Theorem sitgf

Description: The integral for simple functions is itself a function. (Contributed by Thierry Arnoux, 13-Feb-2018)

	Ref	Expression
Hypotheses	sitgval.b	⊢ 𝐵 = ( Base ‘ 𝑊 )
	sitgval.j	⊢ 𝐽 = ( TopOpen ‘ 𝑊 )
	sitgval.s	⊢ 𝑆 = ( sigaGen ‘ 𝐽 )
	sitgval.0	⊢ 0 = ( 0g ‘ 𝑊 )
	sitgval.x	⊢ · = ( ·𝑠 ‘ 𝑊 )
	sitgval.h	⊢ 𝐻 = ( ℝHom ‘ ( Scalar ‘ 𝑊 ) )
	sitgval.1	⊢ ( 𝜑 → 𝑊 ∈ 𝑉 )
	sitgval.2	⊢ ( 𝜑 → 𝑀 ∈ ∪ ran measures )
	sitgf.1	⊢ ( ( 𝜑 ∧ 𝑓 ∈ dom ( 𝑊 sitg 𝑀 ) ) → ( ( 𝑊 sitg 𝑀 ) ‘ 𝑓 ) ∈ 𝐵 )
Assertion	sitgf	⊢ ( 𝜑 → ( 𝑊 sitg 𝑀 ) : dom ( 𝑊 sitg 𝑀 ) ⟶ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		sitgval.b	⊢ 𝐵 = ( Base ‘ 𝑊 )
2		sitgval.j	⊢ 𝐽 = ( TopOpen ‘ 𝑊 )
3		sitgval.s	⊢ 𝑆 = ( sigaGen ‘ 𝐽 )
4		sitgval.0	⊢ 0 = ( 0g ‘ 𝑊 )
5		sitgval.x	⊢ · = ( ·𝑠 ‘ 𝑊 )
6		sitgval.h	⊢ 𝐻 = ( ℝHom ‘ ( Scalar ‘ 𝑊 ) )
7		sitgval.1	⊢ ( 𝜑 → 𝑊 ∈ 𝑉 )
8		sitgval.2	⊢ ( 𝜑 → 𝑀 ∈ ∪ ran measures )
9		sitgf.1	⊢ ( ( 𝜑 ∧ 𝑓 ∈ dom ( 𝑊 sitg 𝑀 ) ) → ( ( 𝑊 sitg 𝑀 ) ‘ 𝑓 ) ∈ 𝐵 )
10		funmpt	⊢ Fun ( 𝑓 ∈ { 𝑔 ∈ ( dom 𝑀 MblFnM 𝑆 ) ∣ ( ran 𝑔 ∈ Fin ∧ ∀ 𝑥 ∈ ( ran 𝑔 ∖ { 0 } ) ( 𝑀 ‘ ( ◡ 𝑔 “ { 𝑥 } ) ) ∈ ( 0 [,) +∞ ) ) } ↦ ( 𝑊 Σg ( 𝑥 ∈ ( ran 𝑓 ∖ { 0 } ) ↦ ( ( 𝐻 ‘ ( 𝑀 ‘ ( ◡ 𝑓 “ { 𝑥 } ) ) ) · 𝑥 ) ) ) )
11	1 2 3 4 5 6 7 8	sitgval	⊢ ( 𝜑 → ( 𝑊 sitg 𝑀 ) = ( 𝑓 ∈ { 𝑔 ∈ ( dom 𝑀 MblFnM 𝑆 ) ∣ ( ran 𝑔 ∈ Fin ∧ ∀ 𝑥 ∈ ( ran 𝑔 ∖ { 0 } ) ( 𝑀 ‘ ( ◡ 𝑔 “ { 𝑥 } ) ) ∈ ( 0 [,) +∞ ) ) } ↦ ( 𝑊 Σg ( 𝑥 ∈ ( ran 𝑓 ∖ { 0 } ) ↦ ( ( 𝐻 ‘ ( 𝑀 ‘ ( ◡ 𝑓 “ { 𝑥 } ) ) ) · 𝑥 ) ) ) ) )
12	11	funeqd	⊢ ( 𝜑 → ( Fun ( 𝑊 sitg 𝑀 ) ↔ Fun ( 𝑓 ∈ { 𝑔 ∈ ( dom 𝑀 MblFnM 𝑆 ) ∣ ( ran 𝑔 ∈ Fin ∧ ∀ 𝑥 ∈ ( ran 𝑔 ∖ { 0 } ) ( 𝑀 ‘ ( ◡ 𝑔 “ { 𝑥 } ) ) ∈ ( 0 [,) +∞ ) ) } ↦ ( 𝑊 Σg ( 𝑥 ∈ ( ran 𝑓 ∖ { 0 } ) ↦ ( ( 𝐻 ‘ ( 𝑀 ‘ ( ◡ 𝑓 “ { 𝑥 } ) ) ) · 𝑥 ) ) ) ) ) )
13	10 12	mpbiri	⊢ ( 𝜑 → Fun ( 𝑊 sitg 𝑀 ) )
14	13	funfnd	⊢ ( 𝜑 → ( 𝑊 sitg 𝑀 ) Fn dom ( 𝑊 sitg 𝑀 ) )
15	9	ralrimiva	⊢ ( 𝜑 → ∀ 𝑓 ∈ dom ( 𝑊 sitg 𝑀 ) ( ( 𝑊 sitg 𝑀 ) ‘ 𝑓 ) ∈ 𝐵 )
16		fnfvrnss	⊢ ( ( ( 𝑊 sitg 𝑀 ) Fn dom ( 𝑊 sitg 𝑀 ) ∧ ∀ 𝑓 ∈ dom ( 𝑊 sitg 𝑀 ) ( ( 𝑊 sitg 𝑀 ) ‘ 𝑓 ) ∈ 𝐵 ) → ran ( 𝑊 sitg 𝑀 ) ⊆ 𝐵 )
17	14 15 16	syl2anc	⊢ ( 𝜑 → ran ( 𝑊 sitg 𝑀 ) ⊆ 𝐵 )
18		df-f	⊢ ( ( 𝑊 sitg 𝑀 ) : dom ( 𝑊 sitg 𝑀 ) ⟶ 𝐵 ↔ ( ( 𝑊 sitg 𝑀 ) Fn dom ( 𝑊 sitg 𝑀 ) ∧ ran ( 𝑊 sitg 𝑀 ) ⊆ 𝐵 ) )
19	14 17 18	sylanbrc	⊢ ( 𝜑 → ( 𝑊 sitg 𝑀 ) : dom ( 𝑊 sitg 𝑀 ) ⟶ 𝐵 )