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Theorem fsumcl2lem 11390
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 (𝜑𝑆 ⊆ ℂ)
fsumcllem.2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
fsumcllem.3 (𝜑𝐴 ∈ Fin)
fsumcllem.4 ((𝜑𝑘𝐴) → 𝐵𝑆)
fsumcl2lem.5 (𝜑𝐴 ≠ ∅)
Assertion
Ref Expression
fsumcl2lem (𝜑 → Σ𝑘𝐴 𝐵𝑆)
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.
StepHypRef Expression
1 fsumcl2lem.5 . . . 4 (𝜑𝐴 ≠ ∅)
21neneqd 2368 . . 3 (𝜑 → ¬ 𝐴 = ∅)
32pm2.21d 619 . 2 (𝜑 → (𝐴 = ∅ → Σ𝑘𝐴 𝐵𝑆))
4 fsumcllem.1 . . . . . . . . . . . 12 (𝜑𝑆 ⊆ ℂ)
54adantr 276 . . . . . . . . . . 11 ((𝜑𝑘𝐴) → 𝑆 ⊆ ℂ)
6 fsumcllem.4 . . . . . . . . . . 11 ((𝜑𝑘𝐴) → 𝐵𝑆)
75, 6sseldd 3156 . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
87ralrimiva 2550 . . . . . . . . 9 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
9 sumfct 11366 . . . . . . . . 9 (∀𝑘𝐴 𝐵 ∈ ℂ → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵)
108, 9syl 14 . . . . . . . 8 (𝜑 → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵)
1110adantr 276 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵)
12 fveq2 5511 . . . . . . . 8 (𝑚 = (𝑓𝑎) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘(𝑓𝑎)))
13 simprl 529 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ ℕ)
14 simprr 531 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)
154ad2antrr 488 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → 𝑆 ⊆ ℂ)
166fmpttd 5667 . . . . . . . . . . 11 (𝜑 → (𝑘𝐴𝐵):𝐴𝑆)
1716adantr 276 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴𝑆)
1817ffvelcdmda 5647 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ 𝑆)
1915, 18sseldd 3156 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
20 f1of 5457 . . . . . . . . . 10 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))⟶𝐴)
2114, 20syl 14 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
22 fvco3 5583 . . . . . . . . 9 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑎 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎) = ((𝑘𝐴𝐵)‘(𝑓𝑎)))
2321, 22sylan 283 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑎 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎) = ((𝑘𝐴𝐵)‘(𝑓𝑎)))
2412, 13, 14, 19, 23fsum3 11379 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎), 0)))‘(♯‘𝐴)))
2511, 24eqtr3d 2212 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑘𝐴 𝐵 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎), 0)))‘(♯‘𝐴)))
26 nnuz 9552 . . . . . . . 8 ℕ = (ℤ‘1)
2713, 26eleqtrdi 2270 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ (ℤ‘1))
28 elnnuz 9553 . . . . . . . . . . 11 (𝑥 ∈ ℕ ↔ 𝑥 ∈ (ℤ‘1))
2928biimpri 133 . . . . . . . . . 10 (𝑥 ∈ (ℤ‘1) → 𝑥 ∈ ℕ)
3029adantl 277 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) → 𝑥 ∈ ℕ)
314ad3antrrr 492 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑆 ⊆ ℂ)
3217ad2antrr 488 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (𝑘𝐴𝐵):𝐴𝑆)
3321ad2antrr 488 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
34 fco 5377 . . . . . . . . . . . . . 14 (((𝑘𝐴𝐵):𝐴𝑆𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶𝑆)
3532, 33, 34syl2anc 411 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶𝑆)
36 1zzd 9269 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 1 ∈ ℤ)
3713ad2antrr 488 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℕ)
3837nnzd 9363 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℤ)
39 simpr 110 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
4039adantr 276 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ∈ ℕ)
4140nnzd 9363 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ∈ ℤ)
4236, 38, 413jca 1177 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑥 ∈ ℤ))
4340nnge1d 8951 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 1 ≤ 𝑥)
44 simpr 110 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ≤ (♯‘𝐴))
4543, 44jca 306 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (1 ≤ 𝑥𝑥 ≤ (♯‘𝐴)))
46 elfz2 10002 . . . . . . . . . . . . . 14 (𝑥 ∈ (1...(♯‘𝐴)) ↔ ((1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑥 ∈ ℤ) ∧ (1 ≤ 𝑥𝑥 ≤ (♯‘𝐴))))
4742, 45, 46sylanbrc 417 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ∈ (1...(♯‘𝐴)))
4835, 47ffvelcdmd 5648 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) ∈ 𝑆)
4931, 48sseldd 3156 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) ∈ ℂ)
50 0cnd 7941 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ≤ (♯‘𝐴)) → 0 ∈ ℂ)
5139nnzd 9363 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℤ)
5213adantr 276 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) → (♯‘𝐴) ∈ ℕ)
5352nnzd 9363 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) → (♯‘𝐴) ∈ ℤ)
54 zdcle 9318 . . . . . . . . . . . 12 ((𝑥 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑥 ≤ (♯‘𝐴))
5551, 53, 54syl2anc 411 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) → DECID 𝑥 ≤ (♯‘𝐴))
5649, 50, 55ifcldadc 3563 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) → if(𝑥 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥), 0) ∈ ℂ)
5730, 56syldan 282 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) → if(𝑥 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥), 0) ∈ ℂ)
58 breq1 4003 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑎 ≤ (♯‘𝐴) ↔ 𝑥 ≤ (♯‘𝐴)))
59 fveq2 5511 . . . . . . . . . . 11 (𝑎 = 𝑥 → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎) = (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥))
6058, 59ifbieq1d 3556 . . . . . . . . . 10 (𝑎 = 𝑥 → if(𝑎 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎), 0) = if(𝑥 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥), 0))
61 eqid 2177 . . . . . . . . . 10 (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎), 0)) = (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎), 0))
6260, 61fvmptg 5588 . . . . . . . . 9 ((𝑥 ∈ ℕ ∧ if(𝑥 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥), 0) ∈ ℂ) → ((𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎), 0))‘𝑥) = if(𝑥 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥), 0))
6330, 57, 62syl2anc 411 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) → ((𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎), 0))‘𝑥) = if(𝑥 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥), 0))
644adantr 276 . . . . . . . . . . . . 13 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑆 ⊆ ℂ)
6517, 64fssd 5374 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
6665ad2antrr 488 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
6721ad2antrr 488 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
68 fco 5377 . . . . . . . . . . 11 (((𝑘𝐴𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
6966, 67, 68syl2anc 411 . . . . . . . . . 10 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
70 1zzd 9269 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → 1 ∈ ℤ)
7113ad2antrr 488 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℕ)
7271nnzd 9363 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℤ)
73 eluzelz 9526 . . . . . . . . . . . . 13 (𝑥 ∈ (ℤ‘1) → 𝑥 ∈ ℤ)
7473ad2antlr 489 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ∈ ℤ)
7570, 72, 743jca 1177 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑥 ∈ ℤ))
7629nnge1d 8951 . . . . . . . . . . . . 13 (𝑥 ∈ (ℤ‘1) → 1 ≤ 𝑥)
7776ad2antlr 489 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → 1 ≤ 𝑥)
78 simpr 110 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ≤ (♯‘𝐴))
7977, 78jca 306 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (1 ≤ 𝑥𝑥 ≤ (♯‘𝐴)))
8075, 79, 46sylanbrc 417 . . . . . . . . . 10 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ∈ (1...(♯‘𝐴)))
8169, 80ffvelcdmd 5648 . . . . . . . . 9 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) ∈ ℂ)
82 0cnd 7941 . . . . . . . . 9 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ ¬ 𝑥 ≤ (♯‘𝐴)) → 0 ∈ ℂ)
8330, 55syldan 282 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) → DECID 𝑥 ≤ (♯‘𝐴))
8481, 82, 83ifcldadc 3563 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) → if(𝑥 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥), 0) ∈ ℂ)
8563, 84eqeltrd 2254 . . . . . . 7 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) → ((𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎), 0))‘𝑥) ∈ ℂ)
86 elfzle2 10014 . . . . . . . . . . 11 (𝑥 ∈ (1...(♯‘𝐴)) → 𝑥 ≤ (♯‘𝐴))
8786adantl 277 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝑥 ≤ (♯‘𝐴))
8887iftrued 3541 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → if(𝑥 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥), 0) = (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥))
89 elfznn 10040 . . . . . . . . . . 11 (𝑥 ∈ (1...(♯‘𝐴)) → 𝑥 ∈ ℕ)
9089anim2i 342 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ))
9190, 87, 48syl2anc 411 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) ∈ 𝑆)
9288, 91eqeltrd 2254 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → if(𝑥 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥), 0) ∈ 𝑆)
9339, 56, 62syl2anc 411 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) → ((𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎), 0))‘𝑥) = if(𝑥 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥), 0))
9493eleq1d 2246 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) → (((𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎), 0))‘𝑥) ∈ 𝑆 ↔ if(𝑥 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥), 0) ∈ 𝑆))
9590, 94syl 14 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎), 0))‘𝑥) ∈ 𝑆 ↔ if(𝑥 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥), 0) ∈ 𝑆))
9692, 95mpbird 167 . . . . . . 7 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → ((𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎), 0))‘𝑥) ∈ 𝑆)
97 fsumcllem.2 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
9897adantlr 477 . . . . . . 7 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
99 addcl 7927 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
10099adantl 277 . . . . . . 7 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ)
10127, 85, 96, 98, 64, 100seq3clss 10453 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎), 0)))‘(♯‘𝐴)) ∈ 𝑆)
10225, 101eqeltrd 2254 . . . . 5 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑘𝐴 𝐵𝑆)
103102expr 375 . . . 4 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → Σ𝑘𝐴 𝐵𝑆))
104103exlimdv 1819 . . 3 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → Σ𝑘𝐴 𝐵𝑆))
105104expimpd 363 . 2 (𝜑 → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → Σ𝑘𝐴 𝐵𝑆))
106 fsumcllem.3 . . 3 (𝜑𝐴 ∈ Fin)
107 fz1f1o 11367 . . 3 (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
108106, 107syl 14 . 2 (𝜑 → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
1093, 105, 108mpjaod 718 1 (𝜑 → Σ𝑘𝐴 𝐵𝑆)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834  w3a 978   = wceq 1353  wex 1492  wcel 2148  wne 2347  wral 2455  wss 3129  c0 3422  ifcif 3534   class class class wbr 4000  cmpt 4061  ccom 4627  wf 5208  1-1-ontowf1o 5211  cfv 5212  (class class class)co 5869  Fincfn 6734  cc 7800  0cc0 7802  1c1 7803   + caddc 7805  cle 7983  cn 8908  cz 9242  cuz 9517  ...cfz 9995  seqcseq 10431  chash 10739  Σcsu 11345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-seqfrec 10432  df-exp 10506  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-sumdc 11346
This theorem is referenced by:  fsumcllem  11391  fsumrpcl  11396
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