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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 (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 2345 . . 3 (𝜑 → ¬ 𝐴 = ∅)
32pm2.21d 609 . 2 (𝜑 → (𝐴 = ∅ → Σ𝑘𝐴 𝐵𝑆))
4 fsumcllem.1 . . . . . . . . . . . 12 (𝜑𝑆 ⊆ ℂ)
54adantr 274 . . . . . . . . . . 11 ((𝜑𝑘𝐴) → 𝑆 ⊆ ℂ)
6 fsumcllem.4 . . . . . . . . . . 11 ((𝜑𝑘𝐴) → 𝐵𝑆)
75, 6sseldd 3125 . . . . . . . . . 10 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
87ralrimiva 2527 . . . . . . . . 9 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
9 sumfct 11248 . . . . . . . . 9 (∀𝑘𝐴 𝐵 ∈ ℂ → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵)
108, 9syl 14 . . . . . . . 8 (𝜑 → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵)
1110adantr 274 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵)
12 fveq2 5461 . . . . . . . 8 (𝑚 = (𝑓𝑎) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘(𝑓𝑎)))
13 simprl 521 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ ℕ)
14 simprr 522 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)
154ad2antrr 480 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → 𝑆 ⊆ ℂ)
166fmpttd 5615 . . . . . . . . . . 11 (𝜑 → (𝑘𝐴𝐵):𝐴𝑆)
1716adantr 274 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴𝑆)
1817ffvelrnda 5595 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ 𝑆)
1915, 18sseldd 3125 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
20 f1of 5407 . . . . . . . . . 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 5532 . . . . . . . . 9 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑎 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎) = ((𝑘𝐴𝐵)‘(𝑓𝑎)))
2321, 22sylan 281 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑎 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎) = ((𝑘𝐴𝐵)‘(𝑓𝑎)))
2412, 13, 14, 19, 23fsum3 11261 . . . . . . 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 2189 . . . . . 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 9453 . . . . . . . 8 ℕ = (ℤ‘1)
2713, 26eleqtrdi 2247 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ (ℤ‘1))
28 elnnuz 9454 . . . . . . . . . . 11 (𝑥 ∈ ℕ ↔ 𝑥 ∈ (ℤ‘1))
2928biimpri 132 . . . . . . . . . 10 (𝑥 ∈ (ℤ‘1) → 𝑥 ∈ ℕ)
3029adantl 275 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) → 𝑥 ∈ ℕ)
314ad3antrrr 484 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑆 ⊆ ℂ)
3217ad2antrr 480 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (𝑘𝐴𝐵):𝐴𝑆)
3321ad2antrr 480 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
34 fco 5328 . . . . . . . . . . . . . 14 (((𝑘𝐴𝐵):𝐴𝑆𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶𝑆)
3532, 33, 34syl2anc 409 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶𝑆)
36 1zzd 9173 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 1 ∈ ℤ)
3713ad2antrr 480 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℕ)
3837nnzd 9264 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℤ)
39 simpr 109 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ)
4039adantr 274 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ∈ ℕ)
4140nnzd 9264 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ∈ ℤ)
4236, 38, 413jca 1162 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑥 ∈ ℤ))
4340nnge1d 8855 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 1 ≤ 𝑥)
44 simpr 109 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ≤ (♯‘𝐴))
4543, 44jca 304 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (1 ≤ 𝑥𝑥 ≤ (♯‘𝐴)))
46 elfz2 9897 . . . . . . . . . . . . . 14 (𝑥 ∈ (1...(♯‘𝐴)) ↔ ((1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑥 ∈ ℤ) ∧ (1 ≤ 𝑥𝑥 ≤ (♯‘𝐴))))
4742, 45, 46sylanbrc 414 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ∈ (1...(♯‘𝐴)))
4835, 47ffvelrnd 5596 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) ∈ 𝑆)
4931, 48sseldd 3125 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ 𝑥 ≤ (♯‘𝐴)) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) ∈ ℂ)
50 0cnd 7850 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) ∧ ¬ 𝑥 ≤ (♯‘𝐴)) → 0 ∈ ℂ)
5139nnzd 9264 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℤ)
5213adantr 274 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) → (♯‘𝐴) ∈ ℕ)
5352nnzd 9264 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) → (♯‘𝐴) ∈ ℤ)
54 zdcle 9219 . . . . . . . . . . . 12 ((𝑥 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑥 ≤ (♯‘𝐴))
5551, 53, 54syl2anc 409 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) → DECID 𝑥 ≤ (♯‘𝐴))
5649, 50, 55ifcldadc 3530 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ ℕ) → if(𝑥 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥), 0) ∈ ℂ)
5730, 56syldan 280 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) → if(𝑥 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥), 0) ∈ ℂ)
58 breq1 3964 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑎 ≤ (♯‘𝐴) ↔ 𝑥 ≤ (♯‘𝐴)))
59 fveq2 5461 . . . . . . . . . . 11 (𝑎 = 𝑥 → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎) = (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥))
6058, 59ifbieq1d 3523 . . . . . . . . . 10 (𝑎 = 𝑥 → if(𝑎 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎), 0) = if(𝑥 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥), 0))
61 eqid 2154 . . . . . . . . . 10 (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎), 0)) = (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎), 0))
6260, 61fvmptg 5537 . . . . . . . . 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 409 . . . . . . . 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 274 . . . . . . . . . . . . 13 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑆 ⊆ ℂ)
6517, 64fssd 5325 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
6665ad2antrr 480 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
6721ad2antrr 480 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
68 fco 5328 . . . . . . . . . . 11 (((𝑘𝐴𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
6966, 67, 68syl2anc 409 . . . . . . . . . 10 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
70 1zzd 9173 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → 1 ∈ ℤ)
7113ad2antrr 480 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℕ)
7271nnzd 9264 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℤ)
73 eluzelz 9427 . . . . . . . . . . . . 13 (𝑥 ∈ (ℤ‘1) → 𝑥 ∈ ℤ)
7473ad2antlr 481 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ∈ ℤ)
7570, 72, 743jca 1162 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑥 ∈ ℤ))
7629nnge1d 8855 . . . . . . . . . . . . 13 (𝑥 ∈ (ℤ‘1) → 1 ≤ 𝑥)
7776ad2antlr 481 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → 1 ≤ 𝑥)
78 simpr 109 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ≤ (♯‘𝐴))
7977, 78jca 304 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (1 ≤ 𝑥𝑥 ≤ (♯‘𝐴)))
8075, 79, 46sylanbrc 414 . . . . . . . . . 10 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → 𝑥 ∈ (1...(♯‘𝐴)))
8169, 80ffvelrnd 5596 . . . . . . . . 9 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ 𝑥 ≤ (♯‘𝐴)) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) ∈ ℂ)
82 0cnd 7850 . . . . . . . . 9 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) ∧ ¬ 𝑥 ≤ (♯‘𝐴)) → 0 ∈ ℂ)
8330, 55syldan 280 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) → DECID 𝑥 ≤ (♯‘𝐴))
8481, 82, 83ifcldadc 3530 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (ℤ‘1)) → if(𝑥 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥), 0) ∈ ℂ)
8563, 84eqeltrd 2231 . . . . . . 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 9908 . . . . . . . . . . 11 (𝑥 ∈ (1...(♯‘𝐴)) → 𝑥 ≤ (♯‘𝐴))
8786adantl 275 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝑥 ≤ (♯‘𝐴))
8887iftrued 3508 . . . . . . . . 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 9934 . . . . . . . . . . 11 (𝑥 ∈ (1...(♯‘𝐴)) → 𝑥 ∈ ℕ)
9089anim2i 340 . . . . . . . . . 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 409 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) ∈ 𝑆)
9288, 91eqeltrd 2231 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → if(𝑥 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥), 0) ∈ 𝑆)
9339, 56, 62syl2anc 409 . . . . . . . . . 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 2223 . . . . . . . . 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 166 . . . . . . 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 469 . . . . . . 7 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
99 addcl 7836 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
10099adantl 275 . . . . . . 7 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ)
10127, 85, 96, 98, 64, 100seq3clss 10344 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑎), 0)))‘(♯‘𝐴)) ∈ 𝑆)
10225, 101eqeltrd 2231 . . . . 5 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑘𝐴 𝐵𝑆)
103102expr 373 . . . 4 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → Σ𝑘𝐴 𝐵𝑆))
104103exlimdv 1796 . . 3 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → Σ𝑘𝐴 𝐵𝑆))
105104expimpd 361 . 2 (𝜑 → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → Σ𝑘𝐴 𝐵𝑆))
106 fsumcllem.3 . . 3 (𝜑𝐴 ∈ Fin)
107 fz1f1o 11249 . . 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 708 1 (𝜑 → Σ𝑘𝐴 𝐵𝑆)
 Colors of variables: wff set class 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
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