Theorem eucalginv 12578

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 12576	. . 3 |- ( X e. ( NN0 X. NN0 ) -> ( E ` X ) = if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) )
3	2	fveq2d 5631	. 2 |- ( X e. ( NN0 X. NN0 ) -> ( gcd ` ( E ` X ) ) = ( gcd ` if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) ) )
4		1st2nd2 6321	. . . . . . . . 9 |- ( X e. ( NN0 X. NN0 ) -> X = <. ( 1st ` X ) , ( 2nd ` X ) >. )
5	4	adantr 276	. . . . . . . 8 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> X = <. ( 1st ` X ) , ( 2nd ` X ) >. )
6	5	fveq2d 5631	. . . . . . 7 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( mod ` X ) = ( mod ` <. ( 1st ` X ) , ( 2nd ` X ) >. ) )
7		df-ov 6004	. . . . . . 7 |- ( ( 1st ` X ) mod ( 2nd ` X ) ) = ( mod ` <. ( 1st ` X ) , ( 2nd ` X ) >. )
8	6, 7	eqtr4di 2280	. . . . . 6 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( mod ` X ) = ( ( 1st ` X ) mod ( 2nd ` X ) ) )
9	8	oveq2d 6017	. . . . 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 9465	. . . . . . 7 |- ( ( 2nd ` X ) e. NN -> ( 2nd ` X ) e. ZZ )
11	10	adantl 277	. . . . . 6 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( 2nd ` X ) e. ZZ )
12		xp1st 6311	. . . . . . . . . 10 |- ( X e. ( NN0 X. NN0 ) -> ( 1st ` X ) e. NN0 )
13	12	adantr 276	. . . . . . . . 9 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( 1st ` X ) e. NN0 )
14	13	nn0zd 9567	. . . . . . . 8 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( 1st ` X ) e. ZZ )
15		zmodcl 10566	. . . . . . . 8 |- ( ( ( 1st ` X ) e. ZZ /\ ( 2nd ` X ) e. NN ) -> ( ( 1st ` X ) mod ( 2nd ` X ) ) e. NN0 )
16	14, 15	sylancom 420	. . . . . . 7 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( ( 1st ` X ) mod ( 2nd ` X ) ) e. NN0 )
17	16	nn0zd 9567	. . . . . 6 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( ( 1st ` X ) mod ( 2nd ` X ) ) e. ZZ )
18		gcdcom 12494	. . . . . 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 411	. . . . 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 12512	. . . . . 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 420	. . . . 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 2266	. . . 4 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( ( 2nd ` X ) gcd ( mod ` X ) ) = ( ( 1st ` X ) gcd ( 2nd ` X ) ) )
23		nnne0 9138	. . . . . . . . 9 |- ( ( 2nd ` X ) e. NN -> ( 2nd ` X ) =/= 0 )
24	23	adantl 277	. . . . . . . 8 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( 2nd ` X ) =/= 0 )
25	24	neneqd 2421	. . . . . . 7 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> -. ( 2nd ` X ) = 0 )
26	25	iffalsed 3612	. . . . . 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 5631	. . . . 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 6004	. . . . 5 |- ( ( 2nd ` X ) gcd ( mod ` X ) ) = ( gcd ` <. ( 2nd ` X ) , ( mod ` X ) >. )
29	27, 28	eqtr4di 2280	. . . 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 5631	. . . . 5 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( gcd ` X ) = ( gcd ` <. ( 1st ` X ) , ( 2nd ` X ) >. ) )
31		df-ov 6004	. . . . 5 |- ( ( 1st ` X ) gcd ( 2nd ` X ) ) = ( gcd ` <. ( 1st ` X ) , ( 2nd ` X ) >. )
32	30, 31	eqtr4di 2280	. . . 4 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( gcd ` X ) = ( ( 1st ` X ) gcd ( 2nd ` X ) ) )
33	22, 29, 32	3eqtr4d 2272	. . 3 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( gcd ` if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) ) = ( gcd ` X ) )
34		iftrue 3607	. . . . 5 |- ( ( 2nd ` X ) = 0 -> if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) = X )
35	34	fveq2d 5631	. . . 4 |- ( ( 2nd ` X ) = 0 -> ( gcd ` if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) ) = ( gcd ` X ) )
36	35	adantl 277	. . 3 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) = 0 ) -> ( gcd ` if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) ) = ( gcd ` X ) )
37		xp2nd 6312	. . . 4 |- ( X e. ( NN0 X. NN0 ) -> ( 2nd ` X ) e. NN0 )
38		elnn0 9371	. . . 4 |- ( ( 2nd ` X ) e. NN0 <-> ( ( 2nd ` X ) e. NN \/ ( 2nd ` X ) = 0 ) )
39	37, 38	sylib 122	. . 3 |- ( X e. ( NN0 X. NN0 ) -> ( ( 2nd ` X ) e. NN \/ ( 2nd ` X ) = 0 ) )
40	33, 36, 39	mpjaodan 803	. 2 |- ( X e. ( NN0 X. NN0 ) -> ( gcd ` if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) ) = ( gcd ` X ) )
41	3, 40	eqtrd 2262	1 |- ( X e. ( NN0 X. NN0 ) -> ( gcd ` ( E ` X ) ) = ( gcd ` X ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   = wceq 1395   e. wcel 2200   =/= wne 2400   ifcif 3602   <.cop 3669   X. cxp 4717   ` cfv 5318  (class class class)co 6001   e. cmpo 6003   1stc1st 6284   2ndc2nd 6285   0cc0 7999   NNcn 9110   NN0cn0 9369   ZZcz 9446   mod cmo 10544   gcd cgcd 12474

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 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-stab 836  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 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-sup 7151  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fz 10205  df-fzo 10339  df-fl 10490  df-mod 10545  df-seqfrec 10670  df-exp 10761  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-dvds 12299  df-gcd 12475

This theorem is referenced by:  eucalg  12581