fsumcl2lem

Description: - Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.) (Contributed by Mario Carneiro, 3-Jun-2014)

	Ref	Expression
Hypotheses	fsumcllem.1	⊢ ( 𝜑 → 𝑆 ⊆ ℂ )
	fsumcllem.2	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 )
	fsumcllem.3	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	fsumcllem.4	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ 𝑆 )
	fsumcl2lem.5	⊢ ( 𝜑 → 𝐴 ≠ ∅ )
Assertion	fsumcl2lem	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		fsumcllem.1	⊢ ( 𝜑 → 𝑆 ⊆ ℂ )
2		fsumcllem.2	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 )
3		fsumcllem.3	⊢ ( 𝜑 → 𝐴 ∈ Fin )
4		fsumcllem.4	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ 𝑆 )
5		fsumcl2lem.5	⊢ ( 𝜑 → 𝐴 ≠ ∅ )
6	5	a1d	⊢ ( 𝜑 → ( ¬ Σ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 → 𝐴 ≠ ∅ ) )
7	6	necon4bd	⊢ ( 𝜑 → ( 𝐴 = ∅ → Σ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ) )
8		sumfc	⊢ Σ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) = Σ 𝑘 ∈ 𝐴 𝐵
9		fveq2	⊢ ( 𝑚 = ( 𝑓 ‘ 𝑥 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑥 ) ) )
10		simprl	⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ♯ ‘ 𝐴 ) ∈ ℕ )
11		simprr	⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 )
12	1	ad2antrr	⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑚 ∈ 𝐴 ) → 𝑆 ⊆ ℂ )
13	4	fmpttd	⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ 𝑆 )
14	13	adantr	⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ 𝑆 )
15	14	ffvelcdmda	⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑚 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) ∈ 𝑆 )
16	12 15	sseldd	⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑚 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) ∈ ℂ )
17		f1of	⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 )
18	11 17	syl	⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 )
19		fvco3	⊢ ( ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ‘ 𝑥 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑥 ) ) )
20	18 19	sylan	⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ‘ 𝑥 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑥 ) ) )
21	9 10 11 16 20	fsum	⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → Σ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) = ( seq 1 ( + , ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) )
22	8 21	eqtr3id	⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → Σ 𝑘 ∈ 𝐴 𝐵 = ( seq 1 ( + , ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) )
23		nnuz	⊢ ℕ = ( ℤ_≥ ‘ 1 )
24	10 23	eleqtrdi	⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ♯ ‘ 𝐴 ) ∈ ( ℤ_≥ ‘ 1 ) )
25		fco	⊢ ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ 𝑆 ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝑆 )
26	14 18 25	syl2anc	⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝑆 )
27	26	ffvelcdmda	⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑥 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ‘ 𝑥 ) ∈ 𝑆 )
28	2	adantlr	⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 )
29	24 27 28	seqcl	⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( seq 1 ( + , ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) ∈ 𝑆 )
30	22 29	eqeltrd	⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → Σ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 )
31	30	expr	⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝐴 ) ∈ ℕ ) → ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → Σ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ) )
32	31	exlimdv	⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝐴 ) ∈ ℕ ) → ( ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → Σ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ) )
33	32	expimpd	⊢ ( 𝜑 → ( ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → Σ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ) )
34		fz1f1o	⊢ ( 𝐴 ∈ Fin → ( 𝐴 = ∅ ∨ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) )
35	3 34	syl	⊢ ( 𝜑 → ( 𝐴 = ∅ ∨ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) )
36	7 33 35	mpjaod	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 )

Metamath Proof Explorer

Theorem fsumcl2lem

Proof