pythagtriplem16 - Metamath Proof Explorer

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Theorem pythagtriplem16 15535

Description: Lemma for pythagtrip 15539. Show the relationship between

, and

. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

**Hypotheses**
Ref	Expression
pythagtriplem15.1
pythagtriplem15.2

**Assertion**
Ref	Expression
pythagtriplem16	$/\$ $/\$ $/\$ $/\$ $/\$

**Proof of Theorem pythagtriplem16**
Step	Hyp	Ref	Expression
1		pythagtriplem15.1	. . . . 5
2		pythagtriplem15.2	. . . . 5
3	1, 2	oveq12i 6662	. . . 4
4		nncn 11028	. . . . . . . . . . . 12
5		nncn 11028	. . . . . . . . . . . 12
6		addcl 10018	. . . . . . . . . . . 12 $/\$
7	4, 5, 6	syl2anr 495	. . . . . . . . . . 11 $/\$
8	7	sqrtcld 14176	. . . . . . . . . 10 $/\$
9		subcl 10280	. . . . . . . . . . . 12 $/\$
10	4, 5, 9	syl2anr 495	. . . . . . . . . . 11 $/\$
11	10	sqrtcld 14176	. . . . . . . . . 10 $/\$
12		addcl 10018	. . . . . . . . . 10 $/\$
13	8, 11, 12	syl2anc 693	. . . . . . . . 9 $/\$
14	13	3adant1 1079	. . . . . . . 8 $/\$ $/\$
15	14	3ad2ant1 1082	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
16		subcl 10280	. . . . . . . . . 10 $/\$
17	8, 11, 16	syl2anc 693	. . . . . . . . 9 $/\$
18	17	3adant1 1079	. . . . . . . 8 $/\$ $/\$
19	18	3ad2ant1 1082	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
20		2cnne0 11242	. . . . . . . 8 $/\$
21		divmuldiv 10725	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
22	20, 20, 21	mpanr12 721	. . . . . . 7 $/\$
23	15, 19, 22	syl2anc 693	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
24	13, 17	mulcld 10060	. . . . . . . . 9 $/\$
25	24	3adant1 1079	. . . . . . . 8 $/\$ $/\$
26	25	3ad2ant1 1082	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
27		divdiv1 10736	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
28	20, 20, 27	mp3an23 1416	. . . . . . 7
29	26, 28	syl 17	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
30	23, 29	eqtr4d 2659	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
31		nnre 11027	. . . . . . . . . . . . 13
32		nnre 11027	. . . . . . . . . . . . 13
33		readdcl 10019	. . . . . . . . . . . . 13 $/\$
34	31, 32, 33	syl2anr 495	. . . . . . . . . . . 12 $/\$
35	34	3adant1 1079	. . . . . . . . . . 11 $/\$ $/\$
36	35	3ad2ant1 1082	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
37	31	adantl 482	. . . . . . . . . . . . . 14 $/\$
38	32	adantr 481	. . . . . . . . . . . . . 14 $/\$
39		nngt0 11049	. . . . . . . . . . . . . . 15
40	39	adantl 482	. . . . . . . . . . . . . 14 $/\$
41		nngt0 11049	. . . . . . . . . . . . . . 15
42	41	adantr 481	. . . . . . . . . . . . . 14 $/\$
43	37, 38, 40, 42	addgt0d 10602	. . . . . . . . . . . . 13 $/\$
44		0re 10040	. . . . . . . . . . . . . 14
45		ltle 10126	. . . . . . . . . . . . . 14 $/\$
46	44, 45	mpan 706	. . . . . . . . . . . . 13
47	34, 43, 46	sylc 65	. . . . . . . . . . . 12 $/\$
48	47	3adant1 1079	. . . . . . . . . . 11 $/\$ $/\$
49	48	3ad2ant1 1082	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
50		resqrtth 13996	. . . . . . . . . 10 $/\$
51	36, 49, 50	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
52		resubcl 10345	. . . . . . . . . . . . 13 $/\$
53	31, 32, 52	syl2anr 495	. . . . . . . . . . . 12 $/\$
54	53	3adant1 1079	. . . . . . . . . . 11 $/\$ $/\$
55	54	3ad2ant1 1082	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
56		pythagtriplem10 15525	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
57	56	3adant3 1081	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
58		ltle 10126	. . . . . . . . . . . 12 $/\$
59	44, 58	mpan 706	. . . . . . . . . . 11
60	55, 57, 59	sylc 65	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
61		resqrtth 13996	. . . . . . . . . 10 $/\$
62	55, 60, 61	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
63	51, 62	oveq12d 6668	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
64	63	oveq1d 6665	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
65		simp12 1092	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
66		simp13 1093	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
67	65, 66, 8	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
68	65, 66, 11	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
69		subsq 12972	. . . . . . . . 9 $/\$
70	67, 68, 69	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
71	70	oveq1d 6665	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
72		pnncan 10322	. . . . . . . . . . . . . 14 $/\$ $/\$
73	72	3anidm23 1385	. . . . . . . . . . . . 13 $/\$
74		2times 11145	. . . . . . . . . . . . . 14
75	74	adantl 482	. . . . . . . . . . . . 13 $/\$
76	73, 75	eqtr4d 2659	. . . . . . . . . . . 12 $/\$
77	4, 5, 76	syl2anr 495	. . . . . . . . . . 11 $/\$
78	77	3adant1 1079	. . . . . . . . . 10 $/\$ $/\$
79	78	3ad2ant1 1082	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
80	79	oveq1d 6665	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
81		2cn 11091	. . . . . . . . . 10
82		2ne0 11113	. . . . . . . . . 10
83		divcan3 10711	. . . . . . . . . 10 $/\$ $/\$
84	81, 82, 83	mp3an23 1416	. . . . . . . . 9
85	65, 5, 84	3syl 18	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
86	80, 85	eqtrd 2656	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
87	64, 71, 86	3eqtr3d 2664	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
88	87	oveq1d 6665	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
89	30, 88	eqtrd 2656	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
90	3, 89	syl5eq 2668	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
91	90	oveq2d 6666	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
92		divcan2 10693	. . . . . 6 $/\$ $/\$
93	81, 82, 92	mp3an23 1416	. . . . 5
94	5, 93	syl 17	. . . 4
95	94	3ad2ant2 1083	. . 3 $/\$ $/\$
96	95	3ad2ant1 1082	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
97	91, 96	eqtr2d 2657	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794 class class class wbr 4653

cfv 5888 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

c1 9937

caddc 9939

cmul 9941

clt 10074

cle 10075

cmin 10266

cdiv 10684

cn 11020

c2 11070

cexp 12860

csqrt 13973

cdvds 14983

cgcd 15216

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976

This theorem is referenced by: pythagtriplem18 15537

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