Theorem eucalglt 12574

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 12571	. . . . . . 7 |- ( X e. ( NN0 X. NN0 ) -> ( E ` X ) = if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) )
3	2	adantr 276	. . . . . 6 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( E ` X ) = if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) )
4		simpr 110	. . . . . . . 8 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` ( E ` X ) ) =/= 0 )
5		iftrue 3607	. . . . . . . . . . . . 13 |- ( ( 2nd ` X ) = 0 -> if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) = X )
6	5	eqeq2d 2241	. . . . . . . . . . . 12 |- ( ( 2nd ` X ) = 0 -> ( ( E ` X ) = if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) <-> ( E ` X ) = X ) )
7		fveq2 5626	. . . . . . . . . . . 12 |- ( ( E ` X ) = X -> ( 2nd ` ( E ` X ) ) = ( 2nd ` X ) )
8	6, 7	biimtrdi 163	. . . . . . . . . . 11 |- ( ( 2nd ` X ) = 0 -> ( ( E ` X ) = if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) -> ( 2nd ` ( E ` X ) ) = ( 2nd ` X ) ) )
9		eqeq2 2239	. . . . . . . . . . 11 |- ( ( 2nd ` X ) = 0 -> ( ( 2nd ` ( E ` X ) ) = ( 2nd ` X ) <-> ( 2nd ` ( E ` X ) ) = 0 ) )
10	8, 9	sylibd 149	. . . . . . . . . 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 2442	. . . . . . . 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 3612	. . . . . 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 2262	. . . . 5 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( E ` X ) = <. ( 2nd ` X ) , ( mod ` X ) >. )
16	15	fveq2d 5630	. . . 4 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` ( E ` X ) ) = ( 2nd ` <. ( 2nd ` X ) , ( mod ` X ) >. ) )
17		xp2nd 6310	. . . . . 6 |- ( X e. ( NN0 X. NN0 ) -> ( 2nd ` X ) e. NN0 )
18	17	adantr 276	. . . . 5 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` X ) e. NN0 )
19		1st2nd2 6319	. . . . . . . . 9 |- ( X e. ( NN0 X. NN0 ) -> X = <. ( 1st ` X ) , ( 2nd ` X ) >. )
20	19	adantr 276	. . . . . . . 8 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> X = <. ( 1st ` X ) , ( 2nd ` X ) >. )
21	20	fveq2d 5630	. . . . . . 7 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( mod ` X ) = ( mod ` <. ( 1st ` X ) , ( 2nd ` X ) >. ) )
22		df-ov 6003	. . . . . . 7 |- ( ( 1st ` X ) mod ( 2nd ` X ) ) = ( mod ` <. ( 1st ` X ) , ( 2nd ` X ) >. )
23	21, 22	eqtr4di 2280	. . . . . 6 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( mod ` X ) = ( ( 1st ` X ) mod ( 2nd ` X ) ) )
24		xp1st 6309	. . . . . . . . 9 |- ( X e. ( NN0 X. NN0 ) -> ( 1st ` X ) e. NN0 )
25	24	adantr 276	. . . . . . . 8 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 1st ` X ) e. NN0 )
26	25	nn0zd 9563	. . . . . . 7 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 1st ` X ) e. ZZ )
27	13	neqned 2407	. . . . . . . 8 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` X ) =/= 0 )
28		elnnne0 9379	. . . . . . . 8 |- ( ( 2nd ` X ) e. NN <-> ( ( 2nd ` X ) e. NN0 /\ ( 2nd ` X ) =/= 0 ) )
29	18, 27, 28	sylanbrc 417	. . . . . . 7 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` X ) e. NN )
30	26, 29	zmodcld 10562	. . . . . 6 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( ( 1st ` X ) mod ( 2nd ` X ) ) e. NN0 )
31	23, 30	eqeltrd 2306	. . . . 5 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( mod ` X ) e. NN0 )
32		op2ndg 6295	. . . . 5 |- ( ( ( 2nd ` X ) e. NN0 /\ ( mod ` X ) e. NN0 ) -> ( 2nd ` <. ( 2nd ` X ) , ( mod ` X ) >. ) = ( mod ` X ) )
33	18, 31, 32	syl2anc 411	. . . 4 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` <. ( 2nd ` X ) , ( mod ` X ) >. ) = ( mod ` X ) )
34	16, 33, 23	3eqtrd 2266	. . 3 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` ( E ` X ) ) = ( ( 1st ` X ) mod ( 2nd ` X ) ) )
35		zq 9817	. . . . 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 9563	. . . . 5 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` X ) e. ZZ )
38		zq 9817	. . . . 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 9150	. . . 4 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> 0 < ( 2nd ` X ) )
41		modqlt 10550	. . . 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 1271	. . 3 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( ( 1st ` X ) mod ( 2nd ` X ) ) < ( 2nd ` X ) )
43	34, 42	eqbrtrd 4104	. 2 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` ( E ` X ) ) < ( 2nd ` X ) )
44	43	ex 115	1 |- ( X e. ( NN0 X. NN0 ) -> ( ( 2nd ` ( E ` X ) ) =/= 0 -> ( 2nd ` ( E ` X ) ) < ( 2nd ` X ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   =/= wne 2400   ifcif 3602   <.cop 3669   class class class wbr 4082   X. cxp 4716   ` cfv 5317  (class class class)co 6000   e. cmpo 6002   1stc1st 6282   2ndc2nd 6283   0cc0 7995   < clt 8177   NNcn 9106   NN0cn0 9365   ZZcz 9442   QQcq 9810   mod cmo 10539

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-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114

This theorem depends on definitions:  df-bi 117  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 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-po 4386  df-iso 4387  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-n0 9366  df-z 9443  df-q 9811  df-rp 9846  df-fl 10485  df-mod 10540

This theorem is referenced by:  eucalgcvga  12575