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Mirrors > Home > MPE Home > Th. List > sqrt2irrlem | Structured version Visualization version GIF version |
Description: Lemma for sqrt2irr 15604. 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 11718 | . . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℂ) | |
2 | 1 | sqsqrtd 14801 | . . . . . . . . . . 11 ⊢ (𝜑 → ((√‘2)↑2) = 2) |
3 | sqrt2irrlem.3 | . . . . . . . . . . . 12 ⊢ (𝜑 → (√‘2) = (𝐴 / 𝐵)) | |
4 | 3 | oveq1d 7173 | . . . . . . . . . . 11 ⊢ (𝜑 → ((√‘2)↑2) = ((𝐴 / 𝐵)↑2)) |
5 | 2, 4 | eqtr3d 2860 | . . . . . . . . . 10 ⊢ (𝜑 → 2 = ((𝐴 / 𝐵)↑2)) |
6 | sqrt2irrlem.1 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
7 | 6 | zcnd 12091 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
8 | sqrt2irrlem.2 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
9 | 8 | nncnd 11656 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
10 | 8 | nnne0d 11690 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ≠ 0) |
11 | 7, 9, 10 | sqdivd 13526 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2))) |
12 | 5, 11 | eqtrd 2858 | . . . . . . . . 9 ⊢ (𝜑 → 2 = ((𝐴↑2) / (𝐵↑2))) |
13 | 12 | oveq1d 7173 | . . . . . . . 8 ⊢ (𝜑 → (2 · (𝐵↑2)) = (((𝐴↑2) / (𝐵↑2)) · (𝐵↑2))) |
14 | 7 | sqcld 13511 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴↑2) ∈ ℂ) |
15 | 8 | nnsqcld 13608 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐵↑2) ∈ ℕ) |
16 | 15 | nncnd 11656 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵↑2) ∈ ℂ) |
17 | 15 | nnne0d 11690 | . . . . . . . . 9 ⊢ (𝜑 → (𝐵↑2) ≠ 0) |
18 | 14, 16, 17 | divcan1d 11419 | . . . . . . . 8 ⊢ (𝜑 → (((𝐴↑2) / (𝐵↑2)) · (𝐵↑2)) = (𝐴↑2)) |
19 | 13, 18 | eqtrd 2858 | . . . . . . 7 ⊢ (𝜑 → (2 · (𝐵↑2)) = (𝐴↑2)) |
20 | 19 | oveq1d 7173 | . . . . . 6 ⊢ (𝜑 → ((2 · (𝐵↑2)) / 2) = ((𝐴↑2) / 2)) |
21 | 2ne0 11744 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
22 | 21 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 2 ≠ 0) |
23 | 16, 1, 22 | divcan3d 11423 | . . . . . 6 ⊢ (𝜑 → ((2 · (𝐵↑2)) / 2) = (𝐵↑2)) |
24 | 20, 23 | eqtr3d 2860 | . . . . 5 ⊢ (𝜑 → ((𝐴↑2) / 2) = (𝐵↑2)) |
25 | 24, 15 | eqeltrd 2915 | . . . 4 ⊢ (𝜑 → ((𝐴↑2) / 2) ∈ ℕ) |
26 | 25 | nnzd 12089 | . . 3 ⊢ (𝜑 → ((𝐴↑2) / 2) ∈ ℤ) |
27 | zesq 13590 | . . . 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 259 | . 2 ⊢ (𝜑 → (𝐴 / 2) ∈ ℤ) |
30 | 1 | sqvald 13510 | . . . . . . . 8 ⊢ (𝜑 → (2↑2) = (2 · 2)) |
31 | 30 | oveq2d 7174 | . . . . . . 7 ⊢ (𝜑 → ((𝐴↑2) / (2↑2)) = ((𝐴↑2) / (2 · 2))) |
32 | 7, 1, 22 | sqdivd 13526 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 / 2)↑2) = ((𝐴↑2) / (2↑2))) |
33 | 14, 1, 1, 22, 22 | divdiv1d 11449 | . . . . . . 7 ⊢ (𝜑 → (((𝐴↑2) / 2) / 2) = ((𝐴↑2) / (2 · 2))) |
34 | 31, 32, 33 | 3eqtr4d 2868 | . . . . . 6 ⊢ (𝜑 → ((𝐴 / 2)↑2) = (((𝐴↑2) / 2) / 2)) |
35 | 24 | oveq1d 7173 | . . . . . 6 ⊢ (𝜑 → (((𝐴↑2) / 2) / 2) = ((𝐵↑2) / 2)) |
36 | 34, 35 | eqtrd 2858 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 2)↑2) = ((𝐵↑2) / 2)) |
37 | zsqcl 13497 | . . . . . 6 ⊢ ((𝐴 / 2) ∈ ℤ → ((𝐴 / 2)↑2) ∈ ℤ) | |
38 | 29, 37 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝐴 / 2)↑2) ∈ ℤ) |
39 | 36, 38 | eqeltrrd 2916 | . . . 4 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℤ) |
40 | 15 | nnrpd 12432 | . . . . . 6 ⊢ (𝜑 → (𝐵↑2) ∈ ℝ+) |
41 | 40 | rphalfcld 12446 | . . . . 5 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℝ+) |
42 | 41 | rpgt0d 12437 | . . . 4 ⊢ (𝜑 → 0 < ((𝐵↑2) / 2)) |
43 | elnnz 11994 | . . . 4 ⊢ (((𝐵↑2) / 2) ∈ ℕ ↔ (((𝐵↑2) / 2) ∈ ℤ ∧ 0 < ((𝐵↑2) / 2))) | |
44 | 39, 42, 43 | sylanbrc 585 | . . 3 ⊢ (𝜑 → ((𝐵↑2) / 2) ∈ ℕ) |
45 | nnesq 13591 | . . . 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 259 | . 2 ⊢ (𝜑 → (𝐵 / 2) ∈ ℕ) |
48 | 29, 47 | jca 514 | 1 ⊢ (𝜑 → ((𝐴 / 2) ∈ ℤ ∧ (𝐵 / 2) ∈ ℕ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 0cc0 10539 · cmul 10544 < clt 10677 / cdiv 11299 ℕcn 11640 2c2 11695 ℤcz 11984 ↑cexp 13432 √csqrt 14594 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 |
This theorem is referenced by: sqrt2irr 15604 |
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