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Mirrors > Home > ILE Home > Th. List > sqrt2irrlem | GIF version |
Description: Lemma for sqrt2irr 12073. 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 8918 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ | |
2 | 0le2 8938 | . . . . . . . . . . . 12 ⊢ 0 ≤ 2 | |
3 | resqrtth 10959 | . . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ 0 ≤ 2) → ((√‘2)↑2) = 2) | |
4 | 1, 2, 3 | mp2an 423 | . . . . . . . . . . 11 ⊢ ((√‘2)↑2) = 2 |
5 | sqrt2irrlem.3 | . . . . . . . . . . . 12 ⊢ (𝜑 → (√‘2) = (𝐴 / 𝐵)) | |
6 | 5 | oveq1d 5851 | . . . . . . . . . . 11 ⊢ (𝜑 → ((√‘2)↑2) = ((𝐴 / 𝐵)↑2)) |
7 | 4, 6 | eqtr3id 2211 | . . . . . . . . . 10 ⊢ (𝜑 → 2 = ((𝐴 / 𝐵)↑2)) |
8 | sqrt2irrlem.1 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
9 | 8 | zcnd 9305 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
10 | sqrt2irrlem.2 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
11 | 10 | nncnd 8862 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
12 | 10 | nnap0d 8894 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 # 0) |
13 | 9, 11, 12 | sqdivapd 10590 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2))) |
14 | 7, 13 | eqtrd 2197 | . . . . . . . . 9 ⊢ (𝜑 → 2 = ((𝐴↑2) / (𝐵↑2))) |
15 | 14 | oveq1d 5851 | . . . . . . . 8 ⊢ (𝜑 → (2 · (𝐵↑2)) = (((𝐴↑2) / (𝐵↑2)) · (𝐵↑2))) |
16 | 9 | sqcld 10575 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴↑2) ∈ ℂ) |
17 | 10 | nnsqcld 10598 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐵↑2) ∈ ℕ) |
18 | 17 | nncnd 8862 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵↑2) ∈ ℂ) |
19 | 17 | nnap0d 8894 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵↑2) # 0) |
20 | 16, 18, 19 | divcanap1d 8678 | . . . . . . . 8 ⊢ (𝜑 → (((𝐴↑2) / (𝐵↑2)) · (𝐵↑2)) = (𝐴↑2)) |
21 | 15, 20 | eqtrd 2197 | . . . . . . 7 ⊢ (𝜑 → (2 · (𝐵↑2)) = (𝐴↑2)) |
22 | 21 | oveq1d 5851 | . . . . . 6 ⊢ (𝜑 → ((2 · (𝐵↑2)) / 2) = ((𝐴↑2) / 2)) |
23 | 2cnd 8921 | . . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ) | |
24 | 2ap0 8941 | . . . . . . . 8 ⊢ 2 # 0 | |
25 | 24 | a1i 9 | . . . . . . 7 ⊢ (𝜑 → 2 # 0) |
26 | 18, 23, 25 | divcanap3d 8682 | . . . . . 6 ⊢ (𝜑 → ((2 · (𝐵↑2)) / 2) = (𝐵↑2)) |
27 | 22, 26 | eqtr3d 2199 | . . . . 5 ⊢ (𝜑 → ((𝐴↑2) / 2) = (𝐵↑2)) |
28 | 27, 17 | eqeltrd 2241 | . . . 4 ⊢ (𝜑 → ((𝐴↑2) / 2) ∈ ℕ) |
29 | 28 | nnzd 9303 | . . 3 ⊢ (𝜑 → ((𝐴↑2) / 2) ∈ ℤ) |
30 | zesq 10562 | . . . 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 166 | . 2 ⊢ (𝜑 → (𝐴 / 2) ∈ ℤ) |
33 | 2cn 8919 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
34 | 33 | sqvali 10524 | . . . . . . . 8 ⊢ (2↑2) = (2 · 2) |
35 | 34 | oveq2i 5847 | . . . . . . 7 ⊢ ((𝐴↑2) / (2↑2)) = ((𝐴↑2) / (2 · 2)) |
36 | 9, 23, 25 | sqdivapd 10590 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 / 2)↑2) = ((𝐴↑2) / (2↑2))) |
37 | 16, 23, 23, 25, 25 | divdivap1d 8709 | . . . . . . 7 ⊢ (𝜑 → (((𝐴↑2) / 2) / 2) = ((𝐴↑2) / (2 · 2))) |
38 | 35, 36, 37 | 3eqtr4a 2223 | . . . . . 6 ⊢ (𝜑 → ((𝐴 / 2)↑2) = (((𝐴↑2) / 2) / 2)) |
39 | 27 | oveq1d 5851 | . . . . . 6 ⊢ (𝜑 → (((𝐴↑2) / 2) / 2) = ((𝐵↑2) / 2)) |
40 | 38, 39 | eqtrd 2197 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 2)↑2) = ((𝐵↑2) / 2)) |
41 | zsqcl 10515 | . . . . . 6 ⊢ ((𝐴 / 2) ∈ ℤ → ((𝐴 / 2)↑2) ∈ ℤ) | |
42 | 32, 41 | syl 14 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 2)↑2) ∈ ℤ) |
43 | 40, 42 | eqeltrrd 2242 | . . . 4 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℤ) |
44 | 17 | nnrpd 9621 | . . . . . 6 ⊢ (𝜑 → (𝐵↑2) ∈ ℝ+) |
45 | 44 | rphalfcld 9636 | . . . . 5 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℝ+) |
46 | 45 | rpgt0d 9626 | . . . 4 ⊢ (𝜑 → 0 < ((𝐵↑2) / 2)) |
47 | elnnz 9192 | . . . 4 ⊢ (((𝐵↑2) / 2) ∈ ℕ ↔ (((𝐵↑2) / 2) ∈ ℤ ∧ 0 < ((𝐵↑2) / 2))) | |
48 | 43, 46, 47 | sylanbrc 414 | . . 3 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℕ) |
49 | nnesq 10563 | . . . 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 166 | . 2 ⊢ (𝜑 → (𝐵 / 2) ∈ ℕ) |
52 | 32, 51 | jca 304 | 1 ⊢ (𝜑 → ((𝐴 / 2) ∈ ℤ ∧ (𝐵 / 2) ∈ ℕ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 class class class wbr 3976 ‘cfv 5182 (class class class)co 5836 ℝcr 7743 0cc0 7744 · cmul 7749 < clt 7924 ≤ cle 7925 # cap 8470 / cdiv 8559 ℕcn 8848 2c2 8899 ℤcz 9182 ↑cexp 10444 √csqrt 10924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-rp 9581 df-seqfrec 10371 df-exp 10445 df-rsqrt 10926 |
This theorem is referenced by: sqrt2irr 12073 |
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