Theorem eucalglt 12040

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 12037	. . . . . . 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 3539	. . . . . . . . . . . . 13 |- ( ( 2nd ` X ) = 0 -> if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) = X )
6	5	eqeq2d 2189	. . . . . . . . . . . 12 |- ( ( 2nd ` X ) = 0 -> ( ( E ` X ) = if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) <-> ( E ` X ) = X ) )
7		fveq2 5511	. . . . . . . . . . . 12 |- ( ( E ` X ) = X -> ( 2nd ` ( E ` X ) ) = ( 2nd ` X ) )
8	6, 7	syl6bi 163	. . . . . . . . . . 11 |- ( ( 2nd ` X ) = 0 -> ( ( E ` X ) = if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) -> ( 2nd ` ( E ` X ) ) = ( 2nd ` X ) ) )
9		eqeq2 2187	. . . . . . . . . . 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 2389	. . . . . . . 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 3544	. . . . . 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 2210	. . . . 5 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( E ` X ) = <. ( 2nd ` X ) , ( mod ` X ) >. )
16	15	fveq2d 5515	. . . 4 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` ( E ` X ) ) = ( 2nd ` <. ( 2nd ` X ) , ( mod ` X ) >. ) )
17		xp2nd 6161	. . . . . 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 6170	. . . . . . . . 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 5515	. . . . . . 7 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( mod ` X ) = ( mod ` <. ( 1st ` X ) , ( 2nd ` X ) >. ) )
22		df-ov 5872	. . . . . . 7 |- ( ( 1st ` X ) mod ( 2nd ` X ) ) = ( mod ` <. ( 1st ` X ) , ( 2nd ` X ) >. )
23	21, 22	eqtr4di 2228	. . . . . 6 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( mod ` X ) = ( ( 1st ` X ) mod ( 2nd ` X ) ) )
24		xp1st 6160	. . . . . . . . 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 9362	. . . . . . 7 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 1st ` X ) e. ZZ )
27	13	neqned 2354	. . . . . . . 8 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` X ) =/= 0 )
28		elnnne0 9179	. . . . . . . 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 10331	. . . . . 6 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( ( 1st ` X ) mod ( 2nd ` X ) ) e. NN0 )
31	23, 30	eqeltrd 2254	. . . . 5 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( mod ` X ) e. NN0 )
32		op2ndg 6146	. . . . 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 2214	. . 3 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` ( E ` X ) ) = ( ( 1st ` X ) mod ( 2nd ` X ) ) )
35		zq 9615	. . . . 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 9362	. . . . 5 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` X ) e. ZZ )
38		zq 9615	. . . . 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 8952	. . . 4 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> 0 < ( 2nd ` X ) )
41		modqlt 10319	. . . 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 1238	. . 3 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( ( 1st ` X ) mod ( 2nd ` X ) ) < ( 2nd ` X ) )
43	34, 42	eqbrtrd 4022	. 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 1353   e. wcel 2148   =/= wne 2347   ifcif 3534   <.cop 3594   class class class wbr 4000   X. cxp 4621   ` cfv 5212  (class class class)co 5869   e. cmpo 5871   1stc1st 6133   2ndc2nd 6134   0cc0 7802   < clt 7982   NNcn 8908   NN0cn0 9165   ZZcz 9242   QQcq 9608   mod cmo 10308

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-po 4293  df-iso 4294  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-n0 9166  df-z 9243  df-q 9609  df-rp 9641  df-fl 10256  df-mod 10309

This theorem is referenced by:  eucalgcvga  12041