Theorem sqrt2irrlem 12704

Description: Lemma for sqrt2irr 12705. 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 9196	. . . . . . . . . . . 12 |- 2 e. RR
2		0le2 9216	. . . . . . . . . . . 12 |- 0 <_ 2
3		resqrtth 11563	. . . . . . . . . . . 12 |- ( ( 2 e. RR /\ 0 <_ 2 ) -> ( ( sqr ` 2 ) ^ 2 ) = 2 )
4	1, 2, 3	mp2an 426	. . . . . . . . . . 11 |- ( ( sqr ` 2 ) ^ 2 ) = 2
5		sqrt2irrlem.3	. . . . . . . . . . . 12 |- ( ph -> ( sqr ` 2 ) = ( A / B ) )
6	5	oveq1d 6025	. . . . . . . . . . 11 |- ( ph -> ( ( sqr ` 2 ) ^ 2 ) = ( ( A / B ) ^ 2 ) )
7	4, 6	eqtr3id 2276	. . . . . . . . . 10 |- ( ph -> 2 = ( ( A / B ) ^ 2 ) )
8		sqrt2irrlem.1	. . . . . . . . . . . 12 |- ( ph -> A e. ZZ )
9	8	zcnd 9586	. . . . . . . . . . 11 |- ( ph -> A e. CC )
10		sqrt2irrlem.2	. . . . . . . . . . . 12 |- ( ph -> B e. NN )
11	10	nncnd 9140	. . . . . . . . . . 11 |- ( ph -> B e. CC )
12	10	nnap0d 9172	. . . . . . . . . . 11 |- ( ph -> B # 0 )
13	9, 11, 12	sqdivapd 10925	. . . . . . . . . 10 |- ( ph -> ( ( A / B ) ^ 2 ) = ( ( A ^ 2 ) / ( B ^ 2 ) ) )
14	7, 13	eqtrd 2262	. . . . . . . . 9 |- ( ph -> 2 = ( ( A ^ 2 ) / ( B ^ 2 ) ) )
15	14	oveq1d 6025	. . . . . . . 8 |- ( ph -> ( 2 x. ( B ^ 2 ) ) = ( ( ( A ^ 2 ) / ( B ^ 2 ) ) x. ( B ^ 2 ) ) )
16	9	sqcld 10910	. . . . . . . . 9 |- ( ph -> ( A ^ 2 ) e. CC )
17	10	nnsqcld 10933	. . . . . . . . . 10 |- ( ph -> ( B ^ 2 ) e. NN )
18	17	nncnd 9140	. . . . . . . . 9 |- ( ph -> ( B ^ 2 ) e. CC )
19	17	nnap0d 9172	. . . . . . . . 9 |- ( ph -> ( B ^ 2 ) # 0 )
20	16, 18, 19	divcanap1d 8954	. . . . . . . 8 |- ( ph -> ( ( ( A ^ 2 ) / ( B ^ 2 ) ) x. ( B ^ 2 ) ) = ( A ^ 2 ) )
21	15, 20	eqtrd 2262	. . . . . . 7 |- ( ph -> ( 2 x. ( B ^ 2 ) ) = ( A ^ 2 ) )
22	21	oveq1d 6025	. . . . . 6 |- ( ph -> ( ( 2 x. ( B ^ 2 ) ) / 2 ) = ( ( A ^ 2 ) / 2 ) )
23		2cnd 9199	. . . . . . 7 |- ( ph -> 2 e. CC )
24		2ap0 9219	. . . . . . . 8 |- 2 # 0
25	24	a1i 9	. . . . . . 7 |- ( ph -> 2 # 0 )
26	18, 23, 25	divcanap3d 8958	. . . . . 6 |- ( ph -> ( ( 2 x. ( B ^ 2 ) ) / 2 ) = ( B ^ 2 ) )
27	22, 26	eqtr3d 2264	. . . . 5 |- ( ph -> ( ( A ^ 2 ) / 2 ) = ( B ^ 2 ) )
28	27, 17	eqeltrd 2306	. . . 4 |- ( ph -> ( ( A ^ 2 ) / 2 ) e. NN )
29	28	nnzd 9584	. . 3 |- ( ph -> ( ( A ^ 2 ) / 2 ) e. ZZ )
30		zesq 10897	. . . 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 167	. 2 |- ( ph -> ( A / 2 ) e. ZZ )
33		2cn 9197	. . . . . . . . 9 |- 2 e. CC
34	33	sqvali 10858	. . . . . . . 8 |- ( 2 ^ 2 ) = ( 2 x. 2 )
35	34	oveq2i 6021	. . . . . . 7 |- ( ( A ^ 2 ) / ( 2 ^ 2 ) ) = ( ( A ^ 2 ) / ( 2 x. 2 ) )
36	9, 23, 25	sqdivapd 10925	. . . . . . 7 |- ( ph -> ( ( A / 2 ) ^ 2 ) = ( ( A ^ 2 ) / ( 2 ^ 2 ) ) )
37	16, 23, 23, 25, 25	divdivap1d 8985	. . . . . . 7 |- ( ph -> ( ( ( A ^ 2 ) / 2 ) / 2 ) = ( ( A ^ 2 ) / ( 2 x. 2 ) ) )
38	35, 36, 37	3eqtr4a 2288	. . . . . 6 |- ( ph -> ( ( A / 2 ) ^ 2 ) = ( ( ( A ^ 2 ) / 2 ) / 2 ) )
39	27	oveq1d 6025	. . . . . 6 |- ( ph -> ( ( ( A ^ 2 ) / 2 ) / 2 ) = ( ( B ^ 2 ) / 2 ) )
40	38, 39	eqtrd 2262	. . . . 5 |- ( ph -> ( ( A / 2 ) ^ 2 ) = ( ( B ^ 2 ) / 2 ) )
41		zsqcl 10849	. . . . . 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 2307	. . . 4 |- ( ph -> ( ( B ^ 2 ) / 2 ) e. ZZ )
44	17	nnrpd 9907	. . . . . 6 |- ( ph -> ( B ^ 2 ) e. RR+ )
45	44	rphalfcld 9922	. . . . 5 |- ( ph -> ( ( B ^ 2 ) / 2 ) e. RR+ )
46	45	rpgt0d 9912	. . . 4 |- ( ph -> 0 < ( ( B ^ 2 ) / 2 ) )
47		elnnz 9472	. . . 4 |- ( ( ( B ^ 2 ) / 2 ) e. NN <-> ( ( ( B ^ 2 ) / 2 ) e. ZZ /\ 0 < ( ( B ^ 2 ) / 2 ) ) )
48	43, 46, 47	sylanbrc 417	. . 3 |- ( ph -> ( ( B ^ 2 ) / 2 ) e. NN )
49		nnesq 10898	. . . 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 167	. 2 |- ( ph -> ( B / 2 ) e. NN )
52	32, 51	jca 306	1 |- ( ph -> ( ( A / 2 ) e. ZZ /\ ( B / 2 ) e. NN ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   class class class wbr 4083   ` cfv 5321  (class class class)co 6010   RRcr 8014   0cc0 8015   x. cmul 8020   < clt 8197   <_ cle 8198   # cap 8744   / cdiv 8835   NNcn 9126   2c2 9177   ZZcz 9462   ^cexp 10777   sqrcsqrt 11528

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-mulrcl 8114  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-precex 8125  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131  ax-pre-mulgt0 8132  ax-pre-mulext 8133  ax-arch 8134  ax-caucvg 8135

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-po 4388  df-iso 4389  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-frec 6548  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-reap 8738  df-ap 8745  df-div 8836  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-n0 9386  df-z 9463  df-uz 9739  df-rp 9867  df-seqfrec 10687  df-exp 10778  df-rsqrt 11530

This theorem is referenced by:  sqrt2irr  12705