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Mirrors > Home > ILE Home > Th. List > sqrt2irrlem | Unicode version |
Description: Lemma for sqrt2irr 11767. This is the core of the proof: - if , then and are even, so and are smaller representatives, which is absurd by the method of infinite descent (here implemented by strong induction). (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.) |
Ref | Expression |
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sqrt2irrlem.1 | |
sqrt2irrlem.2 | |
sqrt2irrlem.3 |
Ref | Expression |
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sqrt2irrlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 8758 | . . . . . . . . . . . 12 | |
2 | 0le2 8778 | . . . . . . . . . . . 12 | |
3 | resqrtth 10771 | . . . . . . . . . . . 12 | |
4 | 1, 2, 3 | mp2an 422 | . . . . . . . . . . 11 |
5 | sqrt2irrlem.3 | . . . . . . . . . . . 12 | |
6 | 5 | oveq1d 5757 | . . . . . . . . . . 11 |
7 | 4, 6 | syl5eqr 2164 | . . . . . . . . . 10 |
8 | sqrt2irrlem.1 | . . . . . . . . . . . 12 | |
9 | 8 | zcnd 9142 | . . . . . . . . . . 11 |
10 | sqrt2irrlem.2 | . . . . . . . . . . . 12 | |
11 | 10 | nncnd 8702 | . . . . . . . . . . 11 |
12 | 10 | nnap0d 8734 | . . . . . . . . . . 11 # |
13 | 9, 11, 12 | sqdivapd 10405 | . . . . . . . . . 10 |
14 | 7, 13 | eqtrd 2150 | . . . . . . . . 9 |
15 | 14 | oveq1d 5757 | . . . . . . . 8 |
16 | 9 | sqcld 10390 | . . . . . . . . 9 |
17 | 10 | nnsqcld 10413 | . . . . . . . . . 10 |
18 | 17 | nncnd 8702 | . . . . . . . . 9 |
19 | 17 | nnap0d 8734 | . . . . . . . . 9 # |
20 | 16, 18, 19 | divcanap1d 8519 | . . . . . . . 8 |
21 | 15, 20 | eqtrd 2150 | . . . . . . 7 |
22 | 21 | oveq1d 5757 | . . . . . 6 |
23 | 2cnd 8761 | . . . . . . 7 | |
24 | 2ap0 8781 | . . . . . . . 8 # | |
25 | 24 | a1i 9 | . . . . . . 7 # |
26 | 18, 23, 25 | divcanap3d 8523 | . . . . . 6 |
27 | 22, 26 | eqtr3d 2152 | . . . . 5 |
28 | 27, 17 | eqeltrd 2194 | . . . 4 |
29 | 28 | nnzd 9140 | . . 3 |
30 | zesq 10378 | . . . 4 | |
31 | 8, 30 | syl 14 | . . 3 |
32 | 29, 31 | mpbird 166 | . 2 |
33 | 2cn 8759 | . . . . . . . . 9 | |
34 | 33 | sqvali 10340 | . . . . . . . 8 |
35 | 34 | oveq2i 5753 | . . . . . . 7 |
36 | 9, 23, 25 | sqdivapd 10405 | . . . . . . 7 |
37 | 16, 23, 23, 25, 25 | divdivap1d 8550 | . . . . . . 7 |
38 | 35, 36, 37 | 3eqtr4a 2176 | . . . . . 6 |
39 | 27 | oveq1d 5757 | . . . . . 6 |
40 | 38, 39 | eqtrd 2150 | . . . . 5 |
41 | zsqcl 10331 | . . . . . 6 | |
42 | 32, 41 | syl 14 | . . . . 5 |
43 | 40, 42 | eqeltrrd 2195 | . . . 4 |
44 | 17 | nnrpd 9450 | . . . . . 6 |
45 | 44 | rphalfcld 9464 | . . . . 5 |
46 | 45 | rpgt0d 9454 | . . . 4 |
47 | elnnz 9032 | . . . 4 | |
48 | 43, 46, 47 | sylanbrc 413 | . . 3 |
49 | nnesq 10379 | . . . 4 | |
50 | 10, 49 | syl 14 | . . 3 |
51 | 48, 50 | mpbird 166 | . 2 |
52 | 32, 51 | jca 304 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 class class class wbr 3899 cfv 5093 (class class class)co 5742 cr 7587 cc0 7588 cmul 7593 clt 7768 cle 7769 # cap 8311 cdiv 8400 cn 8688 c2 8739 cz 9022 cexp 10260 csqrt 10736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 ax-arch 7707 ax-caucvg 7708 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-2 8747 df-3 8748 df-4 8749 df-n0 8946 df-z 9023 df-uz 9295 df-rp 9410 df-seqfrec 10187 df-exp 10261 df-rsqrt 10738 |
This theorem is referenced by: sqrt2irr 11767 |
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