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Theorem fsumss 15074
Description: Change the index set to a subset in a finite sum. (Contributed by Mario Carneiro, 21-Apr-2014.)
Hypotheses
Ref Expression
sumss.1 (𝜑𝐴𝐵)
sumss.2 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
sumss.3 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 0)
fsumss.4 (𝜑𝐵 ∈ Fin)
Assertion
Ref Expression
fsumss (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)
Distinct variable groups:   𝐴,𝑘   𝐵,𝑘   𝜑,𝑘
Allowed substitution hint:   𝐶(𝑘)

Proof of Theorem fsumss
Dummy variables 𝑓 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sumss.1 . . . . 5 (𝜑𝐴𝐵)
21adantr 481 . . . 4 ((𝜑𝐵 = ∅) → 𝐴𝐵)
3 sumss.2 . . . . 5 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
43adantlr 711 . . . 4 (((𝜑𝐵 = ∅) ∧ 𝑘𝐴) → 𝐶 ∈ ℂ)
5 sumss.3 . . . . 5 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 0)
65adantlr 711 . . . 4 (((𝜑𝐵 = ∅) ∧ 𝑘 ∈ (𝐵𝐴)) → 𝐶 = 0)
7 simpr 485 . . . . 5 ((𝜑𝐵 = ∅) → 𝐵 = ∅)
8 0ss 4353 . . . . 5 ∅ ⊆ (ℤ‘0)
97, 8eqsstrdi 4024 . . . 4 ((𝜑𝐵 = ∅) → 𝐵 ⊆ (ℤ‘0))
102, 4, 6, 9sumss 15073 . . 3 ((𝜑𝐵 = ∅) → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)
1110ex 413 . 2 (𝜑 → (𝐵 = ∅ → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶))
12 cnvimass 5946 . . . . . . . . 9 (𝑓𝐴) ⊆ dom 𝑓
13 simprr 769 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)
14 f1of 6611 . . . . . . . . . 10 (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑓:(1...(♯‘𝐵))⟶𝐵)
1513, 14syl 17 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝑓:(1...(♯‘𝐵))⟶𝐵)
1612, 15fssdm 6526 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑓𝐴) ⊆ (1...(♯‘𝐵)))
1715ffnd 6511 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝑓 Fn (1...(♯‘𝐵)))
18 elpreima 6823 . . . . . . . . . . . 12 (𝑓 Fn (1...(♯‘𝐵)) → (𝑛 ∈ (𝑓𝐴) ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴)))
1917, 18syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑛 ∈ (𝑓𝐴) ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴)))
2015ffvelrnda 6846 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (1...(♯‘𝐵))) → (𝑓𝑛) ∈ 𝐵)
2120ex 413 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑛 ∈ (1...(♯‘𝐵)) → (𝑓𝑛) ∈ 𝐵))
2221adantrd 492 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ((𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴) → (𝑓𝑛) ∈ 𝐵))
2319, 22sylbid 241 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑛 ∈ (𝑓𝐴) → (𝑓𝑛) ∈ 𝐵))
2423imp 407 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (𝑓𝐴)) → (𝑓𝑛) ∈ 𝐵)
253ex 413 . . . . . . . . . . . . . 14 (𝜑 → (𝑘𝐴𝐶 ∈ ℂ))
2625adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑘𝐵) → (𝑘𝐴𝐶 ∈ ℂ))
27 eldif 3949 . . . . . . . . . . . . . . 15 (𝑘 ∈ (𝐵𝐴) ↔ (𝑘𝐵 ∧ ¬ 𝑘𝐴))
28 0cn 10625 . . . . . . . . . . . . . . . 16 0 ∈ ℂ
295, 28syl6eqel 2925 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 ∈ ℂ)
3027, 29sylan2br 594 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘𝐵 ∧ ¬ 𝑘𝐴)) → 𝐶 ∈ ℂ)
3130expr 457 . . . . . . . . . . . . 13 ((𝜑𝑘𝐵) → (¬ 𝑘𝐴𝐶 ∈ ℂ))
3226, 31pm2.61d 180 . . . . . . . . . . . 12 ((𝜑𝑘𝐵) → 𝐶 ∈ ℂ)
3332fmpttd 6874 . . . . . . . . . . 11 (𝜑 → (𝑘𝐵𝐶):𝐵⟶ℂ)
3433adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑘𝐵𝐶):𝐵⟶ℂ)
3534ffvelrnda 6846 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ (𝑓𝑛) ∈ 𝐵) → ((𝑘𝐵𝐶)‘(𝑓𝑛)) ∈ ℂ)
3624, 35syldan 591 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (𝑓𝐴)) → ((𝑘𝐵𝐶)‘(𝑓𝑛)) ∈ ℂ)
37 eldifi 4106 . . . . . . . . . . . 12 (𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴)) → 𝑛 ∈ (1...(♯‘𝐵)))
3837, 20sylan2 592 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → (𝑓𝑛) ∈ 𝐵)
39 eldifn 4107 . . . . . . . . . . . . 13 (𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴)) → ¬ 𝑛 ∈ (𝑓𝐴))
4039adantl 482 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ¬ 𝑛 ∈ (𝑓𝐴))
4137adantl 482 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → 𝑛 ∈ (1...(♯‘𝐵)))
4219adantr 481 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → (𝑛 ∈ (𝑓𝐴) ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴)))
4341, 42mpbirand 703 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → (𝑛 ∈ (𝑓𝐴) ↔ (𝑓𝑛) ∈ 𝐴))
4440, 43mtbid 325 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ¬ (𝑓𝑛) ∈ 𝐴)
4538, 44eldifd 3950 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → (𝑓𝑛) ∈ (𝐵𝐴))
46 difss 4111 . . . . . . . . . . . . 13 (𝐵𝐴) ⊆ 𝐵
47 resmpt 5903 . . . . . . . . . . . . 13 ((𝐵𝐴) ⊆ 𝐵 → ((𝑘𝐵𝐶) ↾ (𝐵𝐴)) = (𝑘 ∈ (𝐵𝐴) ↦ 𝐶))
4846, 47ax-mp 5 . . . . . . . . . . . 12 ((𝑘𝐵𝐶) ↾ (𝐵𝐴)) = (𝑘 ∈ (𝐵𝐴) ↦ 𝐶)
4948fveq1i 6667 . . . . . . . . . . 11 (((𝑘𝐵𝐶) ↾ (𝐵𝐴))‘(𝑓𝑛)) = ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛))
50 fvres 6685 . . . . . . . . . . 11 ((𝑓𝑛) ∈ (𝐵𝐴) → (((𝑘𝐵𝐶) ↾ (𝐵𝐴))‘(𝑓𝑛)) = ((𝑘𝐵𝐶)‘(𝑓𝑛)))
5149, 50syl5eqr 2874 . . . . . . . . . 10 ((𝑓𝑛) ∈ (𝐵𝐴) → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) = ((𝑘𝐵𝐶)‘(𝑓𝑛)))
5245, 51syl 17 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) = ((𝑘𝐵𝐶)‘(𝑓𝑛)))
53 c0ex 10627 . . . . . . . . . . . . . . 15 0 ∈ V
5453elsn2 4600 . . . . . . . . . . . . . 14 (𝐶 ∈ {0} ↔ 𝐶 = 0)
555, 54sylibr 235 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 ∈ {0})
5655fmpttd 6874 . . . . . . . . . . . 12 (𝜑 → (𝑘 ∈ (𝐵𝐴) ↦ 𝐶):(𝐵𝐴)⟶{0})
5756ad2antrr 722 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → (𝑘 ∈ (𝐵𝐴) ↦ 𝐶):(𝐵𝐴)⟶{0})
5857, 45ffvelrnd 6847 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) ∈ {0})
59 elsni 4580 . . . . . . . . . 10 (((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) ∈ {0} → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) = 0)
6058, 59syl 17 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) = 0)
6152, 60eqtr3d 2862 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ((𝑘𝐵𝐶)‘(𝑓𝑛)) = 0)
62 fzssuz 12941 . . . . . . . . 9 (1...(♯‘𝐵)) ⊆ (ℤ‘1)
6362a1i 11 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (1...(♯‘𝐵)) ⊆ (ℤ‘1))
6416, 36, 61, 63sumss 15073 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → Σ𝑛 ∈ (𝑓𝐴)((𝑘𝐵𝐶)‘(𝑓𝑛)) = Σ𝑛 ∈ (1...(♯‘𝐵))((𝑘𝐵𝐶)‘(𝑓𝑛)))
651ad2antrr 722 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → 𝐴𝐵)
6665resmptd 5906 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → ((𝑘𝐵𝐶) ↾ 𝐴) = (𝑘𝐴𝐶))
6766fveq1d 6668 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → (((𝑘𝐵𝐶) ↾ 𝐴)‘𝑚) = ((𝑘𝐴𝐶)‘𝑚))
68 fvres 6685 . . . . . . . . . . 11 (𝑚𝐴 → (((𝑘𝐵𝐶) ↾ 𝐴)‘𝑚) = ((𝑘𝐵𝐶)‘𝑚))
6968adantl 482 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → (((𝑘𝐵𝐶) ↾ 𝐴)‘𝑚) = ((𝑘𝐵𝐶)‘𝑚))
7067, 69eqtr3d 2862 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐶)‘𝑚) = ((𝑘𝐵𝐶)‘𝑚))
7170sumeq2dv 15052 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = Σ𝑚𝐴 ((𝑘𝐵𝐶)‘𝑚))
72 fveq2 6666 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐵𝐶)‘𝑚) = ((𝑘𝐵𝐶)‘(𝑓𝑛)))
73 fzfid 13334 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (1...(♯‘𝐵)) ∈ Fin)
7473, 15fisuppfi 8833 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑓𝐴) ∈ Fin)
75 f1of1 6610 . . . . . . . . . . . 12 (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑓:(1...(♯‘𝐵))–1-1𝐵)
7613, 75syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝑓:(1...(♯‘𝐵))–1-1𝐵)
77 f1ores 6625 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝐵))–1-1𝐵 ∧ (𝑓𝐴) ⊆ (1...(♯‘𝐵))) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto→(𝑓 “ (𝑓𝐴)))
7876, 16, 77syl2anc 584 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto→(𝑓 “ (𝑓𝐴)))
79 f1ofo 6618 . . . . . . . . . . . . 13 (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑓:(1...(♯‘𝐵))–onto𝐵)
8013, 79syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝑓:(1...(♯‘𝐵))–onto𝐵)
811adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝐴𝐵)
82 foimacnv 6628 . . . . . . . . . . . 12 ((𝑓:(1...(♯‘𝐵))–onto𝐵𝐴𝐵) → (𝑓 “ (𝑓𝐴)) = 𝐴)
8380, 81, 82syl2anc 584 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑓 “ (𝑓𝐴)) = 𝐴)
8483f1oeq3d 6608 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ((𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto→(𝑓 “ (𝑓𝐴)) ↔ (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto𝐴))
8578, 84mpbid 233 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto𝐴)
86 fvres 6685 . . . . . . . . . 10 (𝑛 ∈ (𝑓𝐴) → ((𝑓 ↾ (𝑓𝐴))‘𝑛) = (𝑓𝑛))
8786adantl 482 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (𝑓𝐴)) → ((𝑓 ↾ (𝑓𝐴))‘𝑛) = (𝑓𝑛))
8881sselda 3970 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → 𝑚𝐵)
8934ffvelrnda 6846 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐵) → ((𝑘𝐵𝐶)‘𝑚) ∈ ℂ)
9088, 89syldan 591 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → ((𝑘𝐵𝐶)‘𝑚) ∈ ℂ)
9172, 74, 85, 87, 90fsumf1o 15072 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → Σ𝑚𝐴 ((𝑘𝐵𝐶)‘𝑚) = Σ𝑛 ∈ (𝑓𝐴)((𝑘𝐵𝐶)‘(𝑓𝑛)))
9271, 91eqtrd 2860 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = Σ𝑛 ∈ (𝑓𝐴)((𝑘𝐵𝐶)‘(𝑓𝑛)))
93 eqidd 2826 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (1...(♯‘𝐵))) → (𝑓𝑛) = (𝑓𝑛))
9472, 73, 13, 93, 89fsumf1o 15072 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → Σ𝑚𝐵 ((𝑘𝐵𝐶)‘𝑚) = Σ𝑛 ∈ (1...(♯‘𝐵))((𝑘𝐵𝐶)‘(𝑓𝑛)))
9564, 92, 943eqtr4d 2870 . . . . . 6 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = Σ𝑚𝐵 ((𝑘𝐵𝐶)‘𝑚))
96 sumfc 15058 . . . . . 6 Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = Σ𝑘𝐴 𝐶
97 sumfc 15058 . . . . . 6 Σ𝑚𝐵 ((𝑘𝐵𝐶)‘𝑚) = Σ𝑘𝐵 𝐶
9895, 96, 973eqtr3g 2883 . . . . 5 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)
9998expr 457 . . . 4 ((𝜑 ∧ (♯‘𝐵) ∈ ℕ) → (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶))
10099exlimdv 1927 . . 3 ((𝜑 ∧ (♯‘𝐵) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶))
101100expimpd 454 . 2 (𝜑 → (((♯‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵) → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶))
102 fsumss.4 . . 3 (𝜑𝐵 ∈ Fin)
103 fz1f1o 15059 . . 3 (𝐵 ∈ Fin → (𝐵 = ∅ ∨ ((♯‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)))
104102, 103syl 17 . 2 (𝜑 → (𝐵 = ∅ ∨ ((♯‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)))
10511, 101, 104mpjaod 856 1 (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 843   = wceq 1530  wex 1773  wcel 2107  cdif 3936  wss 3939  c0 4294  {csn 4563  cmpt 5142  ccnv 5552  cres 5555  cima 5556   Fn wfn 6346  wf 6347  1-1wf1 6348  ontowfo 6349  1-1-ontowf1o 6350  cfv 6351  (class class class)co 7151  Fincfn 8501  cc 10527  0cc0 10529  1c1 10530  cn 11630  cuz 12235  ...cfz 12885  chash 13683  Σcsu 15035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8898  df-oi 8966  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12383  df-fz 12886  df-fzo 13027  df-seq 13363  df-exp 13423  df-hash 13684  df-cj 14451  df-re 14452  df-im 14453  df-sqrt 14587  df-abs 14588  df-clim 14838  df-sum 15036
This theorem is referenced by:  sumss2  15075  rrxmval  23923  rrxmetlem  23925  itg1val2  24200  itg1addlem4  24215  itg1addlem5  24216  ply1termlem  24708  plyaddlem1  24718  plymullem1  24719  coeeulem  24729  coeidlem  24742  coeid3  24745  coefv0  24753  coemulhi  24759  coemulc  24760  dvply1  24788  vieta1lem2  24815  dvtaylp  24873  pserdvlem2  24931  basellem3  25574  musum  25682  muinv  25684  fsumvma  25703  chpub  25710  logexprlim  25715  dchrsum  25759  chebbnd1lem1  25959  rpvmasumlem  25977  dchrisum0fno1  26001  rplogsum  26017  indsum  31166  eulerpartlemgs2  31524  flcidc  39635  fsumsupp0  41720  elaa2lem  42380
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