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Theorem pythagtriplem16 12646

Description: Lemma for pythagtrip 12650. 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 5963	. . . 4
4		simp13 1032	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
5		simp12 1031	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
6	4, 5	nnaddcld 9091	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
7	6	nnrpd 9823	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
8	7	rpsqrtcld 11513	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
9	8	rpcnd 9827	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
10	4	nnzd 9501	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
11	5	nnzd 9501	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
12	10, 11	zsubcld 9507	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
13	12	zred 9502	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
14		pythagtriplem10 12636	. . . . . . . . . . 11 $/\$ $/\$ $/\$
15	14	3adant3 1020	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
16	13, 15	elrpd 9822	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
17	16	rpsqrtcld 11513	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
18	17	rpcnd 9827	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
19	9, 18	addcld 8099	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
20		2cnd 9116	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
21	9, 18	subcld 8390	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
22		2ap0 9136	. . . . . . 7 #
23	22	a1i 9	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ #
24	19, 20, 21, 20, 23, 23	divmuldivapd 8912	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
25	19, 21	mulcld 8100	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
26	25, 20, 20, 23, 23	divdivap1d 8902	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
27		nnre 9050	. . . . . . . . . . . . 13
28		nnre 9050	. . . . . . . . . . . . 13
29		readdcl 8058	. . . . . . . . . . . . 13 $/\$
30	27, 28, 29	syl2anr 290	. . . . . . . . . . . 12 $/\$
31	30	3adant1 1018	. . . . . . . . . . 11 $/\$ $/\$
32	31	3ad2ant1 1021	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
33	27	adantl 277	. . . . . . . . . . . . . 14 $/\$
34	28	adantr 276	. . . . . . . . . . . . . 14 $/\$
35		nngt0 9068	. . . . . . . . . . . . . . 15
36	35	adantl 277	. . . . . . . . . . . . . 14 $/\$
37		nngt0 9068	. . . . . . . . . . . . . . 15
38	37	adantr 276	. . . . . . . . . . . . . 14 $/\$
39	33, 34, 36, 38	addgt0d 8601	. . . . . . . . . . . . 13 $/\$
40		0re 8079	. . . . . . . . . . . . . 14
41		ltle 8167	. . . . . . . . . . . . . 14 $/\$
42	40, 41	mpan 424	. . . . . . . . . . . . 13
43	30, 39, 42	sylc 62	. . . . . . . . . . . 12 $/\$
44	43	3adant1 1018	. . . . . . . . . . 11 $/\$ $/\$
45	44	3ad2ant1 1021	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
46		resqrtth 11386	. . . . . . . . . 10 $/\$
47	32, 45, 46	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
48		resubcl 8343	. . . . . . . . . . . . 13 $/\$
49	27, 28, 48	syl2anr 290	. . . . . . . . . . . 12 $/\$
50	49	3adant1 1018	. . . . . . . . . . 11 $/\$ $/\$
51	50	3ad2ant1 1021	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
52		ltle 8167	. . . . . . . . . . . 12 $/\$
53	40, 52	mpan 424	. . . . . . . . . . 11
54	51, 15, 53	sylc 62	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
55		resqrtth 11386	. . . . . . . . . 10 $/\$
56	51, 54, 55	syl2anc 411	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
57	47, 56	oveq12d 5969	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
58	57	oveq1d 5966	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
59		subsq 10798	. . . . . . . . 9 $/\$
60	9, 18, 59	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
61	60	oveq1d 5966	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
62		nncn 9051	. . . . . . . . . . . 12
63		nncn 9051	. . . . . . . . . . . 12
64		pnncan 8320	. . . . . . . . . . . . . 14 $/\$ $/\$
65	64	3anidm23 1310	. . . . . . . . . . . . 13 $/\$
66		2times 9171	. . . . . . . . . . . . . 14
67	66	adantl 277	. . . . . . . . . . . . 13 $/\$
68	65, 67	eqtr4d 2242	. . . . . . . . . . . 12 $/\$
69	62, 63, 68	syl2anr 290	. . . . . . . . . . 11 $/\$
70	69	3adant1 1018	. . . . . . . . . 10 $/\$ $/\$
71	70	3ad2ant1 1021	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
72	71	oveq1d 5966	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
73	5	nncnd 9057	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
74	73, 20, 23	divcanap3d 8875	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
75	72, 74	eqtrd 2239	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
76	58, 61, 75	3eqtr3d 2247	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	oveq1d 5966	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
78	24, 26, 77	3eqtr2d 2245	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
79	3, 78	eqtrid 2251	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
80	79	oveq2d 5967	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
81	73, 20, 23	divcanap2d 8872	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
82	80, 81	eqtr2d 2240	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104 $/\$ w3a 981

wceq 1373

wcel 2177 class class class wbr 4047

cfv 5276 (class class class)co 5951

cc 7930

cr 7931

cc0 7932

c1 7933

caddc 7935

cmul 7937

clt 8114

cle 8115

cmin 8250 # cap 8661

cdiv 8752

cn 9043

c2 9094

cexp 10690

csqrt 11351

cdvds 12142

cgcd 12318

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4163 ax-sep 4166 ax-nul 4174 ax-pow 4222 ax-pr 4257 ax-un 4484 ax-setind 4589 ax-iinf 4640 ax-cnex 8023 ax-resscn 8024 ax-1cn 8025 ax-1re 8026 ax-icn 8027 ax-addcl 8028 ax-addrcl 8029 ax-mulcl 8030 ax-mulrcl 8031 ax-addcom 8032 ax-mulcom 8033 ax-addass 8034 ax-mulass 8035 ax-distr 8036 ax-i2m1 8037 ax-0lt1 8038 ax-1rid 8039 ax-0id 8040 ax-rnegex 8041 ax-precex 8042 ax-cnre 8043 ax-pre-ltirr 8044 ax-pre-ltwlin 8045 ax-pre-lttrn 8046 ax-pre-apti 8047 ax-pre-ltadd 8048 ax-pre-mulgt0 8049 ax-pre-mulext 8050 ax-arch 8051 ax-caucvg 8052

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3000 df-csb 3095 df-dif 3169 df-un 3171 df-in 3173 df-ss 3180 df-nul 3462 df-if 3573 df-pw 3619 df-sn 3640 df-pr 3641 df-op 3643 df-uni 3853 df-int 3888 df-iun 3931 df-br 4048 df-opab 4110 df-mpt 4111 df-tr 4147 df-id 4344 df-po 4347 df-iso 4348 df-iord 4417 df-on 4419 df-ilim 4420 df-suc 4422 df-iom 4643 df-xp 4685 df-rel 4686 df-cnv 4687 df-co 4688 df-dm 4689 df-rn 4690 df-res 4691 df-ima 4692 df-iota 5237 df-fun 5278 df-fn 5279 df-f 5280 df-f1 5281 df-fo 5282 df-f1o 5283 df-fv 5284 df-riota 5906 df-ov 5954 df-oprab 5955 df-mpo 5956 df-1st 6233 df-2nd 6234 df-recs 6398 df-frec 6484 df-pnf 8116 df-mnf 8117 df-xr 8118 df-ltxr 8119 df-le 8120 df-sub 8252 df-neg 8253 df-reap 8655 df-ap 8662 df-div 8753 df-inn 9044 df-2 9102 df-3 9103 df-4 9104 df-n0 9303 df-z 9380 df-uz 9656 df-rp 9783 df-seqfrec 10600 df-exp 10691 df-rsqrt 11353

This theorem is referenced by: pythagtriplem18 12648

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