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Mirrors > Home > ILE Home > Th. List > lt2msq | Unicode version |
Description: Two nonnegative numbers compare the same as their squares. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
lt2msq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt2msq1 8815 | . . . 4 | |
2 | 1 | 3expia 1205 | . . 3 |
3 | 2 | adantrr 479 | . 2 |
4 | simpr 110 | . . . . . . 7 | |
5 | simpll 527 | . . . . . . 7 | |
6 | lt2msq1 8815 | . . . . . . . 8 | |
7 | 6 | 3expia 1205 | . . . . . . 7 |
8 | 4, 5, 7 | syl2anc 411 | . . . . . 6 |
9 | 8 | con3d 631 | . . . . 5 |
10 | 5, 5 | remulcld 7962 | . . . . . 6 |
11 | simprl 529 | . . . . . . 7 | |
12 | 11, 11 | remulcld 7962 | . . . . . 6 |
13 | 10, 12 | lenltd 8049 | . . . . 5 |
14 | 5, 11 | lenltd 8049 | . . . . 5 |
15 | 9, 13, 14 | 3imtr4d 203 | . . . 4 |
16 | 5 | recnd 7960 | . . . . . 6 |
17 | 11 | recnd 7960 | . . . . . 6 |
18 | mulext 8545 | . . . . . 6 # # # | |
19 | 16, 16, 17, 17, 18 | syl22anc 1239 | . . . . 5 # # # |
20 | oridm 757 | . . . . 5 # # # | |
21 | 19, 20 | syl6ib 161 | . . . 4 # # |
22 | 15, 21 | anim12d 335 | . . 3 # # |
23 | ltleap 8563 | . . . 4 # | |
24 | 10, 12, 23 | syl2anc 411 | . . 3 # |
25 | ltleap 8563 | . . . 4 # | |
26 | 5, 11, 25 | syl2anc 411 | . . 3 # |
27 | 22, 24, 26 | 3imtr4d 203 | . 2 |
28 | 3, 27 | impbid 129 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 104 wb 105 wo 708 wcel 2146 class class class wbr 3998 (class class class)co 5865 cc 7784 cr 7785 cc0 7786 cmul 7791 clt 7966 cle 7967 # cap 8512 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 ax-pre-mulext 7904 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-br 3999 df-opab 4060 df-id 4287 df-po 4290 df-iso 4291 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-iota 5170 df-fun 5210 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-reap 8506 df-ap 8513 |
This theorem is referenced by: le2msq 8831 lt2msqi 8844 lt2sq 10563 |
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