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Theorem pythagtriplem16 12170

Description: Lemma for pythagtrip 12174. 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 5839	. . . 4
4		simp13 1014	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
5		simp12 1013	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
6	4, 5	nnaddcld 8887	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
7	6	nnrpd 9608	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
8	7	rpsqrtcld 11070	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
9	8	rpcnd 9612	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
10	4	nnzd 9291	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
11	5	nnzd 9291	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
12	10, 11	zsubcld 9297	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
13	12	zred 9292	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
14		pythagtriplem10 12160	. . . . . . . . . . 11 $/\$ $/\$ $/\$
15	14	3adant3 1002	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
16	13, 15	elrpd 9607	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
17	16	rpsqrtcld 11070	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
18	17	rpcnd 9612	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
19	9, 18	addcld 7900	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
20		2cnd 8912	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
21	9, 18	subcld 8191	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
22		2ap0 8932	. . . . . . 7 #
23	22	a1i 9	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ #
24	19, 20, 21, 20, 23, 23	divmuldivapd 8710	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
25	19, 21	mulcld 7901	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
26	25, 20, 20, 23, 23	divdivap1d 8700	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
27		nnre 8846	. . . . . . . . . . . . 13
28		nnre 8846	. . . . . . . . . . . . 13
29		readdcl 7861	. . . . . . . . . . . . 13 $/\$
30	27, 28, 29	syl2anr 288	. . . . . . . . . . . 12 $/\$
31	30	3adant1 1000	. . . . . . . . . . 11 $/\$ $/\$
32	31	3ad2ant1 1003	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
33	27	adantl 275	. . . . . . . . . . . . . 14 $/\$
34	28	adantr 274	. . . . . . . . . . . . . 14 $/\$
35		nngt0 8864	. . . . . . . . . . . . . . 15
36	35	adantl 275	. . . . . . . . . . . . . 14 $/\$
37		nngt0 8864	. . . . . . . . . . . . . . 15
38	37	adantr 274	. . . . . . . . . . . . . 14 $/\$
39	33, 34, 36, 38	addgt0d 8401	. . . . . . . . . . . . 13 $/\$
40		0re 7881	. . . . . . . . . . . . . 14
41		ltle 7968	. . . . . . . . . . . . . 14 $/\$
42	40, 41	mpan 421	. . . . . . . . . . . . 13
43	30, 39, 42	sylc 62	. . . . . . . . . . . 12 $/\$
44	43	3adant1 1000	. . . . . . . . . . 11 $/\$ $/\$
45	44	3ad2ant1 1003	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
46		resqrtth 10943	. . . . . . . . . 10 $/\$
47	32, 45, 46	syl2anc 409	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
48		resubcl 8144	. . . . . . . . . . . . 13 $/\$
49	27, 28, 48	syl2anr 288	. . . . . . . . . . . 12 $/\$
50	49	3adant1 1000	. . . . . . . . . . 11 $/\$ $/\$
51	50	3ad2ant1 1003	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
52		ltle 7968	. . . . . . . . . . . 12 $/\$
53	40, 52	mpan 421	. . . . . . . . . . 11
54	51, 15, 53	sylc 62	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
55		resqrtth 10943	. . . . . . . . . 10 $/\$
56	51, 54, 55	syl2anc 409	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
57	47, 56	oveq12d 5845	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
58	57	oveq1d 5842	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
59		subsq 10535	. . . . . . . . 9 $/\$
60	9, 18, 59	syl2anc 409	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
61	60	oveq1d 5842	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
62		nncn 8847	. . . . . . . . . . . 12
63		nncn 8847	. . . . . . . . . . . 12
64		pnncan 8121	. . . . . . . . . . . . . 14 $/\$ $/\$
65	64	3anidm23 1279	. . . . . . . . . . . . 13 $/\$
66		2times 8967	. . . . . . . . . . . . . 14
67	66	adantl 275	. . . . . . . . . . . . 13 $/\$
68	65, 67	eqtr4d 2193	. . . . . . . . . . . 12 $/\$
69	62, 63, 68	syl2anr 288	. . . . . . . . . . 11 $/\$
70	69	3adant1 1000	. . . . . . . . . 10 $/\$ $/\$
71	70	3ad2ant1 1003	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
72	71	oveq1d 5842	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
73	5	nncnd 8853	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
74	73, 20, 23	divcanap3d 8673	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
75	72, 74	eqtrd 2190	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
76	58, 61, 75	3eqtr3d 2198	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	oveq1d 5842	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
78	24, 26, 77	3eqtr2d 2196	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
79	3, 78	syl5eq 2202	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
80	79	oveq2d 5843	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
81	73, 20, 23	divcanap2d 8670	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
82	80, 81	eqtr2d 2191	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103 $/\$ w3a 963

wceq 1335

wcel 2128 class class class wbr 3967

cfv 5173 (class class class)co 5827

cc 7733

cr 7734

cc0 7735

c1 7736

caddc 7738

cmul 7740

clt 7915

cle 7916

cmin 8051 # cap 8461

cdiv 8550

cn 8839

c2 8890

cexp 10428

csqrt 10908

cdvds 11695

cgcd 11842

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-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-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-rp 9568 df-seqfrec 10355 df-exp 10429 df-rsqrt 10910

This theorem is referenced by: pythagtriplem18 12172

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