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Theorem fsumss 14500
 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 480 . . . 4 ((𝜑𝐵 = ∅) → 𝐴𝐵)
3 sumss.2 . . . . 5 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
43adantlr 751 . . . 4 (((𝜑𝐵 = ∅) ∧ 𝑘𝐴) → 𝐶 ∈ ℂ)
5 sumss.3 . . . . 5 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 0)
65adantlr 751 . . . 4 (((𝜑𝐵 = ∅) ∧ 𝑘 ∈ (𝐵𝐴)) → 𝐶 = 0)
7 simpr 476 . . . . 5 ((𝜑𝐵 = ∅) → 𝐵 = ∅)
8 0ss 4005 . . . . 5 ∅ ⊆ (ℤ‘0)
97, 8syl6eqss 3688 . . . 4 ((𝜑𝐵 = ∅) → 𝐵 ⊆ (ℤ‘0))
102, 4, 6, 9sumss 14499 . . 3 ((𝜑𝐵 = ∅) → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)
1110ex 449 . 2 (𝜑 → (𝐵 = ∅ → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶))
12 cnvimass 5520 . . . . . . . . 9 (𝑓𝐴) ⊆ dom 𝑓
13 simprr 811 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)
14 f1of 6175 . . . . . . . . . . 11 (𝑓:(1...(#‘𝐵))–1-1-onto𝐵𝑓:(1...(#‘𝐵))⟶𝐵)
1513, 14syl 17 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → 𝑓:(1...(#‘𝐵))⟶𝐵)
16 fdm 6089 . . . . . . . . . 10 (𝑓:(1...(#‘𝐵))⟶𝐵 → dom 𝑓 = (1...(#‘𝐵)))
1715, 16syl 17 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → dom 𝑓 = (1...(#‘𝐵)))
1812, 17syl5sseq 3686 . . . . . . . 8 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (𝑓𝐴) ⊆ (1...(#‘𝐵)))
19 ffn 6083 . . . . . . . . . . . . 13 (𝑓:(1...(#‘𝐵))⟶𝐵𝑓 Fn (1...(#‘𝐵)))
2015, 19syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → 𝑓 Fn (1...(#‘𝐵)))
21 elpreima 6377 . . . . . . . . . . . 12 (𝑓 Fn (1...(#‘𝐵)) → (𝑛 ∈ (𝑓𝐴) ↔ (𝑛 ∈ (1...(#‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴)))
2220, 21syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (𝑛 ∈ (𝑓𝐴) ↔ (𝑛 ∈ (1...(#‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴)))
2315ffvelrnda 6399 . . . . . . . . . . . . 13 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (1...(#‘𝐵))) → (𝑓𝑛) ∈ 𝐵)
2423ex 449 . . . . . . . . . . . 12 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (𝑛 ∈ (1...(#‘𝐵)) → (𝑓𝑛) ∈ 𝐵))
2524adantrd 483 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → ((𝑛 ∈ (1...(#‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴) → (𝑓𝑛) ∈ 𝐵))
2622, 25sylbid 230 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (𝑛 ∈ (𝑓𝐴) → (𝑓𝑛) ∈ 𝐵))
2726imp 444 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (𝑓𝐴)) → (𝑓𝑛) ∈ 𝐵)
283ex 449 . . . . . . . . . . . . . 14 (𝜑 → (𝑘𝐴𝐶 ∈ ℂ))
2928adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑘𝐵) → (𝑘𝐴𝐶 ∈ ℂ))
30 eldif 3617 . . . . . . . . . . . . . . 15 (𝑘 ∈ (𝐵𝐴) ↔ (𝑘𝐵 ∧ ¬ 𝑘𝐴))
31 0cn 10070 . . . . . . . . . . . . . . . 16 0 ∈ ℂ
325, 31syl6eqel 2738 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 ∈ ℂ)
3330, 32sylan2br 492 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘𝐵 ∧ ¬ 𝑘𝐴)) → 𝐶 ∈ ℂ)
3433expr 642 . . . . . . . . . . . . 13 ((𝜑𝑘𝐵) → (¬ 𝑘𝐴𝐶 ∈ ℂ))
3529, 34pm2.61d 170 . . . . . . . . . . . 12 ((𝜑𝑘𝐵) → 𝐶 ∈ ℂ)
36 eqid 2651 . . . . . . . . . . . 12 (𝑘𝐵𝐶) = (𝑘𝐵𝐶)
3735, 36fmptd 6425 . . . . . . . . . . 11 (𝜑 → (𝑘𝐵𝐶):𝐵⟶ℂ)
3837adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (𝑘𝐵𝐶):𝐵⟶ℂ)
3938ffvelrnda 6399 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ (𝑓𝑛) ∈ 𝐵) → ((𝑘𝐵𝐶)‘(𝑓𝑛)) ∈ ℂ)
4027, 39syldan 486 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (𝑓𝐴)) → ((𝑘𝐵𝐶)‘(𝑓𝑛)) ∈ ℂ)
41 eldifi 3765 . . . . . . . . . . . 12 (𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴)) → 𝑛 ∈ (1...(#‘𝐵)))
4241, 23sylan2 490 . . . . . . . . . . 11 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → (𝑓𝑛) ∈ 𝐵)
43 eldifn 3766 . . . . . . . . . . . . 13 (𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴)) → ¬ 𝑛 ∈ (𝑓𝐴))
4443adantl 481 . . . . . . . . . . . 12 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → ¬ 𝑛 ∈ (𝑓𝐴))
4522adantr 480 . . . . . . . . . . . . 13 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → (𝑛 ∈ (𝑓𝐴) ↔ (𝑛 ∈ (1...(#‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴)))
4641adantl 481 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → 𝑛 ∈ (1...(#‘𝐵)))
4746biantrurd 528 . . . . . . . . . . . . 13 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → ((𝑓𝑛) ∈ 𝐴 ↔ (𝑛 ∈ (1...(#‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴)))
4845, 47bitr4d 271 . . . . . . . . . . . 12 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → (𝑛 ∈ (𝑓𝐴) ↔ (𝑓𝑛) ∈ 𝐴))
4944, 48mtbid 313 . . . . . . . . . . 11 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → ¬ (𝑓𝑛) ∈ 𝐴)
5042, 49eldifd 3618 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → (𝑓𝑛) ∈ (𝐵𝐴))
51 difss 3770 . . . . . . . . . . . . 13 (𝐵𝐴) ⊆ 𝐵
52 resmpt 5484 . . . . . . . . . . . . 13 ((𝐵𝐴) ⊆ 𝐵 → ((𝑘𝐵𝐶) ↾ (𝐵𝐴)) = (𝑘 ∈ (𝐵𝐴) ↦ 𝐶))
5351, 52ax-mp 5 . . . . . . . . . . . 12 ((𝑘𝐵𝐶) ↾ (𝐵𝐴)) = (𝑘 ∈ (𝐵𝐴) ↦ 𝐶)
5453fveq1i 6230 . . . . . . . . . . 11 (((𝑘𝐵𝐶) ↾ (𝐵𝐴))‘(𝑓𝑛)) = ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛))
55 fvres 6245 . . . . . . . . . . 11 ((𝑓𝑛) ∈ (𝐵𝐴) → (((𝑘𝐵𝐶) ↾ (𝐵𝐴))‘(𝑓𝑛)) = ((𝑘𝐵𝐶)‘(𝑓𝑛)))
5654, 55syl5eqr 2699 . . . . . . . . . 10 ((𝑓𝑛) ∈ (𝐵𝐴) → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) = ((𝑘𝐵𝐶)‘(𝑓𝑛)))
5750, 56syl 17 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) = ((𝑘𝐵𝐶)‘(𝑓𝑛)))
58 c0ex 10072 . . . . . . . . . . . . . . 15 0 ∈ V
5958elsn2 4244 . . . . . . . . . . . . . 14 (𝐶 ∈ {0} ↔ 𝐶 = 0)
605, 59sylibr 224 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 ∈ {0})
61 eqid 2651 . . . . . . . . . . . . 13 (𝑘 ∈ (𝐵𝐴) ↦ 𝐶) = (𝑘 ∈ (𝐵𝐴) ↦ 𝐶)
6260, 61fmptd 6425 . . . . . . . . . . . 12 (𝜑 → (𝑘 ∈ (𝐵𝐴) ↦ 𝐶):(𝐵𝐴)⟶{0})
6362ad2antrr 762 . . . . . . . . . . 11 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → (𝑘 ∈ (𝐵𝐴) ↦ 𝐶):(𝐵𝐴)⟶{0})
6463, 50ffvelrnd 6400 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) ∈ {0})
65 elsni 4227 . . . . . . . . . 10 (((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) ∈ {0} → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) = 0)
6664, 65syl 17 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) = 0)
6757, 66eqtr3d 2687 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (𝑓𝐴))) → ((𝑘𝐵𝐶)‘(𝑓𝑛)) = 0)
68 fzssuz 12420 . . . . . . . . 9 (1...(#‘𝐵)) ⊆ (ℤ‘1)
6968a1i 11 . . . . . . . 8 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (1...(#‘𝐵)) ⊆ (ℤ‘1))
7018, 40, 67, 69sumss 14499 . . . . . . 7 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → Σ𝑛 ∈ (𝑓𝐴)((𝑘𝐵𝐶)‘(𝑓𝑛)) = Σ𝑛 ∈ (1...(#‘𝐵))((𝑘𝐵𝐶)‘(𝑓𝑛)))
711ad2antrr 762 . . . . . . . . . . . 12 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → 𝐴𝐵)
7271resmptd 5487 . . . . . . . . . . 11 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → ((𝑘𝐵𝐶) ↾ 𝐴) = (𝑘𝐴𝐶))
7372fveq1d 6231 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → (((𝑘𝐵𝐶) ↾ 𝐴)‘𝑚) = ((𝑘𝐴𝐶)‘𝑚))
74 fvres 6245 . . . . . . . . . . 11 (𝑚𝐴 → (((𝑘𝐵𝐶) ↾ 𝐴)‘𝑚) = ((𝑘𝐵𝐶)‘𝑚))
7574adantl 481 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → (((𝑘𝐵𝐶) ↾ 𝐴)‘𝑚) = ((𝑘𝐵𝐶)‘𝑚))
7673, 75eqtr3d 2687 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐶)‘𝑚) = ((𝑘𝐵𝐶)‘𝑚))
7776sumeq2dv 14477 . . . . . . . 8 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = Σ𝑚𝐴 ((𝑘𝐵𝐶)‘𝑚))
78 fveq2 6229 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐵𝐶)‘𝑚) = ((𝑘𝐵𝐶)‘(𝑓𝑛)))
79 fzfid 12812 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (1...(#‘𝐵)) ∈ Fin)
8079, 15fisuppfi 8324 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (𝑓𝐴) ∈ Fin)
81 f1of1 6174 . . . . . . . . . . . 12 (𝑓:(1...(#‘𝐵))–1-1-onto𝐵𝑓:(1...(#‘𝐵))–1-1𝐵)
8213, 81syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → 𝑓:(1...(#‘𝐵))–1-1𝐵)
83 f1ores 6189 . . . . . . . . . . 11 ((𝑓:(1...(#‘𝐵))–1-1𝐵 ∧ (𝑓𝐴) ⊆ (1...(#‘𝐵))) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto→(𝑓 “ (𝑓𝐴)))
8482, 18, 83syl2anc 694 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto→(𝑓 “ (𝑓𝐴)))
85 f1ofo 6182 . . . . . . . . . . . . 13 (𝑓:(1...(#‘𝐵))–1-1-onto𝐵𝑓:(1...(#‘𝐵))–onto𝐵)
8613, 85syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → 𝑓:(1...(#‘𝐵))–onto𝐵)
871adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → 𝐴𝐵)
88 foimacnv 6192 . . . . . . . . . . . 12 ((𝑓:(1...(#‘𝐵))–onto𝐵𝐴𝐵) → (𝑓 “ (𝑓𝐴)) = 𝐴)
8986, 87, 88syl2anc 694 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (𝑓 “ (𝑓𝐴)) = 𝐴)
90 f1oeq3 6167 . . . . . . . . . . 11 ((𝑓 “ (𝑓𝐴)) = 𝐴 → ((𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto→(𝑓 “ (𝑓𝐴)) ↔ (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto𝐴))
9189, 90syl 17 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → ((𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto→(𝑓 “ (𝑓𝐴)) ↔ (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto𝐴))
9284, 91mpbid 222 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto𝐴)
93 fvres 6245 . . . . . . . . . 10 (𝑛 ∈ (𝑓𝐴) → ((𝑓 ↾ (𝑓𝐴))‘𝑛) = (𝑓𝑛))
9493adantl 481 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (𝑓𝐴)) → ((𝑓 ↾ (𝑓𝐴))‘𝑛) = (𝑓𝑛))
9587sselda 3636 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → 𝑚𝐵)
9638ffvelrnda 6399 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐵) → ((𝑘𝐵𝐶)‘𝑚) ∈ ℂ)
9795, 96syldan 486 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → ((𝑘𝐵𝐶)‘𝑚) ∈ ℂ)
9878, 80, 92, 94, 97fsumf1o 14498 . . . . . . . 8 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → Σ𝑚𝐴 ((𝑘𝐵𝐶)‘𝑚) = Σ𝑛 ∈ (𝑓𝐴)((𝑘𝐵𝐶)‘(𝑓𝑛)))
9977, 98eqtrd 2685 . . . . . . 7 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = Σ𝑛 ∈ (𝑓𝐴)((𝑘𝐵𝐶)‘(𝑓𝑛)))
100 eqidd 2652 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (1...(#‘𝐵))) → (𝑓𝑛) = (𝑓𝑛))
10178, 79, 13, 100, 96fsumf1o 14498 . . . . . . 7 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → Σ𝑚𝐵 ((𝑘𝐵𝐶)‘𝑚) = Σ𝑛 ∈ (1...(#‘𝐵))((𝑘𝐵𝐶)‘(𝑓𝑛)))
10270, 99, 1013eqtr4d 2695 . . . . . 6 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = Σ𝑚𝐵 ((𝑘𝐵𝐶)‘𝑚))
103 sumfc 14484 . . . . . 6 Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = Σ𝑘𝐴 𝐶
104 sumfc 14484 . . . . . 6 Σ𝑚𝐵 ((𝑘𝐵𝐶)‘𝑚) = Σ𝑘𝐵 𝐶
105102, 103, 1043eqtr3g 2708 . . . . 5 ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)) → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)
106105expr 642 . . . 4 ((𝜑 ∧ (#‘𝐵) ∈ ℕ) → (𝑓:(1...(#‘𝐵))–1-1-onto𝐵 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶))
107106exlimdv 1901 . . 3 ((𝜑 ∧ (#‘𝐵) ∈ ℕ) → (∃𝑓 𝑓:(1...(#‘𝐵))–1-1-onto𝐵 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶))
108107expimpd 628 . 2 (𝜑 → (((#‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐵))–1-1-onto𝐵) → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶))
109 fsumss.4 . . 3 (𝜑𝐵 ∈ Fin)
110 fz1f1o 14485 . . 3 (𝐵 ∈ Fin → (𝐵 = ∅ ∨ ((#‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)))
111109, 110syl 17 . 2 (𝜑 → (𝐵 = ∅ ∨ ((#‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐵))–1-1-onto𝐵)))
11211, 108, 111mpjaod 395 1 (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   = wceq 1523  ∃wex 1744   ∈ wcel 2030   ∖ cdif 3604   ⊆ wss 3607  ∅c0 3948  {csn 4210   ↦ cmpt 4762  ◡ccnv 5142  dom cdm 5143   ↾ cres 5145   “ cima 5146   Fn wfn 5921  ⟶wf 5922  –1-1→wf1 5923  –onto→wfo 5924  –1-1-onto→wf1o 5925  ‘cfv 5926  (class class class)co 6690  Fincfn 7997  ℂcc 9972  0cc0 9974  1c1 9975  ℕcn 11058  ℤ≥cuz 11725  ...cfz 12364  #chash 13157  Σcsu 14460 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461 This theorem is referenced by:  sumss2  14501  rrxmval  23234  rrxmetlem  23236  itg1val2  23496  itg1addlem4  23511  itg1addlem5  23512  ply1termlem  24004  plyaddlem1  24014  plymullem1  24015  coeeulem  24025  coeidlem  24038  coeid3  24041  coefv0  24049  coemulhi  24055  coemulc  24056  dvply1  24084  vieta1lem2  24111  dvtaylp  24169  pserdvlem2  24227  basellem3  24854  musum  24962  muinv  24964  fsumvma  24983  chpub  24990  logexprlim  24995  dchrsum  25039  chebbnd1lem1  25203  rpvmasumlem  25221  dchrisum0fno1  25245  rplogsum  25261  indsum  30211  eulerpartlemgs2  30570  flcidc  38061  fsumsupp0  40128  elaa2lem  40768
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