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Theorem pythagtriplem16 12281

Description: Lemma for pythagtrip 12285. 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 5889	. . . 4
4		simp13 1029	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
5		simp12 1028	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
6	4, 5	nnaddcld 8969	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
7	6	nnrpd 9696	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
8	7	rpsqrtcld 11169	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
9	8	rpcnd 9700	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
10	4	nnzd 9376	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
11	5	nnzd 9376	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
12	10, 11	zsubcld 9382	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
13	12	zred 9377	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
14		pythagtriplem10 12271	. . . . . . . . . . 11 $/\$ $/\$ $/\$
15	14	3adant3 1017	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
16	13, 15	elrpd 9695	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
17	16	rpsqrtcld 11169	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
18	17	rpcnd 9700	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
19	9, 18	addcld 7979	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
20		2cnd 8994	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
21	9, 18	subcld 8270	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
22		2ap0 9014	. . . . . . 7 #
23	22	a1i 9	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ #
24	19, 20, 21, 20, 23, 23	divmuldivapd 8791	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
25	19, 21	mulcld 7980	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
26	25, 20, 20, 23, 23	divdivap1d 8781	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
27		nnre 8928	. . . . . . . . . . . . 13
28		nnre 8928	. . . . . . . . . . . . 13
29		readdcl 7939	. . . . . . . . . . . . 13 $/\$
30	27, 28, 29	syl2anr 290	. . . . . . . . . . . 12 $/\$
31	30	3adant1 1015	. . . . . . . . . . 11 $/\$ $/\$
32	31	3ad2ant1 1018	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
33	27	adantl 277	. . . . . . . . . . . . . 14 $/\$
34	28	adantr 276	. . . . . . . . . . . . . 14 $/\$
35		nngt0 8946	. . . . . . . . . . . . . . 15
36	35	adantl 277	. . . . . . . . . . . . . 14 $/\$
37		nngt0 8946	. . . . . . . . . . . . . . 15
38	37	adantr 276	. . . . . . . . . . . . . 14 $/\$
39	33, 34, 36, 38	addgt0d 8480	. . . . . . . . . . . . 13 $/\$
40		0re 7959	. . . . . . . . . . . . . 14
41		ltle 8047	. . . . . . . . . . . . . 14 $/\$
42	40, 41	mpan 424	. . . . . . . . . . . . 13
43	30, 39, 42	sylc 62	. . . . . . . . . . . 12 $/\$
44	43	3adant1 1015	. . . . . . . . . . 11 $/\$ $/\$
45	44	3ad2ant1 1018	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
46		resqrtth 11042	. . . . . . . . . 10 $/\$
47	32, 45, 46	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
48		resubcl 8223	. . . . . . . . . . . . 13 $/\$
49	27, 28, 48	syl2anr 290	. . . . . . . . . . . 12 $/\$
50	49	3adant1 1015	. . . . . . . . . . 11 $/\$ $/\$
51	50	3ad2ant1 1018	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
52		ltle 8047	. . . . . . . . . . . 12 $/\$
53	40, 52	mpan 424	. . . . . . . . . . 11
54	51, 15, 53	sylc 62	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
55		resqrtth 11042	. . . . . . . . . 10 $/\$
56	51, 54, 55	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
57	47, 56	oveq12d 5895	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
58	57	oveq1d 5892	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
59		subsq 10629	. . . . . . . . 9 $/\$
60	9, 18, 59	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
61	60	oveq1d 5892	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
62		nncn 8929	. . . . . . . . . . . 12
63		nncn 8929	. . . . . . . . . . . 12
64		pnncan 8200	. . . . . . . . . . . . . 14 $/\$ $/\$
65	64	3anidm23 1297	. . . . . . . . . . . . 13 $/\$
66		2times 9049	. . . . . . . . . . . . . 14
67	66	adantl 277	. . . . . . . . . . . . 13 $/\$
68	65, 67	eqtr4d 2213	. . . . . . . . . . . 12 $/\$
69	62, 63, 68	syl2anr 290	. . . . . . . . . . 11 $/\$
70	69	3adant1 1015	. . . . . . . . . 10 $/\$ $/\$
71	70	3ad2ant1 1018	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
72	71	oveq1d 5892	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
73	5	nncnd 8935	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
74	73, 20, 23	divcanap3d 8754	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
75	72, 74	eqtrd 2210	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
76	58, 61, 75	3eqtr3d 2218	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	oveq1d 5892	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
78	24, 26, 77	3eqtr2d 2216	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
79	3, 78	eqtrid 2222	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
80	79	oveq2d 5893	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
81	73, 20, 23	divcanap2d 8751	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
82	80, 81	eqtr2d 2211	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $/\$ w3a 978

wceq 1353

wcel 2148 class class class wbr 4005

cfv 5218 (class class class)co 5877

cc 7811

cr 7812

cc0 7813

c1 7814

caddc 7816

cmul 7818

clt 7994

cle 7995

cmin 8130 # cap 8540

cdiv 8631

cn 8921

c2 8972

cexp 10521

csqrt 11007

cdvds 11796

cgcd 11945

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 ax-arch 7932 ax-caucvg 7933

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-frec 6394 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-n0 9179 df-z 9256 df-uz 9531 df-rp 9656 df-seqfrec 10448 df-exp 10522 df-rsqrt 11009

This theorem is referenced by: pythagtriplem18 12283

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