Theorem fsumcl2lem 11167

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 2329	. . 3 ⊢ (𝜑 → ¬ 𝐴 = ∅)
3	2	pm2.21d 608	. 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 3098	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
8	7	ralrimiva 2505	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
9		sumfct 11143	. . . . . . . . 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 5421	. . . . . . . 8 ⊢ (𝑚 = (𝑓‘𝑎) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑎)))
13		simprl 520	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ ℕ)
14		simprr 521	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
15	4	ad2antrr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → 𝑆 ⊆ ℂ)
16	6	fmpttd 5575	. . . . . . . . . . 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 5555	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) ∈ 𝑆)
19	15, 18	sseldd 3098	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) ∈ ℂ)
20		f1of 5367	. . . . . . . . . 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 5492	. . . . . . . . 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 11156	. . . . . . 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 2174	. . . . . 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 9361	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
27	13, 26	eleqtrdi 2232	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ (ℤ≥‘1))
28		elnnuz 9362	. . . . . . . . . . 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 483	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑆 ⊆ ℂ)
32	17	ad2antrr 479	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑆)
33	21	ad2antrr 479	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
34		fco 5288	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝑆 ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶𝑆)
35	32, 33, 34	syl2anc 408	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶𝑆)
36		1zzd 9081	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 1 ∈ ℤ)
37	13	ad2antrr 479	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℕ)
38	37	nnzd 9172	. . . . . . . . . . . . . . 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 9172	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ∈ ℤ)
42	36, 38, 41	3jca 1161	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑥 ∈ ℤ))
43	40	nnge1d 8763	. . . . . . . . . . . . . . 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 9797	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1...(♯‘𝐴)) ↔ ((1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑥 ∈ ℤ) ∧ (1 ≤ 𝑥 ∧ 𝑥 ≤ (♯‘𝐴))))
47	42, 45, 46	sylanbrc 413	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ∈ (1...(♯‘𝐴)))
48	35, 47	ffvelrnd 5556	. . . . . . . . . . . 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 3098	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥) ∈ ℂ)
50		0cnd 7759	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ≤ (♯‘𝐴)) → 0 ∈ ℂ)
51	39	nnzd 9172	. . . . . . . . . . . 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 9172	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) → (♯‘𝐴) ∈ ℤ)
54		zdcle 9127	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑥 ≤ (♯‘𝐴))
55	51, 53, 54	syl2anc 408	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ ℕ) → DECID 𝑥 ≤ (♯‘𝐴))
56	49, 50, 55	ifcldadc 3501	. . . . . . . . . 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 3932	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑎 ≤ (♯‘𝐴) ↔ 𝑥 ≤ (♯‘𝐴)))
59		fveq2 5421	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑎) = (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥))
60	58, 59	ifbieq1d 3494	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → if(𝑎 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑎), 0) = if(𝑥 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥), 0))
61		eqid 2139	. . . . . . . . . 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 5497	. . . . . . . . 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 408	. . . . . . . 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 5285	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
66	65	ad2antrr 479	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
67	21	ad2antrr 479	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
68		fco 5288	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
69	66, 67, 68	syl2anc 408	. . . . . . . . . 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 9081	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → 1 ∈ ℤ)
71	13	ad2antrr 479	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℕ)
72	71	nnzd 9172	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℤ)
73		eluzelz 9335	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (ℤ≥‘1) → 𝑥 ∈ ℤ)
74	73	ad2antlr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ∈ ℤ)
75	70, 72, 74	3jca 1161	. . . . . . . . . . 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 8763	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (ℤ≥‘1) → 1 ≤ 𝑥)
77	76	ad2antlr 480	. . . . . . . . . . . 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 413	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ∈ (1...(♯‘𝐴)))
81	69, 80	ffvelrnd 5556	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (ℤ≥‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥) ∈ ℂ)
82		0cnd 7759	. . . . . . . . 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 3501	. . . . . . . 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 2216	. . . . . . 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 9808	. . . . . . . . . . 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 3481	. . . . . . . . 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 9834	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1...(♯‘𝐴)) → 𝑥 ∈ ℕ)
90	89	anim2i 339	. . . . . . . . . 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 408	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥) ∈ 𝑆)
92	88, 91	eqeltrd 2216	. . . . . . . 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 408	. . . . . . . . . 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 2208	. . . . . . . . 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 468	. . . . . . 7 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
99		addcl 7745	. . . . . . . 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 10240	. . . . . 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 2216	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆)
103	102	expr 372	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆))
104	103	exlimdv 1791	. . 3 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆))
105	104	expimpd 360	. 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 11144	. . 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 707	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697  DECID wdc 819   ∧ w3a 962   = wceq 1331  ∃wex 1468   ∈ wcel 1480   ≠ wne 2308  ∀wral 2416   ⊆ wss 3071  ∅c0 3363  ifcif 3474   class class class wbr 3929   ↦ cmpt 3989   ∘ ccom 4543  ⟶wf 5119  –1-1-onto→wf1o 5122  ‘cfv 5123  (class class class)co 5774  Fincfn 6634  ℂcc 7618  0cc0 7620  1c1 7621   + caddc 7623   ≤ cle 7801  ℕcn 8720  ℤcz 9054  ℤ_≥cuz 9326  ...cfz 9790  seqcseq 10218  ♯chash 10521  Σcsu 11122

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-fz 9791  df-fzo 9920  df-seqfrec 10219  df-exp 10293  df-ihash 10522  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-clim 11048  df-sumdc 11123

This theorem is referenced by:  fsumcllem  11168  fsumrpcl  11173