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Mirrors > Home > MPE Home > Th. List > sqrt2irrlem | Structured version Visualization version GIF version |
Description: Lemma for sqrt2irr 16229. 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 12323 | . . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℂ) | |
2 | 1 | sqsqrtd 15422 | . . . . . . . . . . 11 ⊢ (𝜑 → ((√‘2)↑2) = 2) |
3 | sqrt2irrlem.3 | . . . . . . . . . . . 12 ⊢ (𝜑 → (√‘2) = (𝐴 / 𝐵)) | |
4 | 3 | oveq1d 7434 | . . . . . . . . . . 11 ⊢ (𝜑 → ((√‘2)↑2) = ((𝐴 / 𝐵)↑2)) |
5 | 2, 4 | eqtr3d 2767 | . . . . . . . . . 10 ⊢ (𝜑 → 2 = ((𝐴 / 𝐵)↑2)) |
6 | sqrt2irrlem.1 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
7 | 6 | zcnd 12700 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
8 | sqrt2irrlem.2 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
9 | 8 | nncnd 12261 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
10 | 8 | nnne0d 12295 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ≠ 0) |
11 | 7, 9, 10 | sqdivd 14159 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2))) |
12 | 5, 11 | eqtrd 2765 | . . . . . . . . 9 ⊢ (𝜑 → 2 = ((𝐴↑2) / (𝐵↑2))) |
13 | 12 | oveq1d 7434 | . . . . . . . 8 ⊢ (𝜑 → (2 · (𝐵↑2)) = (((𝐴↑2) / (𝐵↑2)) · (𝐵↑2))) |
14 | 7 | sqcld 14144 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴↑2) ∈ ℂ) |
15 | 8 | nnsqcld 14242 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐵↑2) ∈ ℕ) |
16 | 15 | nncnd 12261 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵↑2) ∈ ℂ) |
17 | 15 | nnne0d 12295 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵↑2) ≠ 0) |
18 | 14, 16, 17 | divcan1d 12024 | . . . . . . . 8 ⊢ (𝜑 → (((𝐴↑2) / (𝐵↑2)) · (𝐵↑2)) = (𝐴↑2)) |
19 | 13, 18 | eqtrd 2765 | . . . . . . 7 ⊢ (𝜑 → (2 · (𝐵↑2)) = (𝐴↑2)) |
20 | 19 | oveq1d 7434 | . . . . . 6 ⊢ (𝜑 → ((2 · (𝐵↑2)) / 2) = ((𝐴↑2) / 2)) |
21 | 2ne0 12349 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
22 | 21 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 2 ≠ 0) |
23 | 16, 1, 22 | divcan3d 12028 | . . . . . 6 ⊢ (𝜑 → ((2 · (𝐵↑2)) / 2) = (𝐵↑2)) |
24 | 20, 23 | eqtr3d 2767 | . . . . 5 ⊢ (𝜑 → ((𝐴↑2) / 2) = (𝐵↑2)) |
25 | 24, 15 | eqeltrd 2825 | . . . 4 ⊢ (𝜑 → ((𝐴↑2) / 2) ∈ ℕ) |
26 | 25 | nnzd 12618 | . . 3 ⊢ (𝜑 → ((𝐴↑2) / 2) ∈ ℤ) |
27 | zesq 14224 | . . . 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 256 | . 2 ⊢ (𝜑 → (𝐴 / 2) ∈ ℤ) |
30 | 1 | sqvald 14143 | . . . . . . . 8 ⊢ (𝜑 → (2↑2) = (2 · 2)) |
31 | 30 | oveq2d 7435 | . . . . . . 7 ⊢ (𝜑 → ((𝐴↑2) / (2↑2)) = ((𝐴↑2) / (2 · 2))) |
32 | 7, 1, 22 | sqdivd 14159 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 / 2)↑2) = ((𝐴↑2) / (2↑2))) |
33 | 14, 1, 1, 22, 22 | divdiv1d 12054 | . . . . . . 7 ⊢ (𝜑 → (((𝐴↑2) / 2) / 2) = ((𝐴↑2) / (2 · 2))) |
34 | 31, 32, 33 | 3eqtr4d 2775 | . . . . . 6 ⊢ (𝜑 → ((𝐴 / 2)↑2) = (((𝐴↑2) / 2) / 2)) |
35 | 24 | oveq1d 7434 | . . . . . 6 ⊢ (𝜑 → (((𝐴↑2) / 2) / 2) = ((𝐵↑2) / 2)) |
36 | 34, 35 | eqtrd 2765 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 2)↑2) = ((𝐵↑2) / 2)) |
37 | zsqcl 14129 | . . . . . 6 ⊢ ((𝐴 / 2) ∈ ℤ → ((𝐴 / 2)↑2) ∈ ℤ) | |
38 | 29, 37 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 2)↑2) ∈ ℤ) |
39 | 36, 38 | eqeltrrd 2826 | . . . 4 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℤ) |
40 | 15 | nnrpd 13049 | . . . . . 6 ⊢ (𝜑 → (𝐵↑2) ∈ ℝ+) |
41 | 40 | rphalfcld 13063 | . . . . 5 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℝ+) |
42 | 41 | rpgt0d 13054 | . . . 4 ⊢ (𝜑 → 0 < ((𝐵↑2) / 2)) |
43 | elnnz 12601 | . . . 4 ⊢ (((𝐵↑2) / 2) ∈ ℕ ↔ (((𝐵↑2) / 2) ∈ ℤ ∧ 0 < ((𝐵↑2) / 2))) | |
44 | 39, 42, 43 | sylanbrc 581 | . . 3 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℕ) |
45 | nnesq 14225 | . . . 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 256 | . 2 ⊢ (𝜑 → (𝐵 / 2) ∈ ℕ) |
48 | 29, 47 | jca 510 | 1 ⊢ (𝜑 → ((𝐴 / 2) ∈ ℤ ∧ (𝐵 / 2) ∈ ℕ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 0cc0 11140 · cmul 11145 < clt 11280 / cdiv 11903 ℕcn 12245 2c2 12300 ℤcz 12591 ↑cexp 14062 √csqrt 15216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9467 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-rp 13010 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 |
This theorem is referenced by: sqrt2irr 16229 |
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