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Mirrors > Home > ILE Home > Th. List > sqrt2irraplemnn | Unicode version |
Description: Lemma for sqrt2irrap 12091. 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 10598 | . . . . . 6 |
3 | 2 | nnred 8861 | . . . . 5 |
4 | 0red 7891 | . . . . . 6 | |
5 | 2 | nngt0d 8892 | . . . . . 6 |
6 | 4, 3, 5 | ltled 8008 | . . . . 5 |
7 | simpr 109 | . . . . . . 7 | |
8 | 7 | nnsqcld 10598 | . . . . . 6 |
9 | 8 | nnrpd 9621 | . . . . 5 |
10 | 3, 6, 9 | sqrtdivd 11096 | . . . 4 |
11 | 1 | nnred 8861 | . . . . . 6 |
12 | 1 | nngt0d 8892 | . . . . . . 7 |
13 | 4, 11, 12 | ltled 8008 | . . . . . 6 |
14 | 11, 13 | sqrtsqd 11093 | . . . . 5 |
15 | 7 | nnred 8861 | . . . . . 6 |
16 | 7 | nngt0d 8892 | . . . . . . 7 |
17 | 4, 15, 16 | ltled 8008 | . . . . . 6 |
18 | 15, 17 | sqrtsqd 11093 | . . . . 5 |
19 | 14, 18 | oveq12d 5854 | . . . 4 |
20 | 10, 19 | eqtrd 2197 | . . 3 |
21 | sqne2sq 12088 | . . . . . 6 | |
22 | 2 | nncnd 8862 | . . . . . . . 8 |
23 | 2cnd 8921 | . . . . . . . 8 | |
24 | 8 | nncnd 8862 | . . . . . . . 8 |
25 | 8 | nnap0d 8894 | . . . . . . . 8 # |
26 | 22, 23, 24, 25 | divmulap3d 8712 | . . . . . . 7 |
27 | 26 | necon3bid 2375 | . . . . . 6 |
28 | 21, 27 | mpbird 166 | . . . . 5 |
29 | 2 | nnzd 9303 | . . . . . . 7 |
30 | znq 9553 | . . . . . . 7 | |
31 | 29, 8, 30 | syl2anc 409 | . . . . . 6 |
32 | 2z 9210 | . . . . . . 7 | |
33 | zq 9555 | . . . . . . 7 | |
34 | 32, 33 | mp1i 10 | . . . . . 6 |
35 | qapne 9568 | . . . . . 6 # | |
36 | 31, 34, 35 | syl2anc 409 | . . . . 5 # |
37 | 28, 36 | mpbird 166 | . . . 4 # |
38 | qre 9554 | . . . . . 6 | |
39 | 31, 38 | syl 14 | . . . . 5 |
40 | 8 | nnred 8861 | . . . . . . 7 |
41 | 8 | nngt0d 8892 | . . . . . . 7 |
42 | 3, 40, 5, 41 | divgt0d 8821 | . . . . . 6 |
43 | 4, 39, 42 | ltled 8008 | . . . . 5 |
44 | 2re 8918 | . . . . . 6 | |
45 | 44 | a1i 9 | . . . . 5 |
46 | 0le2 8938 | . . . . . 6 | |
47 | 46 | a1i 9 | . . . . 5 |
48 | sqrt11ap 10966 | . . . . 5 # # | |
49 | 39, 43, 45, 47, 48 | syl22anc 1228 | . . . 4 # # |
50 | 37, 49 | mpbird 166 | . . 3 # |
51 | 20, 50 | eqbrtrrd 4000 | . 2 # |
52 | nnz 9201 | . . . . 5 | |
53 | znq 9553 | . . . . 5 | |
54 | 52, 53 | sylan 281 | . . . 4 |
55 | qcn 9563 | . . . 4 | |
56 | 54, 55 | syl 14 | . . 3 |
57 | sqrt2re 12074 | . . . . 5 | |
58 | 57 | recni 7902 | . . . 4 |
59 | 58 | a1i 9 | . . 3 |
60 | apsym 8495 | . . 3 # # | |
61 | 56, 59, 60 | syl2anc 409 | . 2 # # |
62 | 51, 61 | mpbid 146 | 1 # |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 2135 wne 2334 class class class wbr 3976 cfv 5182 (class class class)co 5836 cc 7742 cr 7743 cc0 7744 cmul 7749 cle 7925 # cap 8470 cdiv 8559 cn 8848 c2 8899 cz 9182 cq 9548 cexp 10444 csqrt 10924 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-xor 1365 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-1o 6375 df-2o 6376 df-er 6492 df-en 6698 df-sup 6940 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-fz 9936 df-fzo 10068 df-fl 10195 df-mod 10248 df-seqfrec 10371 df-exp 10445 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-dvds 11714 df-gcd 11861 df-prm 12019 |
This theorem is referenced by: sqrt2irrap 12091 |
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