Theorem eucalglt 11989

Description: The second member of the state decreases with each iteration of the step function E for Euclid's Algorithm. (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
eucalglt	|- ( X e. ( NN0 X. NN0 ) -> ( ( 2nd ` ( E ` X ) ) =/= 0 -> ( 2nd ` ( E ` X ) ) < ( 2nd ` X ) ) )

Distinct variable group:   x, y, X

Allowed substitution hints:   E( x, y)

Proof of Theorem eucalglt

Step	Hyp	Ref	Expression
1		eucalgval.1	. . . . . . . 8 |- E = ( x e. NN0 , y e. NN0 |-> if ( y = 0 , <. x , y >. , <. y , ( x mod y ) >. ) )
2	1	eucalgval 11986	. . . . . . 7 |- ( X e. ( NN0 X. NN0 ) -> ( E ` X ) = if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) )
3	2	adantr 274	. . . . . 6 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( E ` X ) = if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) )
4		simpr 109	. . . . . . . 8 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` ( E ` X ) ) =/= 0 )
5		iftrue 3525	. . . . . . . . . . . . 13 |- ( ( 2nd ` X ) = 0 -> if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) = X )
6	5	eqeq2d 2177	. . . . . . . . . . . 12 |- ( ( 2nd ` X ) = 0 -> ( ( E ` X ) = if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) <-> ( E ` X ) = X ) )
7		fveq2 5486	. . . . . . . . . . . 12 |- ( ( E ` X ) = X -> ( 2nd ` ( E ` X ) ) = ( 2nd ` X ) )
8	6, 7	syl6bi 162	. . . . . . . . . . 11 |- ( ( 2nd ` X ) = 0 -> ( ( E ` X ) = if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) -> ( 2nd ` ( E ` X ) ) = ( 2nd ` X ) ) )
9		eqeq2 2175	. . . . . . . . . . 11 |- ( ( 2nd ` X ) = 0 -> ( ( 2nd ` ( E ` X ) ) = ( 2nd ` X ) <-> ( 2nd ` ( E ` X ) ) = 0 ) )
10	8, 9	sylibd 148	. . . . . . . . . 10 |- ( ( 2nd ` X ) = 0 -> ( ( E ` X ) = if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) -> ( 2nd ` ( E ` X ) ) = 0 ) )
11	3, 10	syl5com 29	. . . . . . . . 9 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( ( 2nd ` X ) = 0 -> ( 2nd ` ( E ` X ) ) = 0 ) )
12	11	necon3ad 2378	. . . . . . . 8 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( ( 2nd ` ( E ` X ) ) =/= 0 -> -. ( 2nd ` X ) = 0 ) )
13	4, 12	mpd 13	. . . . . . 7 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> -. ( 2nd ` X ) = 0 )
14	13	iffalsed 3530	. . . . . 6 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) = <. ( 2nd ` X ) , ( mod ` X ) >. )
15	3, 14	eqtrd 2198	. . . . 5 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( E ` X ) = <. ( 2nd ` X ) , ( mod ` X ) >. )
16	15	fveq2d 5490	. . . 4 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` ( E ` X ) ) = ( 2nd ` <. ( 2nd ` X ) , ( mod ` X ) >. ) )
17		xp2nd 6134	. . . . . 6 |- ( X e. ( NN0 X. NN0 ) -> ( 2nd ` X ) e. NN0 )
18	17	adantr 274	. . . . 5 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` X ) e. NN0 )
19		1st2nd2 6143	. . . . . . . . 9 |- ( X e. ( NN0 X. NN0 ) -> X = <. ( 1st ` X ) , ( 2nd ` X ) >. )
20	19	adantr 274	. . . . . . . 8 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> X = <. ( 1st ` X ) , ( 2nd ` X ) >. )
21	20	fveq2d 5490	. . . . . . 7 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( mod ` X ) = ( mod ` <. ( 1st ` X ) , ( 2nd ` X ) >. ) )
22		df-ov 5845	. . . . . . 7 |- ( ( 1st ` X ) mod ( 2nd ` X ) ) = ( mod ` <. ( 1st ` X ) , ( 2nd ` X ) >. )
23	21, 22	eqtr4di 2217	. . . . . 6 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( mod ` X ) = ( ( 1st ` X ) mod ( 2nd ` X ) ) )
24		xp1st 6133	. . . . . . . . 9 |- ( X e. ( NN0 X. NN0 ) -> ( 1st ` X ) e. NN0 )
25	24	adantr 274	. . . . . . . 8 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 1st ` X ) e. NN0 )
26	25	nn0zd 9311	. . . . . . 7 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 1st ` X ) e. ZZ )
27	13	neqned 2343	. . . . . . . 8 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` X ) =/= 0 )
28		elnnne0 9128	. . . . . . . 8 |- ( ( 2nd ` X ) e. NN <-> ( ( 2nd ` X ) e. NN0 /\ ( 2nd ` X ) =/= 0 ) )
29	18, 27, 28	sylanbrc 414	. . . . . . 7 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` X ) e. NN )
30	26, 29	zmodcld 10280	. . . . . 6 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( ( 1st ` X ) mod ( 2nd ` X ) ) e. NN0 )
31	23, 30	eqeltrd 2243	. . . . 5 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( mod ` X ) e. NN0 )
32		op2ndg 6119	. . . . 5 |- ( ( ( 2nd ` X ) e. NN0 /\ ( mod ` X ) e. NN0 ) -> ( 2nd ` <. ( 2nd ` X ) , ( mod ` X ) >. ) = ( mod ` X ) )
33	18, 31, 32	syl2anc 409	. . . 4 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` <. ( 2nd ` X ) , ( mod ` X ) >. ) = ( mod ` X ) )
34	16, 33, 23	3eqtrd 2202	. . 3 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` ( E ` X ) ) = ( ( 1st ` X ) mod ( 2nd ` X ) ) )
35		zq 9564	. . . . 5 |- ( ( 1st ` X ) e. ZZ -> ( 1st ` X ) e. QQ )
36	26, 35	syl 14	. . . 4 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 1st ` X ) e. QQ )
37	18	nn0zd 9311	. . . . 5 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` X ) e. ZZ )
38		zq 9564	. . . . 5 |- ( ( 2nd ` X ) e. ZZ -> ( 2nd ` X ) e. QQ )
39	37, 38	syl 14	. . . 4 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` X ) e. QQ )
40	29	nngt0d 8901	. . . 4 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> 0 < ( 2nd ` X ) )
41		modqlt 10268	. . . 4 |- ( ( ( 1st ` X ) e. QQ /\ ( 2nd ` X ) e. QQ /\ 0 < ( 2nd ` X ) ) -> ( ( 1st ` X ) mod ( 2nd ` X ) ) < ( 2nd ` X ) )
42	36, 39, 40, 41	syl3anc 1228	. . 3 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( ( 1st ` X ) mod ( 2nd ` X ) ) < ( 2nd ` X ) )
43	34, 42	eqbrtrd 4004	. 2 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` ( E ` X ) ) < ( 2nd ` X ) )
44	43	ex 114	1 |- ( X e. ( NN0 X. NN0 ) -> ( ( 2nd ` ( E ` X ) ) =/= 0 -> ( 2nd ` ( E ` X ) ) < ( 2nd ` X ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   =/= wne 2336   ifcif 3520   <.cop 3579   class class class wbr 3982   X. cxp 4602   ` cfv 5188  (class class class)co 5842   e. cmpo 5844   1stc1st 6106   2ndc2nd 6107   0cc0 7753   < clt 7933   NNcn 8857   NN0cn0 9114   ZZcz 9191   QQcq 9557   mod cmo 10257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-n0 9115  df-z 9192  df-q 9558  df-rp 9590  df-fl 10205  df-mod 10258

This theorem is referenced by:  eucalgcvga  11990