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Mirrors > Home > MPE Home > Th. List > sqrt2irrlem | Structured version Visualization version GIF version |
Description: Lemma for sqrt2irr 15594. 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). This is Metamath 100 proof #1. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.) (Proof shortened by JV, 4-Jan-2022.) |
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 | 2cnd 11703 | . . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℂ) | |
2 | 1 | sqsqrtd 14791 | . . . . . . . . . . 11 ⊢ (𝜑 → ((√‘2)↑2) = 2) |
3 | sqrt2irrlem.3 | . . . . . . . . . . . 12 ⊢ (𝜑 → (√‘2) = (𝐴 / 𝐵)) | |
4 | 3 | oveq1d 7150 | . . . . . . . . . . 11 ⊢ (𝜑 → ((√‘2)↑2) = ((𝐴 / 𝐵)↑2)) |
5 | 2, 4 | eqtr3d 2835 | . . . . . . . . . 10 ⊢ (𝜑 → 2 = ((𝐴 / 𝐵)↑2)) |
6 | sqrt2irrlem.1 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
7 | 6 | zcnd 12076 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
8 | sqrt2irrlem.2 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
9 | 8 | nncnd 11641 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
10 | 8 | nnne0d 11675 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ≠ 0) |
11 | 7, 9, 10 | sqdivd 13519 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2))) |
12 | 5, 11 | eqtrd 2833 | . . . . . . . . 9 ⊢ (𝜑 → 2 = ((𝐴↑2) / (𝐵↑2))) |
13 | 12 | oveq1d 7150 | . . . . . . . 8 ⊢ (𝜑 → (2 · (𝐵↑2)) = (((𝐴↑2) / (𝐵↑2)) · (𝐵↑2))) |
14 | 7 | sqcld 13504 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴↑2) ∈ ℂ) |
15 | 8 | nnsqcld 13601 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐵↑2) ∈ ℕ) |
16 | 15 | nncnd 11641 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵↑2) ∈ ℂ) |
17 | 15 | nnne0d 11675 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵↑2) ≠ 0) |
18 | 14, 16, 17 | divcan1d 11406 | . . . . . . . 8 ⊢ (𝜑 → (((𝐴↑2) / (𝐵↑2)) · (𝐵↑2)) = (𝐴↑2)) |
19 | 13, 18 | eqtrd 2833 | . . . . . . 7 ⊢ (𝜑 → (2 · (𝐵↑2)) = (𝐴↑2)) |
20 | 19 | oveq1d 7150 | . . . . . 6 ⊢ (𝜑 → ((2 · (𝐵↑2)) / 2) = ((𝐴↑2) / 2)) |
21 | 2ne0 11729 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
22 | 21 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 2 ≠ 0) |
23 | 16, 1, 22 | divcan3d 11410 | . . . . . 6 ⊢ (𝜑 → ((2 · (𝐵↑2)) / 2) = (𝐵↑2)) |
24 | 20, 23 | eqtr3d 2835 | . . . . 5 ⊢ (𝜑 → ((𝐴↑2) / 2) = (𝐵↑2)) |
25 | 24, 15 | eqeltrd 2890 | . . . 4 ⊢ (𝜑 → ((𝐴↑2) / 2) ∈ ℕ) |
26 | 25 | nnzd 12074 | . . 3 ⊢ (𝜑 → ((𝐴↑2) / 2) ∈ ℤ) |
27 | zesq 13583 | . . . 4 ⊢ (𝐴 ∈ ℤ → ((𝐴 / 2) ∈ ℤ ↔ ((𝐴↑2) / 2) ∈ ℤ)) | |
28 | 6, 27 | syl 17 | . . 3 ⊢ (𝜑 → ((𝐴 / 2) ∈ ℤ ↔ ((𝐴↑2) / 2) ∈ ℤ)) |
29 | 26, 28 | mpbird 260 | . 2 ⊢ (𝜑 → (𝐴 / 2) ∈ ℤ) |
30 | 1 | sqvald 13503 | . . . . . . . 8 ⊢ (𝜑 → (2↑2) = (2 · 2)) |
31 | 30 | oveq2d 7151 | . . . . . . 7 ⊢ (𝜑 → ((𝐴↑2) / (2↑2)) = ((𝐴↑2) / (2 · 2))) |
32 | 7, 1, 22 | sqdivd 13519 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 / 2)↑2) = ((𝐴↑2) / (2↑2))) |
33 | 14, 1, 1, 22, 22 | divdiv1d 11436 | . . . . . . 7 ⊢ (𝜑 → (((𝐴↑2) / 2) / 2) = ((𝐴↑2) / (2 · 2))) |
34 | 31, 32, 33 | 3eqtr4d 2843 | . . . . . 6 ⊢ (𝜑 → ((𝐴 / 2)↑2) = (((𝐴↑2) / 2) / 2)) |
35 | 24 | oveq1d 7150 | . . . . . 6 ⊢ (𝜑 → (((𝐴↑2) / 2) / 2) = ((𝐵↑2) / 2)) |
36 | 34, 35 | eqtrd 2833 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 2)↑2) = ((𝐵↑2) / 2)) |
37 | zsqcl 13490 | . . . . . 6 ⊢ ((𝐴 / 2) ∈ ℤ → ((𝐴 / 2)↑2) ∈ ℤ) | |
38 | 29, 37 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 2)↑2) ∈ ℤ) |
39 | 36, 38 | eqeltrrd 2891 | . . . 4 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℤ) |
40 | 15 | nnrpd 12417 | . . . . . 6 ⊢ (𝜑 → (𝐵↑2) ∈ ℝ+) |
41 | 40 | rphalfcld 12431 | . . . . 5 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℝ+) |
42 | 41 | rpgt0d 12422 | . . . 4 ⊢ (𝜑 → 0 < ((𝐵↑2) / 2)) |
43 | elnnz 11979 | . . . 4 ⊢ (((𝐵↑2) / 2) ∈ ℕ ↔ (((𝐵↑2) / 2) ∈ ℤ ∧ 0 < ((𝐵↑2) / 2))) | |
44 | 39, 42, 43 | sylanbrc 586 | . . 3 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℕ) |
45 | nnesq 13584 | . . . 4 ⊢ (𝐵 ∈ ℕ → ((𝐵 / 2) ∈ ℕ ↔ ((𝐵↑2) / 2) ∈ ℕ)) | |
46 | 8, 45 | syl 17 | . . 3 ⊢ (𝜑 → ((𝐵 / 2) ∈ ℕ ↔ ((𝐵↑2) / 2) ∈ ℕ)) |
47 | 44, 46 | mpbird 260 | . 2 ⊢ (𝜑 → (𝐵 / 2) ∈ ℕ) |
48 | 29, 47 | jca 515 | 1 ⊢ (𝜑 → ((𝐴 / 2) ∈ ℤ ∧ (𝐵 / 2) ∈ ℕ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 0cc0 10526 · cmul 10531 < clt 10664 / cdiv 11286 ℕcn 11625 2c2 11680 ℤcz 11969 ↑cexp 13425 √csqrt 14584 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 |
This theorem is referenced by: sqrt2irr 15594 |
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