 21 | oveq1d 5912 | . . . . . 6 ⊢ (𝜑 → ((2 · (𝐵↑2)) / 2) = ((𝐴↑2) / 2)) |
23 | 2cnd 9023 | . . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ) | |
24 | 2ap0 9043 | . . . . . . . 8 ⊢ 2 # 0 | |
25 | 24 | a1i 9 | . . . . . . 7 ⊢ (𝜑 → 2 # 0) |
26 | 18, 23, 25 | divcanap3d 8783 | . . . . . 6 ⊢ (𝜑 → ((2 · (𝐵↑2)) / 2) = (𝐵↑2)) |
27 | 22, 26 | eqtr3d 2224 | . . . . 5 ⊢ (𝜑 → ((𝐴↑2) / 2) = (𝐵↑2)) |
28 | 27, 17 | eqeltrd 2266 | . . . 4 ⊢ (𝜑 → ((𝐴↑2) / 2) ∈ ℕ) |
29 | 28 | nnzd 9405 | . . 3 ⊢ (𝜑 → ((𝐴↑2) / 2) ∈ ℤ) |
30 | zesq 10673 | . . . 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 167 | . 2 ⊢ (𝜑 → (𝐴 / 2) ∈ ℤ) |
33 | 2cn 9021 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
34 | 33 | sqvali 10634 | . . . . . . . 8 ⊢ (2↑2) = (2 · 2) |
35 | 34 | oveq2i 5908 | . . . . . . 7 ⊢ ((𝐴↑2) / (2↑2)) = ((𝐴↑2) / (2 · 2)) |
36 | 9, 23, 25 | sqdivapd 10701 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 / 2)↑2) = ((𝐴↑2) / (2↑2))) |
37 | 16, 23, 23, 25, 25 | divdivap1d 8810 | . . . . . . 7 ⊢ (𝜑 → (((𝐴↑2) / 2) / 2) = ((𝐴↑2) / (2 · 2))) |
38 | 35, 36, 37 | 3eqtr4a 2248 | . . . . . 6 ⊢ (𝜑 → ((𝐴 / 2)↑2) = (((𝐴↑2) / 2) / 2)) |
39 | 27 | oveq1d 5912 | . . . . . 6 ⊢ (𝜑 → (((𝐴↑2) / 2) / 2) = ((𝐵↑2) / 2)) |
40 | 38, 39 | eqtrd 2222 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 2)↑2) = ((𝐵↑2) / 2)) |
41 | zsqcl 10625 | . . . . . 6 ⊢ ((𝐴 / 2) ∈ ℤ → ((𝐴 / 2)↑2) ∈ ℤ) | |
42 | 32, 41 | syl 14 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 2)↑2) ∈ ℤ) |
43 | 40, 42 | eqeltrrd 2267 | . . . 4 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℤ) |
44 | 17 | nnrpd 9726 | . . . . . 6 ⊢ (𝜑 → (𝐵↑2) ∈ ℝ+) |
45 | 44 | rphalfcld 9741 | . . . . 5 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℝ+) |
46 | 45 | rpgt0d 9731 | . . . 4 ⊢ (𝜑 → 0 < ((𝐵↑2) / 2)) |
47 | elnnz 9294 | . . . 4 ⊢ (((𝐵↑2) / 2) ∈ ℕ ↔ (((𝐵↑2) / 2) ∈ ℤ ∧ 0 < ((𝐵↑2) / 2))) | |
48 | 43, 46, 47 | sylanbrc 417 | . . 3 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℕ) |
49 | nnesq 10674 | . . . 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 167 | . 2 ⊢ (𝜑 → (𝐵 / 2) ∈ ℕ) |
52 | 32, 51 | jca 306 | 1 ⊢ (𝜑 → ((𝐴 / 2) ∈ ℤ ∧ (𝐵 / 2) ∈ ℕ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2160 class class class wbr 4018 ‘cfv 5235 (class class class)co 5897 ℝcr 7841 0cc0 7842 · cmul 7847 < clt 8023 ≤ cle 8024 # cap 8569 / cdiv 8660 ℕcn 8950 2c2 9001 ℤcz 9284 ↑cexp 10553 √csqrt 11040 |
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 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 ax-pre-mulext 7960 ax-arch 7961 ax-caucvg 7962 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 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-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 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-if 3550 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-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 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-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-frec 6417 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-reap 8563 df-ap 8570 df-div 8661 df-inn 8951 df-2 9009 df-3 9010 df-4 9011 df-n0 9208 df-z 9285 df-uz 9560 df-rp 9686 df-seqfrec 10479 df-exp 10554 df-rsqrt 11042 |
This theorem is referenced by: sqrt2irr 12197 |
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