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Theorem pythagtriplem16 12293

Description: Lemma for pythagtrip 12297. 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 5900	. . . 4
4		simp13 1030	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
5		simp12 1029	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
6	4, 5	nnaddcld 8981	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
7	6	nnrpd 9708	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
8	7	rpsqrtcld 11181	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
9	8	rpcnd 9712	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
10	4	nnzd 9388	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
11	5	nnzd 9388	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
12	10, 11	zsubcld 9394	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
13	12	zred 9389	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
14		pythagtriplem10 12283	. . . . . . . . . . 11 $/\$ $/\$ $/\$
15	14	3adant3 1018	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
16	13, 15	elrpd 9707	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
17	16	rpsqrtcld 11181	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
18	17	rpcnd 9712	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
19	9, 18	addcld 7991	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
20		2cnd 9006	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
21	9, 18	subcld 8282	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
22		2ap0 9026	. . . . . . 7 #
23	22	a1i 9	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ #
24	19, 20, 21, 20, 23, 23	divmuldivapd 8803	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
25	19, 21	mulcld 7992	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
26	25, 20, 20, 23, 23	divdivap1d 8793	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
27		nnre 8940	. . . . . . . . . . . . 13
28		nnre 8940	. . . . . . . . . . . . 13
29		readdcl 7951	. . . . . . . . . . . . 13 $/\$
30	27, 28, 29	syl2anr 290	. . . . . . . . . . . 12 $/\$
31	30	3adant1 1016	. . . . . . . . . . 11 $/\$ $/\$
32	31	3ad2ant1 1019	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
33	27	adantl 277	. . . . . . . . . . . . . 14 $/\$
34	28	adantr 276	. . . . . . . . . . . . . 14 $/\$
35		nngt0 8958	. . . . . . . . . . . . . . 15
36	35	adantl 277	. . . . . . . . . . . . . 14 $/\$
37		nngt0 8958	. . . . . . . . . . . . . . 15
38	37	adantr 276	. . . . . . . . . . . . . 14 $/\$
39	33, 34, 36, 38	addgt0d 8492	. . . . . . . . . . . . 13 $/\$
40		0re 7971	. . . . . . . . . . . . . 14
41		ltle 8059	. . . . . . . . . . . . . 14 $/\$
42	40, 41	mpan 424	. . . . . . . . . . . . 13
43	30, 39, 42	sylc 62	. . . . . . . . . . . 12 $/\$
44	43	3adant1 1016	. . . . . . . . . . 11 $/\$ $/\$
45	44	3ad2ant1 1019	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
46		resqrtth 11054	. . . . . . . . . 10 $/\$
47	32, 45, 46	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
48		resubcl 8235	. . . . . . . . . . . . 13 $/\$
49	27, 28, 48	syl2anr 290	. . . . . . . . . . . 12 $/\$
50	49	3adant1 1016	. . . . . . . . . . 11 $/\$ $/\$
51	50	3ad2ant1 1019	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
52		ltle 8059	. . . . . . . . . . . 12 $/\$
53	40, 52	mpan 424	. . . . . . . . . . 11
54	51, 15, 53	sylc 62	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
55		resqrtth 11054	. . . . . . . . . 10 $/\$
56	51, 54, 55	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
57	47, 56	oveq12d 5906	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
58	57	oveq1d 5903	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
59		subsq 10641	. . . . . . . . 9 $/\$
60	9, 18, 59	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
61	60	oveq1d 5903	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
62		nncn 8941	. . . . . . . . . . . 12
63		nncn 8941	. . . . . . . . . . . 12
64		pnncan 8212	. . . . . . . . . . . . . 14 $/\$ $/\$
65	64	3anidm23 1307	. . . . . . . . . . . . 13 $/\$
66		2times 9061	. . . . . . . . . . . . . 14
67	66	adantl 277	. . . . . . . . . . . . 13 $/\$
68	65, 67	eqtr4d 2223	. . . . . . . . . . . 12 $/\$
69	62, 63, 68	syl2anr 290	. . . . . . . . . . 11 $/\$
70	69	3adant1 1016	. . . . . . . . . 10 $/\$ $/\$
71	70	3ad2ant1 1019	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
72	71	oveq1d 5903	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
73	5	nncnd 8947	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
74	73, 20, 23	divcanap3d 8766	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
75	72, 74	eqtrd 2220	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
76	58, 61, 75	3eqtr3d 2228	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	oveq1d 5903	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
78	24, 26, 77	3eqtr2d 2226	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
79	3, 78	eqtrid 2232	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
80	79	oveq2d 5904	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
81	73, 20, 23	divcanap2d 8763	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
82	80, 81	eqtr2d 2221	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $/\$ w3a 979

wceq 1363

wcel 2158 class class class wbr 4015

cfv 5228 (class class class)co 5888

cc 7823

cr 7824

cc0 7825

c1 7826

caddc 7828

cmul 7830

clt 8006

cle 8007

cmin 8142 # cap 8552

cdiv 8643

cn 8933

c2 8984

cexp 10533

csqrt 11019

cdvds 11808

cgcd 11957

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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-mulrcl 7924 ax-addcom 7925 ax-mulcom 7926 ax-addass 7927 ax-mulass 7928 ax-distr 7929 ax-i2m1 7930 ax-0lt1 7931 ax-1rid 7932 ax-0id 7933 ax-rnegex 7934 ax-precex 7935 ax-cnre 7936 ax-pre-ltirr 7937 ax-pre-ltwlin 7938 ax-pre-lttrn 7939 ax-pre-apti 7940 ax-pre-ltadd 7941 ax-pre-mulgt0 7942 ax-pre-mulext 7943 ax-arch 7944 ax-caucvg 7945

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6155 df-2nd 6156 df-recs 6320 df-frec 6406 df-pnf 8008 df-mnf 8009 df-xr 8010 df-ltxr 8011 df-le 8012 df-sub 8144 df-neg 8145 df-reap 8546 df-ap 8553 df-div 8644 df-inn 8934 df-2 8992 df-3 8993 df-4 8994 df-n0 9191 df-z 9268 df-uz 9543 df-rp 9668 df-seqfrec 10460 df-exp 10534 df-rsqrt 11021

This theorem is referenced by: pythagtriplem18 12295

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