Theorem 1p2e3

Description: 1 + 2 = 3. For a shorter proof using addcomli , see 1p2e3ALT . (Contributed by David A. Wheeler, 8-Dec-2018) Reduce dependencies on axioms. (Revised by Steven Nguyen, 12-Dec-2022)

	Ref	Expression
Assertion	1p2e3	⊢ ( 1 + 2 ) = 3

Proof

Step	Hyp	Ref	Expression
1		df-2	⊢ 2 = ( 1 + 1 )
2	1	oveq2i	⊢ ( 1 + 2 ) = ( 1 + ( 1 + 1 ) )
3		ax-1cn	⊢ 1 ∈ ℂ
4	3 3 3	addassi	⊢ ( ( 1 + 1 ) + 1 ) = ( 1 + ( 1 + 1 ) )
5		1p1e2	⊢ ( 1 + 1 ) = 2
6	5	oveq1i	⊢ ( ( 1 + 1 ) + 1 ) = ( 2 + 1 )
7		2p1e3	⊢ ( 2 + 1 ) = 3
8	6 7	eqtri	⊢ ( ( 1 + 1 ) + 1 ) = 3
9	2 4 8	3eqtr2i	⊢ ( 1 + 2 ) = 3