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Mirrors > Home > ILE Home > Th. List > pythagtriplem10 | Unicode version |
Description: Lemma for pythagtrip 12224. Show that is positive. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
pythagtriplem10 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnre 8872 | . . . . . . . 8 | |
2 | 1 | 3ad2ant1 1013 | . . . . . . 7 |
3 | nnap0 8894 | . . . . . . . 8 # | |
4 | 3 | 3ad2ant1 1013 | . . . . . . 7 # |
5 | 2, 4 | sqgt0apd 10624 | . . . . . 6 |
6 | 2 | resqcld 10622 | . . . . . . 7 |
7 | nnre 8872 | . . . . . . . . 9 | |
8 | 7 | 3ad2ant2 1014 | . . . . . . . 8 |
9 | 8 | resqcld 10622 | . . . . . . 7 |
10 | 6, 9 | ltaddpos2d 8436 | . . . . . 6 |
11 | 5, 10 | mpbid 146 | . . . . 5 |
12 | 11 | adantr 274 | . . . 4 |
13 | simpr 109 | . . . 4 | |
14 | 12, 13 | breqtrd 4013 | . . 3 |
15 | 8 | adantr 274 | . . . 4 |
16 | nnre 8872 | . . . . . 6 | |
17 | 16 | 3ad2ant3 1015 | . . . . 5 |
18 | 17 | adantr 274 | . . . 4 |
19 | nnnn0 9129 | . . . . . . 7 | |
20 | 19 | nn0ge0d 9178 | . . . . . 6 |
21 | 20 | 3ad2ant2 1014 | . . . . 5 |
22 | 21 | adantr 274 | . . . 4 |
23 | nnnn0 9129 | . . . . . . 7 | |
24 | 23 | nn0ge0d 9178 | . . . . . 6 |
25 | 24 | 3ad2ant3 1015 | . . . . 5 |
26 | 25 | adantr 274 | . . . 4 |
27 | 15, 18, 22, 26 | lt2sqd 10627 | . . 3 |
28 | 14, 27 | mpbird 166 | . 2 |
29 | 15, 18 | posdifd 8438 | . 2 |
30 | 28, 29 | mpbid 146 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 973 wceq 1348 wcel 2141 class class class wbr 3987 (class class class)co 5850 cr 7760 cc0 7761 caddc 7764 clt 7941 cle 7942 cmin 8077 # cap 8487 cn 8865 c2 8916 cexp 10462 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-mulrcl 7860 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-precex 7871 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-apti 7876 ax-pre-ltadd 7877 ax-pre-mulgt0 7878 ax-pre-mulext 7879 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-recs 6281 df-frec 6367 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-reap 8481 df-ap 8488 df-div 8577 df-inn 8866 df-2 8924 df-n0 9123 df-z 9200 df-uz 9475 df-seqfrec 10389 df-exp 10463 |
This theorem is referenced by: pythagtriplem6 12211 pythagtriplem12 12216 pythagtriplem13 12217 pythagtriplem14 12218 pythagtriplem16 12220 |
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