Theorem sqrt2irrlem 11632

Description: Lemma for sqrt2irr 11633. This is the core of the proof: - if A / B = sqr ( 2 ), then A and B are even, so A / 2 and B / 2 are smaller representatives, which is absurd by the method of infinite descent (here implemented by strong induction). (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.)

Hypotheses

Ref	Expression
sqrt2irrlem.1	|- ( ph -> A e. ZZ )
sqrt2irrlem.2	|- ( ph -> B e. NN )
sqrt2irrlem.3	|- ( ph -> ( sqr ` 2 ) = ( A / B ) )

Assertion

Ref	Expression
sqrt2irrlem	|- ( ph -> ( ( A / 2 ) e. ZZ /\ ( B / 2 ) e. NN ) )

Proof of Theorem sqrt2irrlem

Step	Hyp	Ref	Expression
1		2re 8648	. . . . . . . . . . . 12 |- 2 e. RR
2		0le2 8668	. . . . . . . . . . . 12 |- 0 <_ 2
3		resqrtth 10643	. . . . . . . . . . . 12 |- ( ( 2 e. RR /\ 0 <_ 2 ) -> ( ( sqr ` 2 ) ^ 2 ) = 2 )
4	1, 2, 3	mp2an 420	. . . . . . . . . . 11 |- ( ( sqr ` 2 ) ^ 2 ) = 2
5		sqrt2irrlem.3	. . . . . . . . . . . 12 |- ( ph -> ( sqr ` 2 ) = ( A / B ) )
6	5	oveq1d 5721	. . . . . . . . . . 11 |- ( ph -> ( ( sqr ` 2 ) ^ 2 ) = ( ( A / B ) ^ 2 ) )
7	4, 6	syl5eqr 2146	. . . . . . . . . 10 |- ( ph -> 2 = ( ( A / B ) ^ 2 ) )
8		sqrt2irrlem.1	. . . . . . . . . . . 12 |- ( ph -> A e. ZZ )
9	8	zcnd 9026	. . . . . . . . . . 11 |- ( ph -> A e. CC )
10		sqrt2irrlem.2	. . . . . . . . . . . 12 |- ( ph -> B e. NN )
11	10	nncnd 8592	. . . . . . . . . . 11 |- ( ph -> B e. CC )
12	10	nnap0d 8624	. . . . . . . . . . 11 |- ( ph -> B # 0 )
13	9, 11, 12	sqdivapd 10278	. . . . . . . . . 10 |- ( ph -> ( ( A / B ) ^ 2 ) = ( ( A ^ 2 ) / ( B ^ 2 ) ) )
14	7, 13	eqtrd 2132	. . . . . . . . 9 |- ( ph -> 2 = ( ( A ^ 2 ) / ( B ^ 2 ) ) )
15	14	oveq1d 5721	. . . . . . . 8 |- ( ph -> ( 2 x. ( B ^ 2 ) ) = ( ( ( A ^ 2 ) / ( B ^ 2 ) ) x. ( B ^ 2 ) ) )
16	9	sqcld 10263	. . . . . . . . 9 |- ( ph -> ( A ^ 2 ) e. CC )
17	10	nnsqcld 10286	. . . . . . . . . 10 |- ( ph -> ( B ^ 2 ) e. NN )
18	17	nncnd 8592	. . . . . . . . 9 |- ( ph -> ( B ^ 2 ) e. CC )
19	17	nnap0d 8624	. . . . . . . . 9 |- ( ph -> ( B ^ 2 ) # 0 )
20	16, 18, 19	divcanap1d 8412	. . . . . . . 8 |- ( ph -> ( ( ( A ^ 2 ) / ( B ^ 2 ) ) x. ( B ^ 2 ) ) = ( A ^ 2 ) )
21	15, 20	eqtrd 2132	. . . . . . 7 |- ( ph -> ( 2 x. ( B ^ 2 ) ) = ( A ^ 2 ) )
22	21	oveq1d 5721	. . . . . 6 |- ( ph -> ( ( 2 x. ( B ^ 2 ) ) / 2 ) = ( ( A ^ 2 ) / 2 ) )
23		2cnd 8651	. . . . . . 7 |- ( ph -> 2 e. CC )
24		2ap0 8671	. . . . . . . 8 |- 2 # 0
25	24	a1i 9	. . . . . . 7 |- ( ph -> 2 # 0 )
26	18, 23, 25	divcanap3d 8416	. . . . . 6 |- ( ph -> ( ( 2 x. ( B ^ 2 ) ) / 2 ) = ( B ^ 2 ) )
27	22, 26	eqtr3d 2134	. . . . 5 |- ( ph -> ( ( A ^ 2 ) / 2 ) = ( B ^ 2 ) )
28	27, 17	eqeltrd 2176	. . . 4 |- ( ph -> ( ( A ^ 2 ) / 2 ) e. NN )
29	28	nnzd 9024	. . 3 |- ( ph -> ( ( A ^ 2 ) / 2 ) e. ZZ )
30		zesq 10251	. . . 4 |- ( A e. ZZ -> ( ( A / 2 ) e. ZZ <-> ( ( A ^ 2 ) / 2 ) e. ZZ ) )
31	8, 30	syl 14	. . 3 |- ( ph -> ( ( A / 2 ) e. ZZ <-> ( ( A ^ 2 ) / 2 ) e. ZZ ) )
32	29, 31	mpbird 166	. 2 |- ( ph -> ( A / 2 ) e. ZZ )
33		2cn 8649	. . . . . . . . 9 |- 2 e. CC
34	33	sqvali 10213	. . . . . . . 8 |- ( 2 ^ 2 ) = ( 2 x. 2 )
35	34	oveq2i 5717	. . . . . . 7 |- ( ( A ^ 2 ) / ( 2 ^ 2 ) ) = ( ( A ^ 2 ) / ( 2 x. 2 ) )
36	9, 23, 25	sqdivapd 10278	. . . . . . 7 |- ( ph -> ( ( A / 2 ) ^ 2 ) = ( ( A ^ 2 ) / ( 2 ^ 2 ) ) )
37	16, 23, 23, 25, 25	divdivap1d 8443	. . . . . . 7 |- ( ph -> ( ( ( A ^ 2 ) / 2 ) / 2 ) = ( ( A ^ 2 ) / ( 2 x. 2 ) ) )
38	35, 36, 37	3eqtr4a 2158	. . . . . 6 |- ( ph -> ( ( A / 2 ) ^ 2 ) = ( ( ( A ^ 2 ) / 2 ) / 2 ) )
39	27	oveq1d 5721	. . . . . 6 |- ( ph -> ( ( ( A ^ 2 ) / 2 ) / 2 ) = ( ( B ^ 2 ) / 2 ) )
40	38, 39	eqtrd 2132	. . . . 5 |- ( ph -> ( ( A / 2 ) ^ 2 ) = ( ( B ^ 2 ) / 2 ) )
41		zsqcl 10204	. . . . . 6 |- ( ( A / 2 ) e. ZZ -> ( ( A / 2 ) ^ 2 ) e. ZZ )
42	32, 41	syl 14	. . . . 5 |- ( ph -> ( ( A / 2 ) ^ 2 ) e. ZZ )
43	40, 42	eqeltrrd 2177	. . . 4 |- ( ph -> ( ( B ^ 2 ) / 2 ) e. ZZ )
44	17	nnrpd 9329	. . . . . 6 |- ( ph -> ( B ^ 2 ) e. RR+ )
45	44	rphalfcld 9343	. . . . 5 |- ( ph -> ( ( B ^ 2 ) / 2 ) e. RR+ )
46	45	rpgt0d 9333	. . . 4 |- ( ph -> 0 < ( ( B ^ 2 ) / 2 ) )
47		elnnz 8916	. . . 4 |- ( ( ( B ^ 2 ) / 2 ) e. NN <-> ( ( ( B ^ 2 ) / 2 ) e. ZZ /\ 0 < ( ( B ^ 2 ) / 2 ) ) )
48	43, 46, 47	sylanbrc 411	. . 3 |- ( ph -> ( ( B ^ 2 ) / 2 ) e. NN )
49		nnesq 10252	. . . 4 |- ( B e. NN -> ( ( B / 2 ) e. NN <-> ( ( B ^ 2 ) / 2 ) e. NN ) )
50	10, 49	syl 14	. . 3 |- ( ph -> ( ( B / 2 ) e. NN <-> ( ( B ^ 2 ) / 2 ) e. NN ) )
51	48, 50	mpbird 166	. 2 |- ( ph -> ( B / 2 ) e. NN )
52	32, 51	jca 302	1 |- ( ph -> ( ( A / 2 ) e. ZZ /\ ( B / 2 ) e. NN ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1299   e. wcel 1448   class class class wbr 3875   ` cfv 5059  (class class class)co 5706   RRcr 7499   0cc0 7500   x. cmul 7505   < clt 7672   <_ cle 7673   # cap 8209   / cdiv 8293   NNcn 8578   2c2 8629   ZZcz 8906   ^cexp 10133   sqrcsqrt 10608

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-rp 9292  df-seqfrec 10060  df-exp 10134  df-rsqrt 10610

This theorem is referenced by:  sqrt2irr  11633