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Theorem pythagtriplem16 12854

Description: Lemma for pythagtrip 12858. 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 6030	. . . 4
4		simp13 1055	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
5		simp12 1054	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
6	4, 5	nnaddcld 9191	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
7	6	nnrpd 9929	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
8	7	rpsqrtcld 11720	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
9	8	rpcnd 9933	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
10	4	nnzd 9601	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
11	5	nnzd 9601	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
12	10, 11	zsubcld 9607	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
13	12	zred 9602	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
14		pythagtriplem10 12844	. . . . . . . . . . 11 $/\$ $/\$ $/\$
15	14	3adant3 1043	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
16	13, 15	elrpd 9928	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
17	16	rpsqrtcld 11720	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
18	17	rpcnd 9933	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
19	9, 18	addcld 8199	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
20		2cnd 9216	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
21	9, 18	subcld 8490	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
22		2ap0 9236	. . . . . . 7 #
23	22	a1i 9	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ #
24	19, 20, 21, 20, 23, 23	divmuldivapd 9012	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
25	19, 21	mulcld 8200	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
26	25, 20, 20, 23, 23	divdivap1d 9002	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
27		nnre 9150	. . . . . . . . . . . . 13
28		nnre 9150	. . . . . . . . . . . . 13
29		readdcl 8158	. . . . . . . . . . . . 13 $/\$
30	27, 28, 29	syl2anr 290	. . . . . . . . . . . 12 $/\$
31	30	3adant1 1041	. . . . . . . . . . 11 $/\$ $/\$
32	31	3ad2ant1 1044	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
33	27	adantl 277	. . . . . . . . . . . . . 14 $/\$
34	28	adantr 276	. . . . . . . . . . . . . 14 $/\$
35		nngt0 9168	. . . . . . . . . . . . . . 15
36	35	adantl 277	. . . . . . . . . . . . . 14 $/\$
37		nngt0 9168	. . . . . . . . . . . . . . 15
38	37	adantr 276	. . . . . . . . . . . . . 14 $/\$
39	33, 34, 36, 38	addgt0d 8701	. . . . . . . . . . . . 13 $/\$
40		0re 8179	. . . . . . . . . . . . . 14
41		ltle 8267	. . . . . . . . . . . . . 14 $/\$
42	40, 41	mpan 424	. . . . . . . . . . . . 13
43	30, 39, 42	sylc 62	. . . . . . . . . . . 12 $/\$
44	43	3adant1 1041	. . . . . . . . . . 11 $/\$ $/\$
45	44	3ad2ant1 1044	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
46		resqrtth 11593	. . . . . . . . . 10 $/\$
47	32, 45, 46	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
48		resubcl 8443	. . . . . . . . . . . . 13 $/\$
49	27, 28, 48	syl2anr 290	. . . . . . . . . . . 12 $/\$
50	49	3adant1 1041	. . . . . . . . . . 11 $/\$ $/\$
51	50	3ad2ant1 1044	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
52		ltle 8267	. . . . . . . . . . . 12 $/\$
53	40, 52	mpan 424	. . . . . . . . . . 11
54	51, 15, 53	sylc 62	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
55		resqrtth 11593	. . . . . . . . . 10 $/\$
56	51, 54, 55	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
57	47, 56	oveq12d 6036	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
58	57	oveq1d 6033	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
59		subsq 10909	. . . . . . . . 9 $/\$
60	9, 18, 59	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
61	60	oveq1d 6033	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
62		nncn 9151	. . . . . . . . . . . 12
63		nncn 9151	. . . . . . . . . . . 12
64		pnncan 8420	. . . . . . . . . . . . . 14 $/\$ $/\$
65	64	3anidm23 1333	. . . . . . . . . . . . 13 $/\$
66		2times 9271	. . . . . . . . . . . . . 14
67	66	adantl 277	. . . . . . . . . . . . 13 $/\$
68	65, 67	eqtr4d 2267	. . . . . . . . . . . 12 $/\$
69	62, 63, 68	syl2anr 290	. . . . . . . . . . 11 $/\$
70	69	3adant1 1041	. . . . . . . . . 10 $/\$ $/\$
71	70	3ad2ant1 1044	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
72	71	oveq1d 6033	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
73	5	nncnd 9157	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
74	73, 20, 23	divcanap3d 8975	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
75	72, 74	eqtrd 2264	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
76	58, 61, 75	3eqtr3d 2272	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	oveq1d 6033	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
78	24, 26, 77	3eqtr2d 2270	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
79	3, 78	eqtrid 2276	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
80	79	oveq2d 6034	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
81	73, 20, 23	divcanap2d 8972	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
82	80, 81	eqtr2d 2265	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $/\$ w3a 1004

wceq 1397

wcel 2202 class class class wbr 4088

cfv 5326 (class class class)co 6018

cc 8030

cr 8031

cc0 8032

c1 8033

caddc 8035

cmul 8037

clt 8214

cle 8215

cmin 8350 # cap 8761

cdiv 8852

cn 9143

c2 9194

cexp 10801

csqrt 11558

cdvds 12350

cgcd 12526

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151 ax-caucvg 8152

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-1st 6303 df-2nd 6304 df-recs 6471 df-frec 6557 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-n0 9403 df-z 9480 df-uz 9756 df-rp 9889 df-seqfrec 10711 df-exp 10802 df-rsqrt 11560

This theorem is referenced by: pythagtriplem18 12856

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