Axiom ax-precex 6753

Description: Existence of reciprocal of positive real number. Axiom for real and complex numbers, justified by theorem axprecex 6724. (Contributed by Jim Kingdon, 6-Feb-2020.)

Assertion

Ref	Expression
ax-precex	⊢ ((A ∈ ℝ ∧ 0 <ℝ A) → ∃x ∈ ℝ (0 <ℝ x ∧ (A · x) = 1))

Distinct variable group:   x,A

Detailed syntax breakdown of Axiom ax-precex

Step	Hyp	Ref	Expression
1		cA	. . . 4 class A
2		cr 6670	. . . 4 class ℝ
3	1, 2	wcel 1390	. . 3 wff A ∈ ℝ
4		cc0 6671	. . . 4 class 0
5		cltrr 6675	. . . 4 class <ℝ
6	4, 1, 5	wbr 3755	. . 3 wff 0 <ℝ A
7	3, 6	wa 97	. 2 wff (A ∈ ℝ ∧ 0 <ℝ A)
8		vx	. . . . . 6 setvar x
9	8	cv 1241	. . . . 5 class x
10	4, 9, 5	wbr 3755	. . . 4 wff 0 <ℝ x
11		cmul 6676	. . . . . 6 class ·
12	1, 9, 11	co 5455	. . . . 5 class (A · x)
13		c1 6672	. . . . 5 class 1
14	12, 13	wceq 1242	. . . 4 wff (A · x) = 1
15	10, 14	wa 97	. . 3 wff (0 <ℝ x ∧ (A · x) = 1)
16	15, 8, 2	wrex 2301	. 2 wff ∃x ∈ ℝ (0 <ℝ x ∧ (A · x) = 1)
17	7, 16	wi 4	1 wff ((A ∈ ℝ ∧ 0 <ℝ A) → ∃x ∈ ℝ (0 <ℝ x ∧ (A · x) = 1))

This axiom is referenced by:  recexre  7322  recexgt0  7324