Theorem eucalgval2 15294

Description: The value of the step function E for Euclid's Algorithm on an ordered pair. (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
eucalgval2	|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M E N ) = if ( N = 0 , <. M , N >. , <. N , ( M mod N ) >. ) )

Distinct variable groups:   x, y, M   x, N, y

Allowed substitution hints:   E( x, y)

Proof of Theorem eucalgval2

Step	Hyp	Ref	Expression
1		simpr 477	. . . 4 |- ( ( x = M /\ y = N ) -> y = N )
2	1	eqeq1d 2624	. . 3 |- ( ( x = M /\ y = N ) -> ( y = 0 <-> N = 0 ) )
3		opeq12 4404	. . 3 |- ( ( x = M /\ y = N ) -> <. x , y >. = <. M , N >. )
4		oveq12 6659	. . . 4 |- ( ( x = M /\ y = N ) -> ( x mod y ) = ( M mod N ) )
5	1, 4	opeq12d 4410	. . 3 |- ( ( x = M /\ y = N ) -> <. y , ( x mod y ) >. = <. N , ( M mod N ) >. )
6	2, 3, 5	ifbieq12d 4113	. 2 |- ( ( x = M /\ y = N ) -> if ( y = 0 , <. x , y >. , <. y , ( x mod y ) >. ) = if ( N = 0 , <. M , N >. , <. N , ( M mod N ) >. ) )
7		eucalgval.1	. 2 |- E = ( x e. NN0 , y e. NN0 |-> if ( y = 0 , <. x , y >. , <. y , ( x mod y ) >. ) )
8		opex 4932	. . 3 |- <. M , N >. e. _V
9		opex 4932	. . 3 |- <. N , ( M mod N ) >. e. _V
10	8, 9	ifex 4156	. 2 |- if ( N = 0 , <. M , N >. , <. N , ( M mod N ) >. ) e. _V
11	6, 7, 10	ovmpt2a 6791	1 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( M E N ) = if ( N = 0 , <. M , N >. , <. N , ( M mod N ) >. ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086   <.cop 4183  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   NN0cn0 11292   mod cmo 12668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  eucalgval  15295