Theorem eucalgval 12857

Description: Euclid's Algorithm eucalg 12862 computes the greatest common divisor of two nonnegative integers by repeatedly replacing the larger of them with its remainder modulo the smaller until the remainder is 0.

The value of the step function E for Euclid's Algorithm. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-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
eucalgval	|- ( X e. ( NN0 X. NN0 ) -> ( E ` X ) = if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) )

Distinct variable group:   x, y, X

Allowed substitution hints:   E( x, y)

Proof of Theorem eucalgval

Step	Hyp	Ref	Expression
1		df-ov 6091	. . 3 |- ( ( 1st ` X ) E ( 2nd ` X ) ) = ( E ` <. ( 1st ` X ) , ( 2nd ` X ) >. )
2		xp1st 6403	. . . 4 |- ( X e. ( NN0 X. NN0 ) -> ( 1st ` X ) e. NN0 )
3		xp2nd 6404	. . . 4 |- ( X e. ( NN0 X. NN0 ) -> ( 2nd ` X ) e. NN0 )
4		eucalgval.1	. . . . 5 |- E = ( x e. NN0 , y e. NN0 |-> if ( y = 0 , <. x , y >. , <. y , ( x mod y ) >. ) )
5	4	eucalgval2 12856	. . . 4 |- ( ( ( 1st ` X ) e. NN0 /\ ( 2nd ` X ) e. NN0 ) -> ( ( 1st ` X ) E ( 2nd ` X ) ) = if ( ( 2nd ` X ) = 0 , <. ( 1st ` X ) , ( 2nd ` X ) >. , <. ( 2nd ` X ) , ( ( 1st ` X ) mod ( 2nd ` X ) ) >. ) )
6	2, 3, 5	syl2anc 415	. . 3 |- ( X e. ( NN0 X. NN0 ) -> ( ( 1st ` X ) E ( 2nd ` X ) ) = if ( ( 2nd ` X ) = 0 , <. ( 1st ` X ) , ( 2nd ` X ) >. , <. ( 2nd ` X ) , ( ( 1st ` X ) mod ( 2nd ` X ) ) >. ) )
7	1, 6	eqtr3id 2285	. 2 |- ( X e. ( NN0 X. NN0 ) -> ( E ` <. ( 1st ` X ) , ( 2nd ` X ) >. ) = if ( ( 2nd ` X ) = 0 , <. ( 1st ` X ) , ( 2nd ` X ) >. , <. ( 2nd ` X ) , ( ( 1st ` X ) mod ( 2nd ` X ) ) >. ) )
8		1st2nd2 6413	. . 3 |- ( X e. ( NN0 X. NN0 ) -> X = <. ( 1st ` X ) , ( 2nd ` X ) >. )
9	8	fveq2d 5702	. 2 |- ( X e. ( NN0 X. NN0 ) -> ( E ` X ) = ( E ` <. ( 1st ` X ) , ( 2nd ` X ) >. ) )
10	8	fveq2d 5702	. . . . 5 |- ( X e. ( NN0 X. NN0 ) -> ( mod ` X ) = ( mod ` <. ( 1st ` X ) , ( 2nd ` X ) >. ) )
11		df-ov 6091	. . . . 5 |- ( ( 1st ` X ) mod ( 2nd ` X ) ) = ( mod ` <. ( 1st ` X ) , ( 2nd ` X ) >. )
12	10, 11	eqtr4di 2289	. . . 4 |- ( X e. ( NN0 X. NN0 ) -> ( mod ` X ) = ( ( 1st ` X ) mod ( 2nd ` X ) ) )
13	12	opeq2d 3911	. . 3 |- ( X e. ( NN0 X. NN0 ) -> <. ( 2nd ` X ) , ( mod ` X ) >. = <. ( 2nd ` X ) , ( ( 1st ` X ) mod ( 2nd ` X ) ) >. )
14	8, 13	ifeq12d 3660	. 2 |- ( X e. ( NN0 X. NN0 ) -> if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) = if ( ( 2nd ` X ) = 0 , <. ( 1st ` X ) , ( 2nd ` X ) >. , <. ( 2nd ` X ) , ( ( 1st ` X ) mod ( 2nd ` X ) ) >. ) )
15	7, 9, 14	3eqtr4d 2281	1 |- ( X e. ( NN0 X. NN0 ) -> ( E ` X ) = if ( ( 2nd ` X ) = 0 , X , <. ( 2nd ` X ) , ( mod ` X ) >. ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   ifcif 3638   <.cop 3712   X. cxp 4774   ` cfv 5380  (class class class)co 6088   e. cmpo 6090   1stc1st 6376   2ndc2nd 6377   0cc0 8185   NN0cn0 9574   mod cmo 10780

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-po 4443  df-iso 4444  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-n0 9575  df-z 9656  df-q 10036  df-rp 10072  df-fl 10722  df-mod 10781

This theorem is used by:  eucalginv  12859  eucalglt  12860