Theorem eucalginv 10438

Description: The invariant of the step function E for Euclid's Algorithm is the gcd operator applied to the state. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 29-May-2014.)

Hypothesis

Ref	Expression
eucalgval.1	|- E = ( x e. NN0 , y e. NN0 |-> if ( y = 0 , <. x , y >. , <. y , ( x mod y ) >. ) )

Assertion

Ref	Expression
eucalginv	|- ( X e. ( NN0 X. NN0 ) -> ( gcd ` ( E ` X ) ) = ( gcd ` X ) )

Distinct variable group:   x, y, X

Allowed substitution hints:   E( x, y)

Proof of Theorem eucalginv

Step	Hyp	Ref	Expression
1		eucalgval.1	. . . 4 |- E = ( x e. NN0 , y e. NN0 |-> if ( y = 0 , <. x , y >. , <. y , ( x mod y ) >. ) )
2	1	eucalgval 10436	. . 3 |- ( X e. ( NN0 X. NN0 ) -> ( E ` X ) = if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) )
3	2	fveq2d 5202	. 2 |- ( X e. ( NN0 X. NN0 ) -> ( gcd ` ( E ` X ) ) = ( gcd ` if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) ) )
4		1st2nd2 5821	. . . . . . . . 9 |- ( X e. ( NN0 X. NN0 ) -> X = <. ( 1st ` X ) , ( 2nd ` X ) >. )
5	4	adantr 270	. . . . . . . 8 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> X = <. ( 1st ` X ) , ( 2nd ` X ) >. )
6	5	fveq2d 5202	. . . . . . 7 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( mod ` X ) = ( mod ` <. ( 1st ` X ) , ( 2nd ` X ) >. ) )
7		df-ov 5535	. . . . . . 7 |- ( ( 1st ` X ) mod ( 2nd ` X ) ) = ( mod ` <. ( 1st ` X ) , ( 2nd ` X ) >. )
8	6, 7	syl6eqr 2131	. . . . . 6 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( mod ` X ) = ( ( 1st ` X ) mod ( 2nd ` X ) ) )
9	8	oveq2d 5548	. . . . 5 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( ( 2nd ` X ) gcd ( mod ` X ) ) = ( ( 2nd ` X ) gcd ( ( 1st ` X ) mod ( 2nd ` X ) ) ) )
10		nnz 8370	. . . . . . 7 |- ( ( 2nd ` X ) e. NN -> ( 2nd ` X ) e. ZZ )
11	10	adantl 271	. . . . . 6 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( 2nd ` X ) e. ZZ )
12		xp1st 5812	. . . . . . . . . 10 |- ( X e. ( NN0 X. NN0 ) -> ( 1st ` X ) e. NN0 )
13	12	adantr 270	. . . . . . . . 9 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( 1st ` X ) e. NN0 )
14	13	nn0zd 8467	. . . . . . . 8 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( 1st ` X ) e. ZZ )
15		zmodcl 9346	. . . . . . . 8 |- ( ( ( 1st ` X ) e. ZZ /\ ( 2nd ` X ) e. NN ) -> ( ( 1st ` X ) mod ( 2nd ` X ) ) e. NN0 )
16	14, 15	sylancom 411	. . . . . . 7 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( ( 1st ` X ) mod ( 2nd ` X ) ) e. NN0 )
17	16	nn0zd 8467	. . . . . 6 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( ( 1st ` X ) mod ( 2nd ` X ) ) e. ZZ )
18		gcdcom 10365	. . . . . 6 |- ( ( ( 2nd ` X ) e. ZZ /\ ( ( 1st ` X ) mod ( 2nd ` X ) ) e. ZZ ) -> ( ( 2nd ` X ) gcd ( ( 1st ` X ) mod ( 2nd ` X ) ) ) = ( ( ( 1st ` X ) mod ( 2nd ` X ) ) gcd ( 2nd ` X ) ) )
19	11, 17, 18	syl2anc 403	. . . . 5 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( ( 2nd ` X ) gcd ( ( 1st ` X ) mod ( 2nd ` X ) ) ) = ( ( ( 1st ` X ) mod ( 2nd ` X ) ) gcd ( 2nd ` X ) ) )
20		modgcd 10382	. . . . . 6 |- ( ( ( 1st ` X ) e. ZZ /\ ( 2nd ` X ) e. NN ) -> ( ( ( 1st ` X ) mod ( 2nd ` X ) ) gcd ( 2nd ` X ) ) = ( ( 1st ` X ) gcd ( 2nd ` X ) ) )
21	14, 20	sylancom 411	. . . . 5 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( ( ( 1st ` X ) mod ( 2nd ` X ) ) gcd ( 2nd ` X ) ) = ( ( 1st ` X ) gcd ( 2nd ` X ) ) )
22	9, 19, 21	3eqtrd 2117	. . . 4 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( ( 2nd ` X ) gcd ( mod ` X ) ) = ( ( 1st ` X ) gcd ( 2nd ` X ) ) )
23		nnne0 8067	. . . . . . . . 9 |- ( ( 2nd ` X ) e. NN -> ( 2nd ` X ) =/= 0 )
24	23	adantl 271	. . . . . . . 8 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( 2nd ` X ) =/= 0 )
25	24	neneqd 2266	. . . . . . 7 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> -. ( 2nd ` X ) = 0 )
26	25	iffalsed 3361	. . . . . 6 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) = <. ( 2nd ` X ) , ( mod ` X ) >. )
27	26	fveq2d 5202	. . . . 5 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( gcd ` if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) ) = ( gcd ` <. ( 2nd ` X ) , ( mod ` X ) >. ) )
28		df-ov 5535	. . . . 5 |- ( ( 2nd ` X ) gcd ( mod ` X ) ) = ( gcd ` <. ( 2nd ` X ) , ( mod ` X ) >. )
29	27, 28	syl6eqr 2131	. . . 4 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( gcd ` if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) ) = ( ( 2nd ` X ) gcd ( mod ` X ) ) )
30	5	fveq2d 5202	. . . . 5 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( gcd ` X ) = ( gcd ` <. ( 1st ` X ) , ( 2nd ` X ) >. ) )
31		df-ov 5535	. . . . 5 |- ( ( 1st ` X ) gcd ( 2nd ` X ) ) = ( gcd ` <. ( 1st ` X ) , ( 2nd ` X ) >. )
32	30, 31	syl6eqr 2131	. . . 4 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( gcd ` X ) = ( ( 1st ` X ) gcd ( 2nd ` X ) ) )
33	22, 29, 32	3eqtr4d 2123	. . 3 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( gcd ` if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) ) = ( gcd ` X ) )
34		iftrue 3356	. . . . 5 |- ( ( 2nd ` X ) = 0 -> if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) = X )
35	34	fveq2d 5202	. . . 4 |- ( ( 2nd ` X ) = 0 -> ( gcd ` if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) ) = ( gcd ` X ) )
36	35	adantl 271	. . 3 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) = 0 ) -> ( gcd ` if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) ) = ( gcd ` X ) )
37		xp2nd 5813	. . . 4 |- ( X e. ( NN0 X. NN0 ) -> ( 2nd ` X ) e. NN0 )
38		elnn0 8290	. . . 4 |- ( ( 2nd ` X ) e. NN0 <-> ( ( 2nd ` X ) e. NN \/ ( 2nd ` X ) = 0 ) )
39	37, 38	sylib 120	. . 3 |- ( X e. ( NN0 X. NN0 ) -> ( ( 2nd ` X ) e. NN \/ ( 2nd ` X ) = 0 ) )
40	33, 36, 39	mpjaodan 744	. 2 |- ( X e. ( NN0 X. NN0 ) -> ( gcd ` if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) ) = ( gcd ` X ) )
41	3, 40	eqtrd 2113	1 |- ( X e. ( NN0 X. NN0 ) -> ( gcd ` ( E ` X ) ) = ( gcd ` X ) )

Syntax hints:   -> wi 4   /\ wa 102   \/ wo 661   = wceq 1284   e. wcel 1433   =/= wne 2245   ifcif 3351   <.cop 3401   X. cxp 4361   ` cfv 4922  (class class class)co 5532   |-> cmpt2 5534   1stc1st 5785   2ndc2nd 5786   0cc0 6981   NNcn 8039   NN0cn0 8288   ZZcz 8351   mod cmo 9324   gcd cgcd 10338

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094  ax-arch 7095  ax-caucvg 7096

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-sup 6397  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-2 8098  df-3 8099  df-4 8100  df-n0 8289  df-z 8352  df-uz 8620  df-q 8705  df-rp 8735  df-fz 9030  df-fzo 9153  df-fl 9274  df-mod 9325  df-iseq 9432  df-iexp 9476  df-cj 9729  df-re 9730  df-im 9731  df-rsqrt 9884  df-abs 9885  df-dvds 10196  df-gcd 10339

This theorem is referenced by:  eucialg  10441