Theorem pi11pr ≪ | index | src | ≫

theorem pi11pr (a b c: nat): $ pi11 ((a, b), c) = a $;

Step	Hyp	Ref	Expression
1		eqtr	pi11 ((a, b), c) = fst (a, b) -> fst (a, b) = a -> pi11 ((a, b), c) = a
2		fsteq	fst ((a, b), c) = a, b -> fst (fst ((a, b), c)) = fst (a, b)
3	2	conv pi11	fst ((a, b), c) = a, b -> pi11 ((a, b), c) = fst (a, b)
4		fstpr	fst ((a, b), c) = a, b
5	3, 4	ax_mp	pi11 ((a, b), c) = fst (a, b)
6	1, 5	ax_mp	fst (a, b) = a -> pi11 ((a, b), c) = a
7		fstpr	fst (a, b) = a
8	6, 7	ax_mp	pi11 ((a, b), c) = a

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)