Theorem fsumcl2lem 11272

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.)

Hypotheses

Ref	Expression
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

Ref	Expression
fsumcl2lem	⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆)

Distinct variable groups:   𝐴,𝑘,𝑥,𝑦   𝑥,𝐵,𝑦   𝑆,𝑘,𝑥,𝑦   𝜑,𝑘,𝑥,𝑦

Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem fsumcl2lem

Dummy variables 𝑎 𝑓 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsumcl2lem.5	. . . 4 ⊢ (𝜑 → 𝐴 ≠ ∅)
2	1	neneqd 2345	. . 3 ⊢ (𝜑 → ¬ 𝐴 = ∅)
3	2	pm2.21d 609	. 2 ⊢ (𝜑 → (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆))
4		fsumcllem.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
5	4	adantr 274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ⊆ ℂ)
6		fsumcllem.4	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆)
7	5, 6	sseldd 3125	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
8	7	ralrimiva 2527	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
9		sumfct 11248	. . . . . . . . 9 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐵)
10	8, 9	syl 14	. . . . . . . 8 ⊢ (𝜑 → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐵)
11	10	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐵)
12		fveq2 5461	. . . . . . . 8 ⊢ (𝑚 = (𝑓‘𝑎) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑎)))
13		simprl 521	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ ℕ)
14		simprr 522	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
15	4	ad2antrr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → 𝑆 ⊆ ℂ)
16	6	fmpttd 5615	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑆)
17	16	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑆)
18	17	ffvelrnda 5595	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) ∈ 𝑆)
19	15, 18	sseldd 3125	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) ∈ ℂ)
20		f1of 5407	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(♯‘𝐴))⟶𝐴)
21	14, 20	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
22		fvco3 5532	. . . . . . . . 9 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑎 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑎) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑎)))
23	21, 22	sylan 281	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑎 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑎) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑎)))
24	12, 13, 14, 19, 23	fsum3 11261	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑎), 0)))‘(♯‘𝐴)))
25	11, 24	eqtr3d 2189	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑘 ∈ 𝐴 𝐵 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑎), 0)))‘(♯‘𝐴)))
26		nnuz 9453	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
27	13, 26	eleqtrdi 2247	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ (ℤ≥‘1))
28		elnnuz 9454	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ ↔ 𝑥 ∈ (ℤ≥‘1))
29	28	biimpri 132	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ℤ≥‘1) → 𝑥 ∈ ℕ)
30	29	adantl 275	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) → 𝑥 ∈ ℕ)
31	4	ad3antrrr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑆 ⊆ ℂ)
32	17	ad2antrr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑆)
33	21	ad2antrr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
34		fco 5328	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑆 ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶𝑆)
35	32, 33, 34	syl2anc 409	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶𝑆)
36		1zzd 9173	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 1 ∈ ℤ)
37	13	ad2antrr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℕ)
38	37	nnzd 9264	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℤ)
39		simpr 109	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
40	39	adantr 274	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ∈ ℕ)
41	40	nnzd 9264	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ∈ ℤ)
42	36, 38, 41	3jca 1162	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑥 ∈ ℤ))
43	40	nnge1d 8855	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 1 ≤ 𝑥)
44		simpr 109	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ≤ (♯‘𝐴))
45	43, 44	jca 304	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (1 ≤ 𝑥 ∧ 𝑥 ≤ (♯‘𝐴)))
46		elfz2 9897	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1...(♯‘𝐴)) ↔ ((1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑥 ∈ ℤ) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ (♯‘𝐴))))
47	42, 45, 46	sylanbrc 414	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ∈ (1...(♯‘𝐴)))
48	35, 47	ffvelrnd 5596	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥) ∈ 𝑆)
49	31, 48	sseldd 3125	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥) ∈ ℂ)
50		0cnd 7850	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ≤ (♯‘𝐴)) → 0 ∈ ℂ)
51	39	nnzd 9264	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℤ)
52	13	adantr 274	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) → (♯‘𝐴) ∈ ℕ)
53	52	nnzd 9264	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) → (♯‘𝐴) ∈ ℤ)
54		zdcle 9219	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑥 ≤ (♯‘𝐴))
55	51, 53, 54	syl2anc 409	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) → DECID 𝑥 ≤ (♯‘𝐴))
56	49, 50, 55	ifcldadc 3530	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) → if(𝑥 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥), 0) ∈ ℂ)
57	30, 56	syldan 280	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) → if(𝑥 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥), 0) ∈ ℂ)
58		breq1 3964	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑎 ≤ (♯‘𝐴) ↔ 𝑥 ≤ (♯‘𝐴)))
59		fveq2 5461	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑎) = (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥))
60	58, 59	ifbieq1d 3523	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → if(𝑎 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑎), 0) = if(𝑥 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥), 0))
61		eqid 2154	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑎), 0)) = (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑎), 0))
62	60, 61	fvmptg 5537	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ ∧ if(𝑥 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥), 0) ∈ ℂ) → ((𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑎), 0))‘𝑥) = if(𝑥 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥), 0))
63	30, 57, 62	syl2anc 409	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) → ((𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑎), 0))‘𝑥) = if(𝑥 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥), 0))
64	4	adantr 274	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑆 ⊆ ℂ)
65	17, 64	fssd 5325	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
66	65	ad2antrr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
67	21	ad2antrr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
68		fco 5328	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
69	66, 67, 68	syl2anc 409	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
70		1zzd 9173	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → 1 ∈ ℤ)
71	13	ad2antrr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℕ)
72	71	nnzd 9264	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℤ)
73		eluzelz 9427	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (ℤ≥‘1) → 𝑥 ∈ ℤ)
74	73	ad2antlr 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ∈ ℤ)
75	70, 72, 74	3jca 1162	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑥 ∈ ℤ))
76	29	nnge1d 8855	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (ℤ≥‘1) → 1 ≤ 𝑥)
77	76	ad2antlr 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → 1 ≤ 𝑥)
78		simpr 109	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ≤ (♯‘𝐴))
79	77, 78	jca 304	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (1 ≤ 𝑥 ∧ 𝑥 ≤ (♯‘𝐴)))
80	75, 79, 46	sylanbrc 414	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ∈ (1...(♯‘𝐴)))
81	69, 80	ffvelrnd 5596	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥) ∈ ℂ)
82		0cnd 7850	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) ∧ ¬ 𝑥 ≤ (♯‘𝐴)) → 0 ∈ ℂ)
83	30, 55	syldan 280	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) → DECID 𝑥 ≤ (♯‘𝐴))
84	81, 82, 83	ifcldadc 3530	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) → if(𝑥 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥), 0) ∈ ℂ)
85	63, 84	eqeltrd 2231	. . . . . . 7 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) → ((𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑎), 0))‘𝑥) ∈ ℂ)
86		elfzle2 9908	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1...(♯‘𝐴)) → 𝑥 ≤ (♯‘𝐴))
87	86	adantl 275	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝑥 ≤ (♯‘𝐴))
88	87	iftrued 3508	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → if(𝑥 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥), 0) = (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥))
89		elfznn 9934	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1...(♯‘𝐴)) → 𝑥 ∈ ℕ)
90	89	anim2i 340	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ))
91	90, 87, 48	syl2anc 409	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥) ∈ 𝑆)
92	88, 91	eqeltrd 2231	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → if(𝑥 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥), 0) ∈ 𝑆)
93	39, 56, 62	syl2anc 409	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) → ((𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑎), 0))‘𝑥) = if(𝑥 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥), 0))
94	93	eleq1d 2223	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) → (((𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑎), 0))‘𝑥) ∈ 𝑆 ↔ if(𝑥 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥), 0) ∈ 𝑆))
95	90, 94	syl 14	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑎), 0))‘𝑥) ∈ 𝑆 ↔ if(𝑥 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥), 0) ∈ 𝑆))
96	92, 95	mpbird 166	. . . . . . 7 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → ((𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑎), 0))‘𝑥) ∈ 𝑆)
97		fsumcllem.2	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
98	97	adantlr 469	. . . . . . 7 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
99		addcl 7836	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
100	99	adantl 275	. . . . . . 7 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ)
101	27, 85, 96, 98, 64, 100	seq3clss 10344	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑎), 0)))‘(♯‘𝐴)) ∈ 𝑆)
102	25, 101	eqeltrd 2231	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆)
103	102	expr 373	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆))
104	103	exlimdv 1796	. . 3 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆))
105	104	expimpd 361	. 2 ⊢ (𝜑 → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆))
106		fsumcllem.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
107		fz1f1o 11249	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
108	106, 107	syl 14	. 2 ⊢ (𝜑 → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
109	3, 105, 108	mpjaod 708	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698  DECID wdc 820   ∧ w3a 963   = wceq 1332  ∃wex 1469   ∈ wcel 2125   ≠ wne 2324  ∀wral 2432   ⊆ wss 3098  ∅c0 3390  ifcif 3501   class class class wbr 3961   ↦ cmpt 4021   ∘ ccom 4583  ⟶wf 5159  –1-1-onto→wf1o 5162  ‘cfv 5163  (class class class)co 5814  Fincfn 6674  ℂcc 7709  0cc0 7711  1c1 7712   + caddc 7714   ≤ cle 7892  ℕcn 8812  ℤcz 9146  ℤ_≥cuz 9418  ...cfz 9890  seqcseq 10322  ♯chash 10626  Σcsu 11227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-coll 4075  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-iinf 4541  ax-cnex 7802  ax-resscn 7803  ax-1cn 7804  ax-1re 7805  ax-icn 7806  ax-addcl 7807  ax-addrcl 7808  ax-mulcl 7809  ax-mulrcl 7810  ax-addcom 7811  ax-mulcom 7812  ax-addass 7813  ax-mulass 7814  ax-distr 7815  ax-i2m1 7816  ax-0lt1 7817  ax-1rid 7818  ax-0id 7819  ax-rnegex 7820  ax-precex 7821  ax-cnre 7822  ax-pre-ltirr 7823  ax-pre-ltwlin 7824  ax-pre-lttrn 7825  ax-pre-apti 7826  ax-pre-ltadd 7827  ax-pre-mulgt0 7828  ax-pre-mulext 7829  ax-arch 7830  ax-caucvg 7831

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-nel 2420  df-ral 2437  df-rex 2438  df-reu 2439  df-rmo 2440  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-if 3502  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-tr 4059  df-id 4248  df-po 4251  df-iso 4252  df-iord 4321  df-on 4323  df-ilim 4324  df-suc 4326  df-iom 4544  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-isom 5172  df-riota 5770  df-ov 5817  df-oprab 5818  df-mpo 5819  df-1st 6078  df-2nd 6079  df-recs 6242  df-irdg 6307  df-frec 6328  df-1o 6353  df-oadd 6357  df-er 6469  df-en 6675  df-dom 6676  df-fin 6677  df-pnf 7893  df-mnf 7894  df-xr 7895  df-ltxr 7896  df-le 7897  df-sub 8027  df-neg 8028  df-reap 8429  df-ap 8436  df-div 8525  df-inn 8813  df-2 8871  df-3 8872  df-4 8873  df-n0 9070  df-z 9147  df-uz 9419  df-q 9507  df-rp 9539  df-fz 9891  df-fzo 10020  df-seqfrec 10323  df-exp 10397  df-ihash 10627  df-cj 10719  df-re 10720  df-im 10721  df-rsqrt 10875  df-abs 10876  df-clim 11153  df-sumdc 11228

This theorem is referenced by:  fsumcllem  11273  fsumrpcl  11278