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Theorem pythagtriplem16 12262

Description: Lemma for pythagtrip 12266. 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 5881	. . . 4
4		simp13 1029	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
5		simp12 1028	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
6	4, 5	nnaddcld 8956	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
7	6	nnrpd 9681	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
8	7	rpsqrtcld 11151	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
9	8	rpcnd 9685	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
10	4	nnzd 9363	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
11	5	nnzd 9363	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
12	10, 11	zsubcld 9369	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
13	12	zred 9364	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
14		pythagtriplem10 12252	. . . . . . . . . . 11 $/\$ $/\$ $/\$
15	14	3adant3 1017	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
16	13, 15	elrpd 9680	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
17	16	rpsqrtcld 11151	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
18	17	rpcnd 9685	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
19	9, 18	addcld 7967	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
20		2cnd 8981	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
21	9, 18	subcld 8258	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
22		2ap0 9001	. . . . . . 7 #
23	22	a1i 9	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ #
24	19, 20, 21, 20, 23, 23	divmuldivapd 8778	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
25	19, 21	mulcld 7968	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
26	25, 20, 20, 23, 23	divdivap1d 8768	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
27		nnre 8915	. . . . . . . . . . . . 13
28		nnre 8915	. . . . . . . . . . . . 13
29		readdcl 7928	. . . . . . . . . . . . 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 8933	. . . . . . . . . . . . . . 15
36	35	adantl 277	. . . . . . . . . . . . . 14 $/\$
37		nngt0 8933	. . . . . . . . . . . . . . 15
38	37	adantr 276	. . . . . . . . . . . . . 14 $/\$
39	33, 34, 36, 38	addgt0d 8468	. . . . . . . . . . . . 13 $/\$
40		0re 7948	. . . . . . . . . . . . . 14
41		ltle 8035	. . . . . . . . . . . . . 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 11024	. . . . . . . . . 10 $/\$
47	32, 45, 46	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
48		resubcl 8211	. . . . . . . . . . . . 13 $/\$
49	27, 28, 48	syl2anr 290	. . . . . . . . . . . 12 $/\$
50	49	3adant1 1015	. . . . . . . . . . 11 $/\$ $/\$
51	50	3ad2ant1 1018	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
52		ltle 8035	. . . . . . . . . . . 12 $/\$
53	40, 52	mpan 424	. . . . . . . . . . 11
54	51, 15, 53	sylc 62	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
55		resqrtth 11024	. . . . . . . . . 10 $/\$
56	51, 54, 55	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
57	47, 56	oveq12d 5887	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
58	57	oveq1d 5884	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
59		subsq 10612	. . . . . . . . 9 $/\$
60	9, 18, 59	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
61	60	oveq1d 5884	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
62		nncn 8916	. . . . . . . . . . . 12
63		nncn 8916	. . . . . . . . . . . 12
64		pnncan 8188	. . . . . . . . . . . . . 14 $/\$ $/\$
65	64	3anidm23 1297	. . . . . . . . . . . . 13 $/\$
66		2times 9036	. . . . . . . . . . . . . 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 5884	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
73	5	nncnd 8922	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
74	73, 20, 23	divcanap3d 8741	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
75	72, 74	eqtrd 2210	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
76	58, 61, 75	3eqtr3d 2218	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	oveq1d 5884	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
78	24, 26, 77	3eqtr2d 2216	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
79	3, 78	eqtrid 2222	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
80	79	oveq2d 5885	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
81	73, 20, 23	divcanap2d 8738	. 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 4000

cfv 5212 (class class class)co 5869

cc 7800

cr 7801

cc0 7802

c1 7803

caddc 7805

cmul 7807

clt 7982

cle 7983

cmin 8118 # cap 8528

cdiv 8618

cn 8908

c2 8959

cexp 10505

csqrt 10989

cdvds 11778

cgcd 11926

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 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922

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 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-n0 9166 df-z 9243 df-uz 9518 df-rp 9641 df-seqfrec 10432 df-exp 10506 df-rsqrt 10991

This theorem is referenced by: pythagtriplem18 12264

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