ax-hvmulid - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > ax-hvmulid		Structured version Visualization version GIF version

Axiom ax-hvmulid 30523

Description: Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)

**Assertion**
Ref	Expression
ax-hvmulid	⊢ (𝐴 ∈ ℋ → (1 ·_ℎ 𝐴) = 𝐴)

**Detailed syntax breakdown of Axiom ax-hvmulid**
Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		chba 30436	. . 3 class ℋ
3	1, 2	wcel 2105	. 2 wff 𝐴 ∈ ℋ
4		c1 11114	. . . 4 class 1
5		csm 30438	. . . 4 class ·_ℎ
6	4, 1, 5	co 7412	. . 3 class (1 ·_ℎ 𝐴)
7	6, 1	wceq 1540	. 2 wff (1 ·_ℎ 𝐴) = 𝐴
8	3, 7	wi 4	1 wff (𝐴 ∈ ℋ → (1 ·_ℎ 𝐴) = 𝐴)

Colors of variables: wff setvar class

This axiom is referenced by: hvmul0or 30542 hvsubid 30543 hvaddsubval 30550 hv2times 30578 hvnegdii 30579 hilvc 30679 hhssnv 30781 h1de2bi 31071 h1datomi 31098 mayete3i 31245 homullid 31317 lnop0 31483 lnopaddi 31488 lnophmlem2 31534 lnfn0i 31559 lnfnaddi 31560 strlem1 31767