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Mirrors > Home > MPE Home > Th. List > sqr2irrlem | Structured version Visualization version GIF version |
Description: Lemma for irrationality 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). This is Metamath 100 proof #1. (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 | 2cn 10968 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℂ | |
2 | sqrtth 13952 | . . . . . . . . . . . 12 ⊢ (2 ∈ ℂ → ((√‘2)↑2) = 2) | |
3 | 1, 2 | ax-mp 5 | . . . . . . . . . . 11 ⊢ ((√‘2)↑2) = 2 |
4 | sqrt2irrlem.3 | . . . . . . . . . . . 12 ⊢ (𝜑 → (√‘2) = (𝐴 / 𝐵)) | |
5 | 4 | oveq1d 6564 | . . . . . . . . . . 11 ⊢ (𝜑 → ((√‘2)↑2) = ((𝐴 / 𝐵)↑2)) |
6 | 3, 5 | syl5eqr 2658 | . . . . . . . . . 10 ⊢ (𝜑 → 2 = ((𝐴 / 𝐵)↑2)) |
7 | sqr2irrlem.1 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
8 | 7 | zcnd 11359 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
9 | sqr2irrlem.2 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
10 | 9 | nncnd 10913 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
11 | 9 | nnne0d 10942 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ≠ 0) |
12 | 8, 10, 11 | sqdivd 12883 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2))) |
13 | 6, 12 | eqtrd 2644 | . . . . . . . . 9 ⊢ (𝜑 → 2 = ((𝐴↑2) / (𝐵↑2))) |
14 | 13 | oveq1d 6564 | . . . . . . . 8 ⊢ (𝜑 → (2 · (𝐵↑2)) = (((𝐴↑2) / (𝐵↑2)) · (𝐵↑2))) |
15 | 8 | sqcld 12868 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴↑2) ∈ ℂ) |
16 | 9 | nnsqcld 12891 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐵↑2) ∈ ℕ) |
17 | 16 | nncnd 10913 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵↑2) ∈ ℂ) |
18 | 16 | nnne0d 10942 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵↑2) ≠ 0) |
19 | 15, 17, 18 | divcan1d 10681 | . . . . . . . 8 ⊢ (𝜑 → (((𝐴↑2) / (𝐵↑2)) · (𝐵↑2)) = (𝐴↑2)) |
20 | 14, 19 | eqtrd 2644 | . . . . . . 7 ⊢ (𝜑 → (2 · (𝐵↑2)) = (𝐴↑2)) |
21 | 20 | oveq1d 6564 | . . . . . 6 ⊢ (𝜑 → ((2 · (𝐵↑2)) / 2) = ((𝐴↑2) / 2)) |
22 | 1 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ) |
23 | 2ne0 10990 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
24 | 23 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 2 ≠ 0) |
25 | 17, 22, 24 | divcan3d 10685 | . . . . . 6 ⊢ (𝜑 → ((2 · (𝐵↑2)) / 2) = (𝐵↑2)) |
26 | 21, 25 | eqtr3d 2646 | . . . . 5 ⊢ (𝜑 → ((𝐴↑2) / 2) = (𝐵↑2)) |
27 | 26, 16 | eqeltrd 2688 | . . . 4 ⊢ (𝜑 → ((𝐴↑2) / 2) ∈ ℕ) |
28 | 27 | nnzd 11357 | . . 3 ⊢ (𝜑 → ((𝐴↑2) / 2) ∈ ℤ) |
29 | zesq 12849 | . . . 4 ⊢ (𝐴 ∈ ℤ → ((𝐴 / 2) ∈ ℤ ↔ ((𝐴↑2) / 2) ∈ ℤ)) | |
30 | 7, 29 | syl 17 | . . 3 ⊢ (𝜑 → ((𝐴 / 2) ∈ ℤ ↔ ((𝐴↑2) / 2) ∈ ℤ)) |
31 | 28, 30 | mpbird 246 | . 2 ⊢ (𝜑 → (𝐴 / 2) ∈ ℤ) |
32 | 1 | sqvali 12805 | . . . . . . . 8 ⊢ (2↑2) = (2 · 2) |
33 | 32 | oveq2i 6560 | . . . . . . 7 ⊢ ((𝐴↑2) / (2↑2)) = ((𝐴↑2) / (2 · 2)) |
34 | 8, 22, 24 | sqdivd 12883 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 / 2)↑2) = ((𝐴↑2) / (2↑2))) |
35 | 15, 22, 22, 24, 24 | divdiv1d 10711 | . . . . . . 7 ⊢ (𝜑 → (((𝐴↑2) / 2) / 2) = ((𝐴↑2) / (2 · 2))) |
36 | 33, 34, 35 | 3eqtr4a 2670 | . . . . . 6 ⊢ (𝜑 → ((𝐴 / 2)↑2) = (((𝐴↑2) / 2) / 2)) |
37 | 26 | oveq1d 6564 | . . . . . 6 ⊢ (𝜑 → (((𝐴↑2) / 2) / 2) = ((𝐵↑2) / 2)) |
38 | 36, 37 | eqtrd 2644 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 2)↑2) = ((𝐵↑2) / 2)) |
39 | zsqcl 12796 | . . . . . 6 ⊢ ((𝐴 / 2) ∈ ℤ → ((𝐴 / 2)↑2) ∈ ℤ) | |
40 | 31, 39 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 2)↑2) ∈ ℤ) |
41 | 38, 40 | eqeltrrd 2689 | . . . 4 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℤ) |
42 | 16 | nnrpd 11746 | . . . . . 6 ⊢ (𝜑 → (𝐵↑2) ∈ ℝ+) |
43 | 42 | rphalfcld 11760 | . . . . 5 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℝ+) |
44 | 43 | rpgt0d 11751 | . . . 4 ⊢ (𝜑 → 0 < ((𝐵↑2) / 2)) |
45 | elnnz 11264 | . . . 4 ⊢ (((𝐵↑2) / 2) ∈ ℕ ↔ (((𝐵↑2) / 2) ∈ ℤ ∧ 0 < ((𝐵↑2) / 2))) | |
46 | 41, 44, 45 | sylanbrc 695 | . . 3 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℕ) |
47 | nnesq 12850 | . . . 4 ⊢ (𝐵 ∈ ℕ → ((𝐵 / 2) ∈ ℕ ↔ ((𝐵↑2) / 2) ∈ ℕ)) | |
48 | 9, 47 | syl 17 | . . 3 ⊢ (𝜑 → ((𝐵 / 2) ∈ ℕ ↔ ((𝐵↑2) / 2) ∈ ℕ)) |
49 | 46, 48 | mpbird 246 | . 2 ⊢ (𝜑 → (𝐵 / 2) ∈ ℕ) |
50 | 31, 49 | jca 553 | 1 ⊢ (𝜑 → ((𝐴 / 2) ∈ ℤ ∧ (𝐵 / 2) ∈ ℕ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 · cmul 9820 < clt 9953 / cdiv 10563 ℕcn 10897 2c2 10947 ℤcz 11254 ↑cexp 12722 √csqrt 13821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 |
This theorem is referenced by: sqrt2irr 14817 |
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