Theorem eucalginv 12075

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 12073	. . 3 |- ( X e. ( NN0 X. NN0 ) -> ( E ` X ) = if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) )
3	2	fveq2d 5534	. 2 |- ( X e. ( NN0 X. NN0 ) -> ( gcd ` ( E ` X ) ) = ( gcd ` if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) ) )
4		1st2nd2 6194	. . . . . . . . 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 5534	. . . . . . 7 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( mod ` X ) = ( mod ` <. ( 1st ` X ) , ( 2nd ` X ) >. ) )
7		df-ov 5894	. . . . . . 7 |- ( ( 1st ` X ) mod ( 2nd ` X ) ) = ( mod ` <. ( 1st ` X ) , ( 2nd ` X ) >. )
8	6, 7	eqtr4di 2240	. . . . . 6 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( mod ` X ) = ( ( 1st ` X ) mod ( 2nd ` X ) ) )
9	8	oveq2d 5907	. . . . 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 9291	. . . . . . 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 6184	. . . . . . . . . 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 9392	. . . . . . . 8 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( 1st ` X ) e. ZZ )
15		zmodcl 10363	. . . . . . . 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 9392	. . . . . 6 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( ( 1st ` X ) mod ( 2nd ` X ) ) e. ZZ )
18		gcdcom 11993	. . . . . 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 12011	. . . . . 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 2226	. . . 4 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( ( 2nd ` X ) gcd ( mod ` X ) ) = ( ( 1st ` X ) gcd ( 2nd ` X ) ) )
23		nnne0 8966	. . . . . . . . 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 2381	. . . . . . 7 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> -. ( 2nd ` X ) = 0 )
26	25	iffalsed 3559	. . . . . 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 5534	. . . . 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 5894	. . . . 5 |- ( ( 2nd ` X ) gcd ( mod ` X ) ) = ( gcd ` <. ( 2nd ` X ) , ( mod ` X ) >. )
29	27, 28	eqtr4di 2240	. . . 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 5534	. . . . 5 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( gcd ` X ) = ( gcd ` <. ( 1st ` X ) , ( 2nd ` X ) >. ) )
31		df-ov 5894	. . . . 5 |- ( ( 1st ` X ) gcd ( 2nd ` X ) ) = ( gcd ` <. ( 1st ` X ) , ( 2nd ` X ) >. )
32	30, 31	eqtr4di 2240	. . . 4 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( gcd ` X ) = ( ( 1st ` X ) gcd ( 2nd ` X ) ) )
33	22, 29, 32	3eqtr4d 2232	. . 3 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` X ) e. NN ) -> ( gcd ` if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) ) = ( gcd ` X ) )
34		iftrue 3554	. . . . 5 |- ( ( 2nd ` X ) = 0 -> if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) = X )
35	34	fveq2d 5534	. . . 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 6185	. . . 4 |- ( X e. ( NN0 X. NN0 ) -> ( 2nd ` X ) e. NN0 )
38		elnn0 9197	. . . 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 799	. 2 |- ( X e. ( NN0 X. NN0 ) -> ( gcd ` if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) ) = ( gcd ` X ) )
41	3, 40	eqtrd 2222	1 |- ( X e. ( NN0 X. NN0 ) -> ( gcd ` ( E ` X ) ) = ( gcd ` X ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   e. wcel 2160   =/= wne 2360   ifcif 3549   <.cop 3610   X. cxp 4639   ` cfv 5231  (class class class)co 5891   e. cmpo 5893   1stc1st 6157   2ndc2nd 6158   0cc0 7830   NNcn 8938   NN0cn0 9195   ZZcz 9272   mod cmo 10341   gcd cgcd 11962

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-mulrcl 7929  ax-addcom 7930  ax-mulcom 7931  ax-addass 7932  ax-mulass 7933  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-1rid 7937  ax-0id 7938  ax-rnegex 7939  ax-precex 7940  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-apti 7945  ax-pre-ltadd 7946  ax-pre-mulgt0 7947  ax-pre-mulext 7948  ax-arch 7949  ax-caucvg 7950

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-frec 6410  df-sup 7002  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-reap 8551  df-ap 8558  df-div 8649  df-inn 8939  df-2 8997  df-3 8998  df-4 8999  df-n0 9196  df-z 9273  df-uz 9548  df-q 9639  df-rp 9673  df-fz 10028  df-fzo 10162  df-fl 10289  df-mod 10342  df-seqfrec 10465  df-exp 10539  df-cj 10870  df-re 10871  df-im 10872  df-rsqrt 11026  df-abs 11027  df-dvds 11814  df-gcd 11963

This theorem is referenced by:  eucalg  12078