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Mirrors > Home > ILE Home > Th. List > sqrt2irraplemnn | Unicode version |
Description: Lemma for sqrt2irrap 11847. The square root of 2 is apart from a positive rational expressed as a numerator and denominator. (Contributed by Jim Kingdon, 2-Oct-2021.) |
Ref | Expression |
---|---|
sqrt2irraplemnn | # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . . . . 7 | |
2 | 1 | nnsqcld 10438 | . . . . . 6 |
3 | 2 | nnred 8726 | . . . . 5 |
4 | 0red 7760 | . . . . . 6 | |
5 | 2 | nngt0d 8757 | . . . . . 6 |
6 | 4, 3, 5 | ltled 7874 | . . . . 5 |
7 | simpr 109 | . . . . . . 7 | |
8 | 7 | nnsqcld 10438 | . . . . . 6 |
9 | 8 | nnrpd 9475 | . . . . 5 |
10 | 3, 6, 9 | sqrtdivd 10933 | . . . 4 |
11 | 1 | nnred 8726 | . . . . . 6 |
12 | 1 | nngt0d 8757 | . . . . . . 7 |
13 | 4, 11, 12 | ltled 7874 | . . . . . 6 |
14 | 11, 13 | sqrtsqd 10930 | . . . . 5 |
15 | 7 | nnred 8726 | . . . . . 6 |
16 | 7 | nngt0d 8757 | . . . . . . 7 |
17 | 4, 15, 16 | ltled 7874 | . . . . . 6 |
18 | 15, 17 | sqrtsqd 10930 | . . . . 5 |
19 | 14, 18 | oveq12d 5785 | . . . 4 |
20 | 10, 19 | eqtrd 2170 | . . 3 |
21 | sqne2sq 11844 | . . . . . 6 | |
22 | 2 | nncnd 8727 | . . . . . . . 8 |
23 | 2cnd 8786 | . . . . . . . 8 | |
24 | 8 | nncnd 8727 | . . . . . . . 8 |
25 | 8 | nnap0d 8759 | . . . . . . . 8 # |
26 | 22, 23, 24, 25 | divmulap3d 8578 | . . . . . . 7 |
27 | 26 | necon3bid 2347 | . . . . . 6 |
28 | 21, 27 | mpbird 166 | . . . . 5 |
29 | 2 | nnzd 9165 | . . . . . . 7 |
30 | znq 9409 | . . . . . . 7 | |
31 | 29, 8, 30 | syl2anc 408 | . . . . . 6 |
32 | 2z 9075 | . . . . . . 7 | |
33 | zq 9411 | . . . . . . 7 | |
34 | 32, 33 | mp1i 10 | . . . . . 6 |
35 | qapne 9424 | . . . . . 6 # | |
36 | 31, 34, 35 | syl2anc 408 | . . . . 5 # |
37 | 28, 36 | mpbird 166 | . . . 4 # |
38 | qre 9410 | . . . . . 6 | |
39 | 31, 38 | syl 14 | . . . . 5 |
40 | 8 | nnred 8726 | . . . . . . 7 |
41 | 8 | nngt0d 8757 | . . . . . . 7 |
42 | 3, 40, 5, 41 | divgt0d 8686 | . . . . . 6 |
43 | 4, 39, 42 | ltled 7874 | . . . . 5 |
44 | 2re 8783 | . . . . . 6 | |
45 | 44 | a1i 9 | . . . . 5 |
46 | 0le2 8803 | . . . . . 6 | |
47 | 46 | a1i 9 | . . . . 5 |
48 | sqrt11ap 10803 | . . . . 5 # # | |
49 | 39, 43, 45, 47, 48 | syl22anc 1217 | . . . 4 # # |
50 | 37, 49 | mpbird 166 | . . 3 # |
51 | 20, 50 | eqbrtrrd 3947 | . 2 # |
52 | nnz 9066 | . . . . 5 | |
53 | znq 9409 | . . . . 5 | |
54 | 52, 53 | sylan 281 | . . . 4 |
55 | qcn 9419 | . . . 4 | |
56 | 54, 55 | syl 14 | . . 3 |
57 | sqrt2re 11830 | . . . . 5 | |
58 | 57 | recni 7771 | . . . 4 |
59 | 58 | a1i 9 | . . 3 |
60 | apsym 8361 | . . 3 # # | |
61 | 56, 59, 60 | syl2anc 408 | . 2 # # |
62 | 51, 61 | mpbid 146 | 1 # |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 1480 wne 2306 class class class wbr 3924 cfv 5118 (class class class)co 5767 cc 7611 cr 7612 cc0 7613 cmul 7618 cle 7794 # cap 8336 cdiv 8425 cn 8713 c2 8764 cz 9047 cq 9404 cexp 10285 csqrt 10761 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 ax-caucvg 7733 |
This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-xor 1354 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-1o 6306 df-2o 6307 df-er 6422 df-en 6628 df-sup 6864 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-n0 8971 df-z 9048 df-uz 9320 df-q 9405 df-rp 9435 df-fz 9784 df-fzo 9913 df-fl 10036 df-mod 10089 df-seqfrec 10212 df-exp 10286 df-cj 10607 df-re 10608 df-im 10609 df-rsqrt 10763 df-abs 10764 df-dvds 11483 df-gcd 11625 df-prm 11778 |
This theorem is referenced by: sqrt2irrap 11847 |
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