Theorem sqrt2irrlem 12966

Description: Lemma for sqrt2irr 12967. 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 9384	. . . . . . . . . . . 12 |- 2 e. RR
2		0le2 9404	. . . . . . . . . . . 12 |- 0 <_ 2
3		resqrtth 11820	. . . . . . . . . . . 12 |- ( ( 2 e. RR /\ 0 <_ 2 ) -> ( ( sqr ` 2 ) ^ 2 ) = 2 )
4	1, 2, 3	mp2an 430	. . . . . . . . . . 11 |- ( ( sqr ` 2 ) ^ 2 ) = 2
5		sqrt2irrlem.3	. . . . . . . . . . . 12 |- ( ph -> ( sqr ` 2 ) = ( A / B ) )
6	5	oveq1d 6103	. . . . . . . . . . 11 |- ( ph -> ( ( sqr ` 2 ) ^ 2 ) = ( ( A / B ) ^ 2 ) )
7	4, 6	eqtr3id 2285	. . . . . . . . . 10 |- ( ph -> 2 = ( ( A / B ) ^ 2 ) )
8		sqrt2irrlem.1	. . . . . . . . . . . 12 |- ( ph -> A e. ZZ )
9	8	zcnd 9781	. . . . . . . . . . 11 |- ( ph -> A e. CC )
10		sqrt2irrlem.2	. . . . . . . . . . . 12 |- ( ph -> B e. NN )
11	10	nncnd 9328	. . . . . . . . . . 11 |- ( ph -> B e. CC )
12	10	nnap0d 9360	. . . . . . . . . . 11 |- ( ph -> B # 0 )
13	9, 11, 12	sqdivapd 11146	. . . . . . . . . 10 |- ( ph -> ( ( A / B ) ^ 2 ) = ( ( A ^ 2 ) / ( B ^ 2 ) ) )
14	7, 13	eqtrd 2271	. . . . . . . . 9 |- ( ph -> 2 = ( ( A ^ 2 ) / ( B ^ 2 ) ) )
15	14	oveq1d 6103	. . . . . . . 8 |- ( ph -> ( 2 x. ( B ^ 2 ) ) = ( ( ( A ^ 2 ) / ( B ^ 2 ) ) x. ( B ^ 2 ) ) )
16	9	sqcld 11131	. . . . . . . . 9 |- ( ph -> ( A ^ 2 ) e. CC )
17	10	nnsqcld 11154	. . . . . . . . . 10 |- ( ph -> ( B ^ 2 ) e. NN )
18	17	nncnd 9328	. . . . . . . . 9 |- ( ph -> ( B ^ 2 ) e. CC )
19	17	nnap0d 9360	. . . . . . . . 9 |- ( ph -> ( B ^ 2 ) # 0 )
20	16, 18, 19	divcanap1d 9130	. . . . . . . 8 |- ( ph -> ( ( ( A ^ 2 ) / ( B ^ 2 ) ) x. ( B ^ 2 ) ) = ( A ^ 2 ) )
21	15, 20	eqtrd 2271	. . . . . . 7 |- ( ph -> ( 2 x. ( B ^ 2 ) ) = ( A ^ 2 ) )
22	21	oveq1d 6103	. . . . . 6 |- ( ph -> ( ( 2 x. ( B ^ 2 ) ) / 2 ) = ( ( A ^ 2 ) / 2 ) )
23		2cnd 9387	. . . . . . 7 |- ( ph -> 2 e. CC )
24		2ap0 9407	. . . . . . . 8 |- 2 # 0
25	24	a1i 9	. . . . . . 7 |- ( ph -> 2 # 0 )
26	18, 23, 25	divcanap3d 9134	. . . . . 6 |- ( ph -> ( ( 2 x. ( B ^ 2 ) ) / 2 ) = ( B ^ 2 ) )
27	22, 26	eqtr3d 2273	. . . . 5 |- ( ph -> ( ( A ^ 2 ) / 2 ) = ( B ^ 2 ) )
28	27, 17	eqeltrd 2315	. . . 4 |- ( ph -> ( ( A ^ 2 ) / 2 ) e. NN )
29	28	nnzd 9779	. . 3 |- ( ph -> ( ( A ^ 2 ) / 2 ) e. ZZ )
30		zesq 11118	. . . 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 9385	. . . . . . . . 9 |- 2 e. CC
34	33	sqvali 11078	. . . . . . . 8 |- ( 2 ^ 2 ) = ( 2 x. 2 )
35	34	oveq2i 6099	. . . . . . 7 |- ( ( A ^ 2 ) / ( 2 ^ 2 ) ) = ( ( A ^ 2 ) / ( 2 x. 2 ) )
36	9, 23, 25	sqdivapd 11146	. . . . . . 7 |- ( ph -> ( ( A / 2 ) ^ 2 ) = ( ( A ^ 2 ) / ( 2 ^ 2 ) ) )
37	16, 23, 23, 25, 25	divdivap1d 9161	. . . . . . 7 |- ( ph -> ( ( ( A ^ 2 ) / 2 ) / 2 ) = ( ( A ^ 2 ) / ( 2 x. 2 ) ) )
38	35, 36, 37	3eqtr4a 2297	. . . . . 6 |- ( ph -> ( ( A / 2 ) ^ 2 ) = ( ( ( A ^ 2 ) / 2 ) / 2 ) )
39	27	oveq1d 6103	. . . . . 6 |- ( ph -> ( ( ( A ^ 2 ) / 2 ) / 2 ) = ( ( B ^ 2 ) / 2 ) )
40	38, 39	eqtrd 2271	. . . . 5 |- ( ph -> ( ( A / 2 ) ^ 2 ) = ( ( B ^ 2 ) / 2 ) )
41		zsqcl 11069	. . . . . 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 2316	. . . 4 |- ( ph -> ( ( B ^ 2 ) / 2 ) e. ZZ )
44	17	nnrpd 10113	. . . . . 6 |- ( ph -> ( B ^ 2 ) e. RR+ )
45	44	rphalfcld 10128	. . . . 5 |- ( ph -> ( ( B ^ 2 ) / 2 ) e. RR+ )
46	45	rpgt0d 10118	. . . 4 |- ( ph -> 0 < ( ( B ^ 2 ) / 2 ) )
47		elnnz 9666	. . . 4 |- ( ( ( B ^ 2 ) / 2 ) e. NN <-> ( ( ( B ^ 2 ) / 2 ) e. ZZ /\ 0 < ( ( B ^ 2 ) / 2 ) ) )
48	43, 46, 47	sylanbrc 421	. . 3 |- ( ph -> ( ( B ^ 2 ) / 2 ) e. NN )
49		nnesq 11119	. . . 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   RRcr 8185   0cc0 8186   x. cmul 8191   < clt 8367   <_ cle 8368   # cap 8918   / cdiv 9011   NNcn 9314   2c2 9365   ZZcz 9656   ^cexp 10997   sqrcsqrt 11785

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305  ax-caucvg 8306

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-n0 9576  df-z 9657  df-uz 9939  df-rp 10073  df-seqfrec 10907  df-exp 10998  df-rsqrt 11787

This theorem is used by:  sqrt2irr  12967