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Mirrors > Home > ILE Home > Th. List > pythagtriplem4 | Unicode version |
Description: Lemma for pythagtrip 12174. Show that and are relatively prime. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
pythagtriplem4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3r 1011 | . . 3 | |
2 | nnz 9192 | . . . . . . . . . . . . 13 | |
3 | nnz 9192 | . . . . . . . . . . . . 13 | |
4 | zsubcl 9214 | . . . . . . . . . . . . 13 | |
5 | 2, 3, 4 | syl2anr 288 | . . . . . . . . . . . 12 |
6 | 5 | 3adant1 1000 | . . . . . . . . . . 11 |
7 | 6 | 3ad2ant1 1003 | . . . . . . . . . 10 |
8 | simp13 1014 | . . . . . . . . . . . 12 | |
9 | simp12 1013 | . . . . . . . . . . . 12 | |
10 | 8, 9 | nnaddcld 8887 | . . . . . . . . . . 11 |
11 | 10 | nnzd 9291 | . . . . . . . . . 10 |
12 | gcddvds 11863 | . . . . . . . . . 10 | |
13 | 7, 11, 12 | syl2anc 409 | . . . . . . . . 9 |
14 | 13 | simprd 113 | . . . . . . . 8 |
15 | breq1 3970 | . . . . . . . . 9 | |
16 | 15 | biimpd 143 | . . . . . . . 8 |
17 | 14, 16 | mpan9 279 | . . . . . . 7 |
18 | 2z 9201 | . . . . . . . 8 | |
19 | simpl13 1059 | . . . . . . . . . 10 | |
20 | 19 | nnzd 9291 | . . . . . . . . 9 |
21 | simpl12 1058 | . . . . . . . . . 10 | |
22 | 21 | nnzd 9291 | . . . . . . . . 9 |
23 | 20, 22 | zaddcld 9296 | . . . . . . . 8 |
24 | 20, 22 | zsubcld 9297 | . . . . . . . 8 |
25 | dvdsmultr1 11738 | . . . . . . . 8 | |
26 | 18, 23, 24, 25 | mp3an2i 1324 | . . . . . . 7 |
27 | 17, 26 | mpd 13 | . . . . . 6 |
28 | 19 | nncnd 8853 | . . . . . . 7 |
29 | 21 | nncnd 8853 | . . . . . . 7 |
30 | subsq 10535 | . . . . . . 7 | |
31 | 28, 29, 30 | syl2anc 409 | . . . . . 6 |
32 | 27, 31 | breqtrrd 3995 | . . . . 5 |
33 | simpl2 986 | . . . . . . 7 | |
34 | 33 | oveq1d 5842 | . . . . . 6 |
35 | simpl11 1057 | . . . . . . . . 9 | |
36 | 35 | nnsqcld 10582 | . . . . . . . 8 |
37 | 36 | nncnd 8853 | . . . . . . 7 |
38 | 21 | nnsqcld 10582 | . . . . . . . 8 |
39 | 38 | nncnd 8853 | . . . . . . 7 |
40 | 37, 39 | pncand 8192 | . . . . . 6 |
41 | 34, 40 | eqtr3d 2192 | . . . . 5 |
42 | 32, 41 | breqtrd 3993 | . . . 4 |
43 | nnz 9192 | . . . . . . . 8 | |
44 | 43 | 3ad2ant1 1003 | . . . . . . 7 |
45 | 44 | 3ad2ant1 1003 | . . . . . 6 |
46 | 45 | adantr 274 | . . . . 5 |
47 | 2prm 12020 | . . . . . 6 | |
48 | 2nn 9000 | . . . . . 6 | |
49 | prmdvdsexp 12039 | . . . . . 6 | |
50 | 47, 48, 49 | mp3an13 1310 | . . . . 5 |
51 | 46, 50 | syl 14 | . . . 4 |
52 | 42, 51 | mpbid 146 | . . 3 |
53 | 1, 52 | mtand 655 | . 2 |
54 | neg1z 9205 | . . . . . . . 8 | |
55 | gcdaddm 11884 | . . . . . . . 8 | |
56 | 54, 7, 11, 55 | mp3an2i 1324 | . . . . . . 7 |
57 | 8 | nncnd 8853 | . . . . . . . 8 |
58 | 9 | nncnd 8853 | . . . . . . . 8 |
59 | pnncan 8121 | . . . . . . . . . . 11 | |
60 | 59 | 3anidm23 1279 | . . . . . . . . . 10 |
61 | subcl 8079 | . . . . . . . . . . . . 13 | |
62 | 61 | mulm1d 8290 | . . . . . . . . . . . 12 |
63 | 62 | oveq2d 5843 | . . . . . . . . . . 11 |
64 | addcl 7860 | . . . . . . . . . . . 12 | |
65 | 64, 61 | negsubd 8197 | . . . . . . . . . . 11 |
66 | 63, 65 | eqtrd 2190 | . . . . . . . . . 10 |
67 | 2times 8967 | . . . . . . . . . . 11 | |
68 | 67 | adantl 275 | . . . . . . . . . 10 |
69 | 60, 66, 68 | 3eqtr4d 2200 | . . . . . . . . 9 |
70 | 69 | oveq2d 5843 | . . . . . . . 8 |
71 | 57, 58, 70 | syl2anc 409 | . . . . . . 7 |
72 | 56, 71 | eqtrd 2190 | . . . . . 6 |
73 | 9 | nnzd 9291 | . . . . . . . . 9 |
74 | zmulcl 9226 | . . . . . . . . 9 | |
75 | 18, 73, 74 | sylancr 411 | . . . . . . . 8 |
76 | gcddvds 11863 | . . . . . . . 8 | |
77 | 7, 75, 76 | syl2anc 409 | . . . . . . 7 |
78 | 77 | simprd 113 | . . . . . 6 |
79 | 72, 78 | eqbrtrd 3989 | . . . . 5 |
80 | 1z 9199 | . . . . . . . 8 | |
81 | gcdaddm 11884 | . . . . . . . 8 | |
82 | 80, 7, 11, 81 | mp3an2i 1324 | . . . . . . 7 |
83 | ppncan 8122 | . . . . . . . . . . 11 | |
84 | 83 | 3anidm13 1278 | . . . . . . . . . 10 |
85 | 61 | mulid2d 7899 | . . . . . . . . . . 11 |
86 | 85 | oveq2d 5843 | . . . . . . . . . 10 |
87 | 2times 8967 | . . . . . . . . . . 11 | |
88 | 87 | adantr 274 | . . . . . . . . . 10 |
89 | 84, 86, 88 | 3eqtr4d 2200 | . . . . . . . . 9 |
90 | 57, 58, 89 | syl2anc 409 | . . . . . . . 8 |
91 | 90 | oveq2d 5843 | . . . . . . 7 |
92 | 82, 91 | eqtrd 2190 | . . . . . 6 |
93 | 8 | nnzd 9291 | . . . . . . . . 9 |
94 | zmulcl 9226 | . . . . . . . . 9 | |
95 | 18, 93, 94 | sylancr 411 | . . . . . . . 8 |
96 | gcddvds 11863 | . . . . . . . 8 | |
97 | 7, 95, 96 | syl2anc 409 | . . . . . . 7 |
98 | 97 | simprd 113 | . . . . . 6 |
99 | 92, 98 | eqbrtrd 3989 | . . . . 5 |
100 | nnaddcl 8859 | . . . . . . . . . . . . . 14 | |
101 | 100 | nnne0d 8884 | . . . . . . . . . . . . 13 |
102 | 101 | ancoms 266 | . . . . . . . . . . . 12 |
103 | 102 | 3adant1 1000 | . . . . . . . . . . 11 |
104 | 103 | 3ad2ant1 1003 | . . . . . . . . . 10 |
105 | 104 | neneqd 2348 | . . . . . . . . 9 |
106 | 105 | intnand 917 | . . . . . . . 8 |
107 | gcdn0cl 11862 | . . . . . . . 8 | |
108 | 7, 11, 106, 107 | syl21anc 1219 | . . . . . . 7 |
109 | 108 | nnzd 9291 | . . . . . 6 |
110 | dvdsgcd 11912 | . . . . . 6 | |
111 | 109, 75, 95, 110 | syl3anc 1220 | . . . . 5 |
112 | 79, 99, 111 | mp2and 430 | . . . 4 |
113 | 2nn0 9113 | . . . . . 6 | |
114 | mulgcd 11916 | . . . . . 6 | |
115 | 113, 73, 93, 114 | mp3an2i 1324 | . . . . 5 |
116 | pythagtriplem3 12158 | . . . . . . 7 | |
117 | 116 | oveq2d 5843 | . . . . . 6 |
118 | 2t1e2 8992 | . . . . . 6 | |
119 | 117, 118 | eqtrdi 2206 | . . . . 5 |
120 | 115, 119 | eqtrd 2190 | . . . 4 |
121 | 112, 120 | breqtrd 3993 | . . 3 |
122 | dvdsprime 12015 | . . . 4 | |
123 | 47, 108, 122 | sylancr 411 | . . 3 |
124 | 121, 123 | mpbid 146 | . 2 |
125 | orel1 715 | . 2 | |
126 | 53, 124, 125 | sylc 62 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 w3a 963 wceq 1335 wcel 2128 wne 2327 class class class wbr 3967 (class class class)co 5827 cc 7733 cc0 7735 c1 7736 caddc 7738 cmul 7740 cmin 8051 cneg 8052 cn 8839 c2 8890 cn0 9096 cz 9173 cexp 10428 cdvds 11695 cgcd 11842 cprime 12000 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4082 ax-sep 4085 ax-nul 4093 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 ax-iinf 4550 ax-cnex 7826 ax-resscn 7827 ax-1cn 7828 ax-1re 7829 ax-icn 7830 ax-addcl 7831 ax-addrcl 7832 ax-mulcl 7833 ax-mulrcl 7834 ax-addcom 7835 ax-mulcom 7836 ax-addass 7837 ax-mulass 7838 ax-distr 7839 ax-i2m1 7840 ax-0lt1 7841 ax-1rid 7842 ax-0id 7843 ax-rnegex 7844 ax-precex 7845 ax-cnre 7846 ax-pre-ltirr 7847 ax-pre-ltwlin 7848 ax-pre-lttrn 7849 ax-pre-apti 7850 ax-pre-ltadd 7851 ax-pre-mulgt0 7852 ax-pre-mulext 7853 ax-arch 7854 ax-caucvg 7855 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-if 3507 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4029 df-mpt 4030 df-tr 4066 df-id 4256 df-po 4259 df-iso 4260 df-iord 4329 df-on 4331 df-ilim 4332 df-suc 4334 df-iom 4553 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-f1 5178 df-fo 5179 df-f1o 5180 df-fv 5181 df-riota 5783 df-ov 5830 df-oprab 5831 df-mpo 5832 df-1st 6091 df-2nd 6092 df-recs 6255 df-frec 6341 df-1o 6366 df-2o 6367 df-er 6483 df-en 6689 df-sup 6931 df-pnf 7917 df-mnf 7918 df-xr 7919 df-ltxr 7920 df-le 7921 df-sub 8053 df-neg 8054 df-reap 8455 df-ap 8462 df-div 8551 df-inn 8840 df-2 8898 df-3 8899 df-4 8900 df-n0 9097 df-z 9174 df-uz 9446 df-q 9536 df-rp 9568 df-fz 9920 df-fzo 10052 df-fl 10179 df-mod 10232 df-seqfrec 10355 df-exp 10429 df-cj 10754 df-re 10755 df-im 10756 df-rsqrt 10910 df-abs 10911 df-dvds 11696 df-gcd 11843 df-prm 12001 |
This theorem is referenced by: pythagtriplem6 12161 pythagtriplem7 12162 |
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