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Theorem pythagtriplem2 12266

Description: Lemma for pythagtrip 12283. Prove the full version of one direction of the theorem. (Contributed by Scott Fenton, 28-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

**Assertion**
Ref	Expression
pythagtriplem2	$/\$ $/\$

Distinct variable groups:

**Proof of Theorem pythagtriplem2**
Step	Hyp	Ref	Expression
1		simplll 533	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
2		simpllr 534	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
3		nnz 9272	. . . . . . . . . 10
4	3	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
5		simplrr 536	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
6	5	nnzd 9374	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
7		zsqcl 10591	. . . . . . . . . . 11
8	6, 7	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
9		simplrl 535	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
10	9	nnzd 9374	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
11		zsqcl 10591	. . . . . . . . . . 11
12	10, 11	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
13	8, 12	zsubcld 9380	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
14	4, 13	zmulcld 9381	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
15		2z 9281	. . . . . . . . . . 11
16	15	a1i 9	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
17	6, 10	zmulcld 9381	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
18	16, 17	zmulcld 9381	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
19	4, 18	zmulcld 9381	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
20		preq12bg 3774	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
21	1, 2, 14, 19, 20	syl22anc 1239	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$
22	21	anbi1d 465	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
23		andir 819	. . . . . . 7 $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
24		df-3an 980	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
25		df-3an 980	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
26	24, 25	orbi12i 764	. . . . . . 7 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
27	23, 26	bitr4i 187	. . . . . 6 $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
28	22, 27	bitrdi 196	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
29	28	rexbidva 2474	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
30	29	2rexbidva 2500	. . 3 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
31		r19.43 2635	. . . . 5 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
32	31	2rexbii 2486	. . . 4 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
33		r19.43 2635	. . . . 5 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
34	33	rexbii 2484	. . . 4 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
35		r19.43 2635	. . . 4 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
36	32, 34, 35	3bitri 206	. . 3 $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
37	30, 36	bitrdi 196	. 2 $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
38		pythagtriplem1 12265	. . . 4 $/\$ $/\$
39	38	a1i 9	. . 3 $/\$ $/\$ $/\$
40		3ancoma 985	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
41	40	rexbii 2484	. . . . . 6 $/\$ $/\$ $/\$ $/\$
42	41	2rexbii 2486	. . . . 5 $/\$ $/\$ $/\$ $/\$
43		pythagtriplem1 12265	. . . . 5 $/\$ $/\$
44	42, 43	sylbi 121	. . . 4 $/\$ $/\$
45		nncn 8927	. . . . . . 7
46	45	sqcld 10652	. . . . . 6
47		nncn 8927	. . . . . . 7
48	47	sqcld 10652	. . . . . 6
49		addcom 8094	. . . . . 6 $/\$
50	46, 48, 49	syl2an 289	. . . . 5 $/\$
51	50	eqeq1d 2186	. . . 4 $/\$
52	44, 51	imbitrrid 156	. . 3 $/\$ $/\$ $/\$
53	39, 52	jaod 717	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$
54	37, 53	sylbid 150	1 $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 708 $/\$ w3a 978

wceq 1353

wcel 2148

wrex 2456

cpr 3594 (class class class)co 5875

cc 7809

caddc 7814

cmul 7816

cmin 8128

cn 8919

c2 8970

cz 9253

cexp 10519

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 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-mulrcl 7910 ax-addcom 7911 ax-mulcom 7912 ax-addass 7913 ax-mulass 7914 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-1rid 7918 ax-0id 7919 ax-rnegex 7920 ax-precex 7921 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-apti 7926 ax-pre-ltadd 7927 ax-pre-mulgt0 7928 ax-pre-mulext 7929

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 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-po 4297 df-iso 4298 df-iord 4367 df-on 4369 df-ilim 4370 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-recs 6306 df-frec 6392 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-reap 8532 df-ap 8539 df-div 8630 df-inn 8920 df-2 8978 df-3 8979 df-4 8980 df-n0 9177 df-z 9254 df-uz 9529 df-seqfrec 10446 df-exp 10520

This theorem is referenced by: pythagtrip 12283

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