Theorem copbezout ≪ | index | src | ≫

theorem copbezout (G: wff) (a b: nat) {x y: nat}:
  $ G -> coprime a b $ >
  $ G -> a != 0 $ >
  $ G -> E. x E. y x * a = y * b + 1 $;

Step	Hyp	Ref	Expression
1		hyp h1	G -> coprime a b
2	1	conv coprime	G -> gcd a b = 1
3	2	addeq2d	G -> y * b + gcd a b = y * b + 1
4	3	eqeq2d	G -> (x * a = y * b + gcd a b <-> x * a = y * b + 1)
5	4	exeqd	G -> (E. y x * a = y * b + gcd a b <-> E. y x * a = y * b + 1)
6	5	exeqd	G -> (E. x E. y x * a = y * b + gcd a b <-> E. x E. y x * a = y * b + 1)
7		bezout	a != 0 -> E. x E. y x * a = y * b + gcd a b
8		hyp h2	G -> a != 0
9	7, 8	syl	G -> E. x E. y x * a = y * b + gcd a b
10	6, 9	mpbid	G -> E. x E. y x * a = y * b + 1

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12), axs_set (elab, ax_8), axs_the (theid, the0), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)