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| Mirrors > Home > MPE Home > Th. List > sqrt2irrlem | Structured version Visualization version GIF version | ||
| Description: Lemma for sqrt2irr 16285. 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 12344 | . . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℂ) | |
| 2 | 1 | sqsqrtd 15478 | . . . . . . . . . . 11 ⊢ (𝜑 → ((√‘2)↑2) = 2) |
| 3 | sqrt2irrlem.3 | . . . . . . . . . . . 12 ⊢ (𝜑 → (√‘2) = (𝐴 / 𝐵)) | |
| 4 | 3 | oveq1d 7446 | . . . . . . . . . . 11 ⊢ (𝜑 → ((√‘2)↑2) = ((𝐴 / 𝐵)↑2)) |
| 5 | 2, 4 | eqtr3d 2779 | . . . . . . . . . 10 ⊢ (𝜑 → 2 = ((𝐴 / 𝐵)↑2)) |
| 6 | sqrt2irrlem.1 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
| 7 | 6 | zcnd 12723 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 8 | sqrt2irrlem.2 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
| 9 | 8 | nncnd 12282 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
| 10 | 8 | nnne0d 12316 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ≠ 0) |
| 11 | 7, 9, 10 | sqdivd 14199 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2))) |
| 12 | 5, 11 | eqtrd 2777 | . . . . . . . . 9 ⊢ (𝜑 → 2 = ((𝐴↑2) / (𝐵↑2))) |
| 13 | 12 | oveq1d 7446 | . . . . . . . 8 ⊢ (𝜑 → (2 · (𝐵↑2)) = (((𝐴↑2) / (𝐵↑2)) · (𝐵↑2))) |
| 14 | 7 | sqcld 14184 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴↑2) ∈ ℂ) |
| 15 | 8 | nnsqcld 14283 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐵↑2) ∈ ℕ) |
| 16 | 15 | nncnd 12282 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵↑2) ∈ ℂ) |
| 17 | 15 | nnne0d 12316 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵↑2) ≠ 0) |
| 18 | 14, 16, 17 | divcan1d 12044 | . . . . . . . 8 ⊢ (𝜑 → (((𝐴↑2) / (𝐵↑2)) · (𝐵↑2)) = (𝐴↑2)) |
| 19 | 13, 18 | eqtrd 2777 | . . . . . . 7 ⊢ (𝜑 → (2 · (𝐵↑2)) = (𝐴↑2)) |
| 20 | 19 | oveq1d 7446 | . . . . . 6 ⊢ (𝜑 → ((2 · (𝐵↑2)) / 2) = ((𝐴↑2) / 2)) |
| 21 | 2ne0 12370 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
| 22 | 21 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 2 ≠ 0) |
| 23 | 16, 1, 22 | divcan3d 12048 | . . . . . 6 ⊢ (𝜑 → ((2 · (𝐵↑2)) / 2) = (𝐵↑2)) |
| 24 | 20, 23 | eqtr3d 2779 | . . . . 5 ⊢ (𝜑 → ((𝐴↑2) / 2) = (𝐵↑2)) |
| 25 | 24, 15 | eqeltrd 2841 | . . . 4 ⊢ (𝜑 → ((𝐴↑2) / 2) ∈ ℕ) |
| 26 | 25 | nnzd 12640 | . . 3 ⊢ (𝜑 → ((𝐴↑2) / 2) ∈ ℤ) |
| 27 | zesq 14265 | . . . 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 257 | . 2 ⊢ (𝜑 → (𝐴 / 2) ∈ ℤ) |
| 30 | 1 | sqvald 14183 | . . . . . . . 8 ⊢ (𝜑 → (2↑2) = (2 · 2)) |
| 31 | 30 | oveq2d 7447 | . . . . . . 7 ⊢ (𝜑 → ((𝐴↑2) / (2↑2)) = ((𝐴↑2) / (2 · 2))) |
| 32 | 7, 1, 22 | sqdivd 14199 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 / 2)↑2) = ((𝐴↑2) / (2↑2))) |
| 33 | 14, 1, 1, 22, 22 | divdiv1d 12074 | . . . . . . 7 ⊢ (𝜑 → (((𝐴↑2) / 2) / 2) = ((𝐴↑2) / (2 · 2))) |
| 34 | 31, 32, 33 | 3eqtr4d 2787 | . . . . . 6 ⊢ (𝜑 → ((𝐴 / 2)↑2) = (((𝐴↑2) / 2) / 2)) |
| 35 | 24 | oveq1d 7446 | . . . . . 6 ⊢ (𝜑 → (((𝐴↑2) / 2) / 2) = ((𝐵↑2) / 2)) |
| 36 | 34, 35 | eqtrd 2777 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 2)↑2) = ((𝐵↑2) / 2)) |
| 37 | zsqcl 14169 | . . . . . 6 ⊢ ((𝐴 / 2) ∈ ℤ → ((𝐴 / 2)↑2) ∈ ℤ) | |
| 38 | 29, 37 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 2)↑2) ∈ ℤ) |
| 39 | 36, 38 | eqeltrrd 2842 | . . . 4 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℤ) |
| 40 | 15 | nnrpd 13075 | . . . . . 6 ⊢ (𝜑 → (𝐵↑2) ∈ ℝ+) |
| 41 | 40 | rphalfcld 13089 | . . . . 5 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℝ+) |
| 42 | 41 | rpgt0d 13080 | . . . 4 ⊢ (𝜑 → 0 < ((𝐵↑2) / 2)) |
| 43 | elnnz 12623 | . . . 4 ⊢ (((𝐵↑2) / 2) ∈ ℕ ↔ (((𝐵↑2) / 2) ∈ ℤ ∧ 0 < ((𝐵↑2) / 2))) | |
| 44 | 39, 42, 43 | sylanbrc 583 | . . 3 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℕ) |
| 45 | nnesq 14266 | . . . 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 257 | . 2 ⊢ (𝜑 → (𝐵 / 2) ∈ ℕ) |
| 48 | 29, 47 | jca 511 | 1 ⊢ (𝜑 → ((𝐴 / 2) ∈ ℤ ∧ (𝐵 / 2) ∈ ℕ)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 0cc0 11155 · cmul 11160 < clt 11295 / cdiv 11920 ℕcn 12266 2c2 12321 ℤcz 12613 ↑cexp 14102 √csqrt 15272 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 |
| This theorem is referenced by: sqrt2irr 16285 |
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