Theorem eucalglt 12754

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 12751	. . . . . . 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 3627	. . . . . . . . . . . . 13 |- ( ( 2nd ` X ) = 0 -> if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) = X )
6	5	eqeq2d 2244	. . . . . . . . . . . 12 |- ( ( 2nd ` X ) = 0 -> ( ( E ` X ) = if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) <-> ( E ` X ) = X ) )
7		fveq2 5670	. . . . . . . . . . . 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 2242	. . . . . . . . . . 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 2454	. . . . . . . 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 3632	. . . . . 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 2265	. . . . 5 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( E ` X ) = <. ( 2nd ` X ) , ( mod ` X ) >. )
16	15	fveq2d 5674	. . . 4 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` ( E ` X ) ) = ( 2nd ` <. ( 2nd ` X ) , ( mod ` X ) >. ) )
17		xp2nd 6360	. . . . . 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 6369	. . . . . . . . 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 5674	. . . . . . 7 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( mod ` X ) = ( mod ` <. ( 1st ` X ) , ( 2nd ` X ) >. ) )
22		df-ov 6053	. . . . . . 7 |- ( ( 1st ` X ) mod ( 2nd ` X ) ) = ( mod ` <. ( 1st ` X ) , ( 2nd ` X ) >. )
23	21, 22	eqtr4di 2283	. . . . . 6 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( mod ` X ) = ( ( 1st ` X ) mod ( 2nd ` X ) ) )
24		xp1st 6359	. . . . . . . . 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 9698	. . . . . . 7 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 1st ` X ) e. ZZ )
27	13	neqned 2419	. . . . . . . 8 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` X ) =/= 0 )
28		elnnne0 9510	. . . . . . . 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 10707	. . . . . 6 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( ( 1st ` X ) mod ( 2nd ` X ) ) e. NN0 )
31	23, 30	eqeltrd 2309	. . . . 5 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( mod ` X ) e. NN0 )
32		op2ndg 6345	. . . . 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 2269	. . 3 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` ( E ` X ) ) = ( ( 1st ` X ) mod ( 2nd ` X ) ) )
35		zq 9958	. . . . 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 9698	. . . . 5 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( 2nd ` X ) e. ZZ )
38		zq 9958	. . . . 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 9281	. . . 4 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> 0 < ( 2nd ` X ) )
41		modqlt 10695	. . . 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 1274	. . 3 |- ( ( X e. ( NN0 X. NN0 ) /\ ( 2nd ` ( E ` X ) ) =/= 0 ) -> ( ( 1st ` X ) mod ( 2nd ` X ) ) < ( 2nd ` X ) )
43	34, 42	eqbrtrd 4131	. 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 1398   e. wcel 2203   =/= wne 2412   ifcif 3620   <.cop 3692   class class class wbr 4109   X. cxp 4747   ` cfv 5352  (class class class)co 6050   e. cmpo 6052   1stc1st 6332   2ndc2nd 6333   0cc0 8127   < clt 8308   NNcn 9237   NN0cn0 9496   ZZcz 9577   QQcq 9951   mod cmo 10684

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-n0 9497  df-z 9578  df-q 9952  df-rp 9987  df-fl 10630  df-mod 10685

This theorem is referenced by:  eucalgcvga  12755