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Theorem fsumcl2lem 11441
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 2381 . . 3 (𝜑 → ¬ 𝐴 = ∅)
32pm2.21d 620 . 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 3171 . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
87ralrimiva 2563 . . . . . . . . 9 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
9 sumfct 11417 . . . . . . . . 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 5534 . . . . . . . 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 5692 . . . . . . . . . . 11 (𝜑 → (𝑘𝐴𝐵):𝐴𝑆)
1716adantr 276 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴𝑆)
1817ffvelcdmda 5672 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ 𝑆)
1915, 18sseldd 3171 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
20 f1of 5480 . . . . . . . . . 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 5608 . . . . . . . . 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 11430 . . . . . . 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 2224 . . . . . 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 9595 . . . . . . . 8 ℕ = (ℤ‘1)
2713, 26eleqtrdi 2282 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ (ℤ‘1))
28 elnnuz 9596 . . . . . . . . . . 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 5400 . . . . . . . . . . . . . 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 9311 . . . . . . . . . . . . . . 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 9405 . . . . . . . . . . . . . . 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 9405 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ∈ ℤ)
4236, 38, 413jca 1179 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑥 ∈ ℤ))
4340nnge1d 8993 . . . . . . . . . . . . . . 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 10047 . . . . . . . . . . . . . 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 5673 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) ∈ 𝑆)
4931, 48sseldd 3171 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) ∈ ℂ)
50 0cnd 7981 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ≤ (♯‘𝐴)) → 0 ∈ ℂ)
5139nnzd 9405 . . . . . . . . . . . 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 9405 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) → (♯‘𝐴) ∈ ℤ)
54 zdcle 9360 . . . . . . . . . . . 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 3578 . . . . . . . . . 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 4021 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑎 ≤ (♯‘𝐴) ↔ 𝑥 ≤ (♯‘𝐴)))
59 fveq2 5534 . . . . . . . . . . 11 (𝑎 = 𝑥 → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎) = (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥))
6058, 59ifbieq1d 3571 . . . . . . . . . 10 (𝑎 = 𝑥 → if(𝑎 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎), 0) = if(𝑥 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥), 0))
61 eqid 2189 . . . . . . . . . 10 (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎), 0)) = (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎), 0))
6260, 61fvmptg 5613 . . . . . . . . 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 5397 . . . . . . . . . . . 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 5400 . . . . . . . . . . 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 9311 . . . . . . . . . . . 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 9405 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℤ)
73 eluzelz 9568 . . . . . . . . . . . . 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 1179 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑥 ∈ ℤ))
7629nnge1d 8993 . . . . . . . . . . . . 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 5673 . . . . . . . . 9 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) ∈ ℂ)
82 0cnd 7981 . . . . . . . . 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 3578 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) → if(𝑥 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥), 0) ∈ ℂ)
8563, 84eqeltrd 2266 . . . . . . 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 10060 . . . . . . . . . . 11 (𝑥 ∈ (1...(♯‘𝐴)) → 𝑥 ≤ (♯‘𝐴))
8786adantl 277 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝑥 ≤ (♯‘𝐴))
8887iftrued 3556 . . . . . . . . 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 10086 . . . . . . . . . . 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 2266 . . . . . . . 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 2258 . . . . . . . . 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 7967 . . . . . . . 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 10500 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎), 0)))‘(♯‘𝐴)) ∈ 𝑆)
10225, 101eqeltrd 2266 . . . . 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 1830 . . 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 11418 . . 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 719 1 (𝜑 → Σ𝑘𝐴 𝐵𝑆)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 709  DECID wdc 835  w3a 980   = wceq 1364  wex 1503  wcel 2160  wne 2360  wral 2468  wss 3144  c0 3437  ifcif 3549   class class class wbr 4018  cmpt 4079  ccom 4648  wf 5231  1-1-ontowf1o 5234  cfv 5235  (class class class)co 5897  Fincfn 6767  cc 7840  0cc0 7842  1c1 7843   + caddc 7845  cle 8024  cn 8950  cz 9284  cuz 9559  ...cfz 10040  seqcseq 10478  chash 10790  Σcsu 11396
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960  ax-arch 7961  ax-caucvg 7962
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-frec 6417  df-1o 6442  df-oadd 6446  df-er 6560  df-en 6768  df-dom 6769  df-fin 6770  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661  df-inn 8951  df-2 9009  df-3 9010  df-4 9011  df-n0 9208  df-z 9285  df-uz 9560  df-q 9652  df-rp 9686  df-fz 10041  df-fzo 10175  df-seqfrec 10479  df-exp 10554  df-ihash 10791  df-cj 10886  df-re 10887  df-im 10888  df-rsqrt 11042  df-abs 11043  df-clim 11322  df-sumdc 11397
This theorem is referenced by:  fsumcllem  11442  fsumrpcl  11447
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