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Theorem fisumss 10748
Description: Change the index set to a subset in a finite sum. (Contributed by Mario Carneiro, 21-Apr-2014.) (Revised by Jim Kingdon, 23-Sep-2022.)
Hypotheses
Ref Expression
fsumss.1 (𝜑𝐴𝐵)
fsumss.2 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
fsumss.3 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 0)
fisumss.adc (𝜑 → ∀𝑗𝐵 DECID 𝑗𝐴)
fsumss.4 (𝜑𝐵 ∈ Fin)
Assertion
Ref Expression
fisumss (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)
Distinct variable groups:   𝐴,𝑗,𝑘   𝐵,𝑗,𝑘   𝜑,𝑘
Allowed substitution hints:   𝜑(𝑗)   𝐶(𝑗,𝑘)

Proof of Theorem fisumss
Dummy variables 𝑓 𝑢 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fsumss.1 . . . . . 6 (𝜑𝐴𝐵)
2 sseq0 3321 . . . . . 6 ((𝐴𝐵𝐵 = ∅) → 𝐴 = ∅)
31, 2sylan 277 . . . . 5 ((𝜑𝐵 = ∅) → 𝐴 = ∅)
43sumeq1d 10719 . . . 4 ((𝜑𝐵 = ∅) → Σ𝑘𝐴 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
5 simpr 108 . . . . 5 ((𝜑𝐵 = ∅) → 𝐵 = ∅)
65sumeq1d 10719 . . . 4 ((𝜑𝐵 = ∅) → Σ𝑘𝐵 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
74, 6eqtr4d 2123 . . 3 ((𝜑𝐵 = ∅) → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)
87ex 113 . 2 (𝜑 → (𝐵 = ∅ → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶))
9 cnvimass 4782 . . . . . . . . 9 (𝑓𝐴) ⊆ dom 𝑓
10 simprr 499 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)
11 f1of 5237 . . . . . . . . . 10 (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑓:(1...(♯‘𝐵))⟶𝐵)
1210, 11syl 14 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝑓:(1...(♯‘𝐵))⟶𝐵)
139, 12fssdm 5160 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑓𝐴) ⊆ (1...(♯‘𝐵)))
1412ffnd 5148 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝑓 Fn (1...(♯‘𝐵)))
15 elpreima 5402 . . . . . . . . . . . 12 (𝑓 Fn (1...(♯‘𝐵)) → (𝑛 ∈ (𝑓𝐴) ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴)))
1614, 15syl 14 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑛 ∈ (𝑓𝐴) ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴)))
1712ffvelrnda 5418 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (1...(♯‘𝐵))) → (𝑓𝑛) ∈ 𝐵)
1817ex 113 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑛 ∈ (1...(♯‘𝐵)) → (𝑓𝑛) ∈ 𝐵))
1918adantrd 273 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ((𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴) → (𝑓𝑛) ∈ 𝐵))
2016, 19sylbid 148 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑛 ∈ (𝑓𝐴) → (𝑓𝑛) ∈ 𝐵))
2120imp 122 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (𝑓𝐴)) → (𝑓𝑛) ∈ 𝐵)
22 fsumss.2 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
2322ex 113 . . . . . . . . . . . . . 14 (𝜑 → (𝑘𝐴𝐶 ∈ ℂ))
2423adantr 270 . . . . . . . . . . . . 13 ((𝜑𝑘𝐵) → (𝑘𝐴𝐶 ∈ ℂ))
25 eldif 3006 . . . . . . . . . . . . . . 15 (𝑘 ∈ (𝐵𝐴) ↔ (𝑘𝐵 ∧ ¬ 𝑘𝐴))
26 fsumss.3 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 0)
27 0cn 7459 . . . . . . . . . . . . . . . 16 0 ∈ ℂ
2826, 27syl6eqel 2178 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 ∈ ℂ)
2925, 28sylan2br 282 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘𝐵 ∧ ¬ 𝑘𝐴)) → 𝐶 ∈ ℂ)
3029expr 367 . . . . . . . . . . . . 13 ((𝜑𝑘𝐵) → (¬ 𝑘𝐴𝐶 ∈ ℂ))
31 eleq1w 2148 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑘 → (𝑗𝐴𝑘𝐴))
3231dcbid 786 . . . . . . . . . . . . . . 15 (𝑗 = 𝑘 → (DECID 𝑗𝐴DECID 𝑘𝐴))
33 fisumss.adc . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑗𝐵 DECID 𝑗𝐴)
3433adantr 270 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐵) → ∀𝑗𝐵 DECID 𝑗𝐴)
35 simpr 108 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐵) → 𝑘𝐵)
3632, 34, 35rspcdva 2727 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐵) → DECID 𝑘𝐴)
37 exmiddc 782 . . . . . . . . . . . . . 14 (DECID 𝑘𝐴 → (𝑘𝐴 ∨ ¬ 𝑘𝐴))
3836, 37syl 14 . . . . . . . . . . . . 13 ((𝜑𝑘𝐵) → (𝑘𝐴 ∨ ¬ 𝑘𝐴))
3924, 30, 38mpjaod 673 . . . . . . . . . . . 12 ((𝜑𝑘𝐵) → 𝐶 ∈ ℂ)
4039fmpttd 5437 . . . . . . . . . . 11 (𝜑 → (𝑘𝐵𝐶):𝐵⟶ℂ)
4140adantr 270 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑘𝐵𝐶):𝐵⟶ℂ)
4241ffvelrnda 5418 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ (𝑓𝑛) ∈ 𝐵) → ((𝑘𝐵𝐶)‘(𝑓𝑛)) ∈ ℂ)
4321, 42syldan 276 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (𝑓𝐴)) → ((𝑘𝐵𝐶)‘(𝑓𝑛)) ∈ ℂ)
44 eldifi 3120 . . . . . . . . . . . 12 (𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴)) → 𝑛 ∈ (1...(♯‘𝐵)))
4544, 17sylan2 280 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → (𝑓𝑛) ∈ 𝐵)
46 eldifn 3121 . . . . . . . . . . . . 13 (𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴)) → ¬ 𝑛 ∈ (𝑓𝐴))
4746adantl 271 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ¬ 𝑛 ∈ (𝑓𝐴))
4816adantr 270 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → (𝑛 ∈ (𝑓𝐴) ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴)))
4944adantl 271 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → 𝑛 ∈ (1...(♯‘𝐵)))
5049biantrurd 299 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ((𝑓𝑛) ∈ 𝐴 ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓𝑛) ∈ 𝐴)))
5148, 50bitr4d 189 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → (𝑛 ∈ (𝑓𝐴) ↔ (𝑓𝑛) ∈ 𝐴))
5247, 51mtbid 632 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ¬ (𝑓𝑛) ∈ 𝐴)
5345, 52eldifd 3007 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → (𝑓𝑛) ∈ (𝐵𝐴))
54 difss 3124 . . . . . . . . . . . . 13 (𝐵𝐴) ⊆ 𝐵
55 resmpt 4747 . . . . . . . . . . . . 13 ((𝐵𝐴) ⊆ 𝐵 → ((𝑘𝐵𝐶) ↾ (𝐵𝐴)) = (𝑘 ∈ (𝐵𝐴) ↦ 𝐶))
5654, 55ax-mp 7 . . . . . . . . . . . 12 ((𝑘𝐵𝐶) ↾ (𝐵𝐴)) = (𝑘 ∈ (𝐵𝐴) ↦ 𝐶)
5756fveq1i 5290 . . . . . . . . . . 11 (((𝑘𝐵𝐶) ↾ (𝐵𝐴))‘(𝑓𝑛)) = ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛))
58 fvres 5313 . . . . . . . . . . 11 ((𝑓𝑛) ∈ (𝐵𝐴) → (((𝑘𝐵𝐶) ↾ (𝐵𝐴))‘(𝑓𝑛)) = ((𝑘𝐵𝐶)‘(𝑓𝑛)))
5957, 58syl5eqr 2134 . . . . . . . . . 10 ((𝑓𝑛) ∈ (𝐵𝐴) → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) = ((𝑘𝐵𝐶)‘(𝑓𝑛)))
6053, 59syl 14 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) = ((𝑘𝐵𝐶)‘(𝑓𝑛)))
61 c0ex 7461 . . . . . . . . . . . . . . 15 0 ∈ V
6261elsn2 3473 . . . . . . . . . . . . . 14 (𝐶 ∈ {0} ↔ 𝐶 = 0)
6326, 62sylibr 132 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 ∈ {0})
6463fmpttd 5437 . . . . . . . . . . . 12 (𝜑 → (𝑘 ∈ (𝐵𝐴) ↦ 𝐶):(𝐵𝐴)⟶{0})
6564ad2antrr 472 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → (𝑘 ∈ (𝐵𝐴) ↦ 𝐶):(𝐵𝐴)⟶{0})
6665, 53ffvelrnd 5419 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) ∈ {0})
67 elsni 3459 . . . . . . . . . 10 (((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) ∈ {0} → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) = 0)
6866, 67syl 14 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ((𝑘 ∈ (𝐵𝐴) ↦ 𝐶)‘(𝑓𝑛)) = 0)
6960, 68eqtr3d 2122 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (𝑓𝐴))) → ((𝑘𝐵𝐶)‘(𝑓𝑛)) = 0)
70 eleq1 2150 . . . . . . . . . . . . 13 (𝑗 = (𝑓𝑢) → (𝑗𝐴 ↔ (𝑓𝑢) ∈ 𝐴))
7170dcbid 786 . . . . . . . . . . . 12 (𝑗 = (𝑓𝑢) → (DECID 𝑗𝐴DECID (𝑓𝑢) ∈ 𝐴))
7233ad3antrrr 476 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → ∀𝑗𝐵 DECID 𝑗𝐴)
7312ad2antrr 472 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → 𝑓:(1...(♯‘𝐵))⟶𝐵)
74 simpr 108 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → 𝑢 ∈ (1...(♯‘𝐵)))
7573, 74ffvelrnd 5419 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → (𝑓𝑢) ∈ 𝐵)
7671, 72, 75rspcdva 2727 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → DECID (𝑓𝑢) ∈ 𝐴)
7710ad2antrr 472 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)
78 f1ofun 5239 . . . . . . . . . . . . . 14 (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵 → Fun 𝑓)
7977, 78syl 14 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → Fun 𝑓)
80 f1odm 5241 . . . . . . . . . . . . . . 15 (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵 → dom 𝑓 = (1...(♯‘𝐵)))
8177, 80syl 14 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → dom 𝑓 = (1...(♯‘𝐵)))
8274, 81eleqtrrd 2167 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → 𝑢 ∈ dom 𝑓)
83 fvimacnv 5398 . . . . . . . . . . . . 13 ((Fun 𝑓𝑢 ∈ dom 𝑓) → ((𝑓𝑢) ∈ 𝐴𝑢 ∈ (𝑓𝐴)))
8479, 82, 83syl2anc 403 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → ((𝑓𝑢) ∈ 𝐴𝑢 ∈ (𝑓𝐴)))
8584dcbid 786 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → (DECID (𝑓𝑢) ∈ 𝐴DECID 𝑢 ∈ (𝑓𝐴)))
8676, 85mpbid 145 . . . . . . . . . 10 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → DECID 𝑢 ∈ (𝑓𝐴))
87 elpreima 5402 . . . . . . . . . . . . . . . . 17 (𝑓 Fn (1...(♯‘𝐵)) → (𝑢 ∈ (𝑓𝐴) ↔ (𝑢 ∈ (1...(♯‘𝐵)) ∧ (𝑓𝑢) ∈ 𝐴)))
88 simpl 107 . . . . . . . . . . . . . . . . 17 ((𝑢 ∈ (1...(♯‘𝐵)) ∧ (𝑓𝑢) ∈ 𝐴) → 𝑢 ∈ (1...(♯‘𝐵)))
8987, 88syl6bi 161 . . . . . . . . . . . . . . . 16 (𝑓 Fn (1...(♯‘𝐵)) → (𝑢 ∈ (𝑓𝐴) → 𝑢 ∈ (1...(♯‘𝐵))))
9089con3d 596 . . . . . . . . . . . . . . 15 (𝑓 Fn (1...(♯‘𝐵)) → (¬ 𝑢 ∈ (1...(♯‘𝐵)) → ¬ 𝑢 ∈ (𝑓𝐴)))
9114, 90syl 14 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (¬ 𝑢 ∈ (1...(♯‘𝐵)) → ¬ 𝑢 ∈ (𝑓𝐴)))
9291adantr 270 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) → (¬ 𝑢 ∈ (1...(♯‘𝐵)) → ¬ 𝑢 ∈ (𝑓𝐴)))
9392imp 122 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ ¬ 𝑢 ∈ (1...(♯‘𝐵))) → ¬ 𝑢 ∈ (𝑓𝐴))
9493olcd 688 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ ¬ 𝑢 ∈ (1...(♯‘𝐵))) → (𝑢 ∈ (𝑓𝐴) ∨ ¬ 𝑢 ∈ (𝑓𝐴)))
95 df-dc 781 . . . . . . . . . . 11 (DECID 𝑢 ∈ (𝑓𝐴) ↔ (𝑢 ∈ (𝑓𝐴) ∨ ¬ 𝑢 ∈ (𝑓𝐴)))
9694, 95sylibr 132 . . . . . . . . . 10 ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) ∧ ¬ 𝑢 ∈ (1...(♯‘𝐵))) → DECID 𝑢 ∈ (𝑓𝐴))
97 eluzelz 8997 . . . . . . . . . . . . 13 (𝑢 ∈ (ℤ‘1) → 𝑢 ∈ ℤ)
9897adantl 271 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) → 𝑢 ∈ ℤ)
99 1zzd 8747 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) → 1 ∈ ℤ)
100 simplrl 502 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) → (♯‘𝐵) ∈ ℕ)
101100nnzd 8837 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) → (♯‘𝐵) ∈ ℤ)
102 fzdcel 9423 . . . . . . . . . . . 12 ((𝑢 ∈ ℤ ∧ 1 ∈ ℤ ∧ (♯‘𝐵) ∈ ℤ) → DECID 𝑢 ∈ (1...(♯‘𝐵)))
10398, 99, 101, 102syl3anc 1174 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) → DECID 𝑢 ∈ (1...(♯‘𝐵)))
104 exmiddc 782 . . . . . . . . . . 11 (DECID 𝑢 ∈ (1...(♯‘𝐵)) → (𝑢 ∈ (1...(♯‘𝐵)) ∨ ¬ 𝑢 ∈ (1...(♯‘𝐵))))
105103, 104syl 14 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) → (𝑢 ∈ (1...(♯‘𝐵)) ∨ ¬ 𝑢 ∈ (1...(♯‘𝐵))))
10686, 96, 105mpjaodan 747 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑢 ∈ (ℤ‘1)) → DECID 𝑢 ∈ (𝑓𝐴))
107106ralrimiva 2446 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∀𝑢 ∈ (ℤ‘1)DECID 𝑢 ∈ (𝑓𝐴))
108 1zzd 8747 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 1 ∈ ℤ)
109 fzssuz 9447 . . . . . . . . 9 (1...(♯‘𝐵)) ⊆ (ℤ‘1)
110109a1i 9 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (1...(♯‘𝐵)) ⊆ (ℤ‘1))
111103ralrimiva 2446 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ∀𝑢 ∈ (ℤ‘1)DECID 𝑢 ∈ (1...(♯‘𝐵)))
11213, 43, 69, 107, 108, 110, 111isumss 10747 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → Σ𝑛 ∈ (𝑓𝐴)((𝑘𝐵𝐶)‘(𝑓𝑛)) = Σ𝑛 ∈ (1...(♯‘𝐵))((𝑘𝐵𝐶)‘(𝑓𝑛)))
1131ad2antrr 472 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → 𝐴𝐵)
114113resmptd 4750 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → ((𝑘𝐵𝐶) ↾ 𝐴) = (𝑘𝐴𝐶))
115114fveq1d 5291 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → (((𝑘𝐵𝐶) ↾ 𝐴)‘𝑚) = ((𝑘𝐴𝐶)‘𝑚))
116 fvres 5313 . . . . . . . . . . 11 (𝑚𝐴 → (((𝑘𝐵𝐶) ↾ 𝐴)‘𝑚) = ((𝑘𝐵𝐶)‘𝑚))
117116adantl 271 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → (((𝑘𝐵𝐶) ↾ 𝐴)‘𝑚) = ((𝑘𝐵𝐶)‘𝑚))
118115, 117eqtr3d 2122 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐶)‘𝑚) = ((𝑘𝐵𝐶)‘𝑚))
119118sumeq2dv 10721 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = Σ𝑚𝐴 ((𝑘𝐵𝐶)‘𝑚))
120 fveq2 5289 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐵𝐶)‘𝑚) = ((𝑘𝐵𝐶)‘(𝑓𝑛)))
1211adantr 270 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝐴𝐵)
122 fsumss.4 . . . . . . . . . . . 12 (𝜑𝐵 ∈ Fin)
123 ssfidc 6623 . . . . . . . . . . . 12 ((𝐵 ∈ Fin ∧ 𝐴𝐵 ∧ ∀𝑗𝐵 DECID 𝑗𝐴) → 𝐴 ∈ Fin)
124122, 1, 33, 123syl3anc 1174 . . . . . . . . . . 11 (𝜑𝐴 ∈ Fin)
125124adantr 270 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝐴 ∈ Fin)
126121, 10, 125preimaf1ofi 6639 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑓𝐴) ∈ Fin)
127 f1of1 5236 . . . . . . . . . . . 12 (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑓:(1...(♯‘𝐵))–1-1𝐵)
12810, 127syl 14 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝑓:(1...(♯‘𝐵))–1-1𝐵)
129 f1ores 5252 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝐵))–1-1𝐵 ∧ (𝑓𝐴) ⊆ (1...(♯‘𝐵))) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto→(𝑓 “ (𝑓𝐴)))
130128, 13, 129syl2anc 403 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto→(𝑓 “ (𝑓𝐴)))
131 f1ofo 5244 . . . . . . . . . . . . 13 (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵𝑓:(1...(♯‘𝐵))–onto𝐵)
13210, 131syl 14 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → 𝑓:(1...(♯‘𝐵))–onto𝐵)
133 foimacnv 5255 . . . . . . . . . . . 12 ((𝑓:(1...(♯‘𝐵))–onto𝐵𝐴𝐵) → (𝑓 “ (𝑓𝐴)) = 𝐴)
134132, 121, 133syl2anc 403 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑓 “ (𝑓𝐴)) = 𝐴)
135 f1oeq3 5230 . . . . . . . . . . 11 ((𝑓 “ (𝑓𝐴)) = 𝐴 → ((𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto→(𝑓 “ (𝑓𝐴)) ↔ (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto𝐴))
136134, 135syl 14 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → ((𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto→(𝑓 “ (𝑓𝐴)) ↔ (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto𝐴))
137130, 136mpbid 145 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto𝐴)
138 fvres 5313 . . . . . . . . . 10 (𝑛 ∈ (𝑓𝐴) → ((𝑓 ↾ (𝑓𝐴))‘𝑛) = (𝑓𝑛))
139138adantl 271 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (𝑓𝐴)) → ((𝑓 ↾ (𝑓𝐴))‘𝑛) = (𝑓𝑛))
140121sselda 3023 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → 𝑚𝐵)
14141ffvelrnda 5418 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐵) → ((𝑘𝐵𝐶)‘𝑚) ∈ ℂ)
142140, 141syldan 276 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑚𝐴) → ((𝑘𝐵𝐶)‘𝑚) ∈ ℂ)
143120, 126, 137, 139, 142fsumf1o 10746 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → Σ𝑚𝐴 ((𝑘𝐵𝐶)‘𝑚) = Σ𝑛 ∈ (𝑓𝐴)((𝑘𝐵𝐶)‘(𝑓𝑛)))
144119, 143eqtrd 2120 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = Σ𝑛 ∈ (𝑓𝐴)((𝑘𝐵𝐶)‘(𝑓𝑛)))
145 simprl 498 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (♯‘𝐵) ∈ ℕ)
146145nnzd 8837 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (♯‘𝐵) ∈ ℤ)
147108, 146fzfigd 9803 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → (1...(♯‘𝐵)) ∈ Fin)
148 eqidd 2089 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) ∧ 𝑛 ∈ (1...(♯‘𝐵))) → (𝑓𝑛) = (𝑓𝑛))
149120, 147, 10, 148, 141fsumf1o 10746 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → Σ𝑚𝐵 ((𝑘𝐵𝐶)‘𝑚) = Σ𝑛 ∈ (1...(♯‘𝐵))((𝑘𝐵𝐶)‘(𝑓𝑛)))
150112, 144, 1493eqtr4d 2130 . . . . . 6 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = Σ𝑚𝐵 ((𝑘𝐵𝐶)‘𝑚))
15122ralrimiva 2446 . . . . . . . 8 (𝜑 → ∀𝑘𝐴 𝐶 ∈ ℂ)
152 sumfct 10727 . . . . . . . 8 (∀𝑘𝐴 𝐶 ∈ ℂ → Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = Σ𝑘𝐴 𝐶)
153151, 152syl 14 . . . . . . 7 (𝜑 → Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = Σ𝑘𝐴 𝐶)
154153adantr 270 . . . . . 6 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = Σ𝑘𝐴 𝐶)
15522adantlr 461 . . . . . . . . . 10 (((𝜑𝑘𝐵) ∧ 𝑘𝐴) → 𝐶 ∈ ℂ)
156 simpll 496 . . . . . . . . . . . 12 (((𝜑𝑘𝐵) ∧ ¬ 𝑘𝐴) → 𝜑)
157 simplr 497 . . . . . . . . . . . . 13 (((𝜑𝑘𝐵) ∧ ¬ 𝑘𝐴) → 𝑘𝐵)
158 simpr 108 . . . . . . . . . . . . 13 (((𝜑𝑘𝐵) ∧ ¬ 𝑘𝐴) → ¬ 𝑘𝐴)
159157, 158eldifd 3007 . . . . . . . . . . . 12 (((𝜑𝑘𝐵) ∧ ¬ 𝑘𝐴) → 𝑘 ∈ (𝐵𝐴))
160156, 159, 26syl2anc 403 . . . . . . . . . . 11 (((𝜑𝑘𝐵) ∧ ¬ 𝑘𝐴) → 𝐶 = 0)
161 0cnd 7460 . . . . . . . . . . 11 (((𝜑𝑘𝐵) ∧ ¬ 𝑘𝐴) → 0 ∈ ℂ)
162160, 161eqeltrd 2164 . . . . . . . . . 10 (((𝜑𝑘𝐵) ∧ ¬ 𝑘𝐴) → 𝐶 ∈ ℂ)
163155, 162, 38mpjaodan 747 . . . . . . . . 9 ((𝜑𝑘𝐵) → 𝐶 ∈ ℂ)
164163ralrimiva 2446 . . . . . . . 8 (𝜑 → ∀𝑘𝐵 𝐶 ∈ ℂ)
165 sumfct 10727 . . . . . . . 8 (∀𝑘𝐵 𝐶 ∈ ℂ → Σ𝑚𝐵 ((𝑘𝐵𝐶)‘𝑚) = Σ𝑘𝐵 𝐶)
166164, 165syl 14 . . . . . . 7 (𝜑 → Σ𝑚𝐵 ((𝑘𝐵𝐶)‘𝑚) = Σ𝑘𝐵 𝐶)
167166adantr 270 . . . . . 6 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → Σ𝑚𝐵 ((𝑘𝐵𝐶)‘𝑚) = Σ𝑘𝐵 𝐶)
168150, 154, 1673eqtr3d 2128 . . . . 5 ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)) → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)
169168expr 367 . . . 4 ((𝜑 ∧ (♯‘𝐵) ∈ ℕ) → (𝑓:(1...(♯‘𝐵))–1-1-onto𝐵 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶))
170169exlimdv 1747 . . 3 ((𝜑 ∧ (♯‘𝐵) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶))
171170expimpd 355 . 2 (𝜑 → (((♯‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵) → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶))
172 fz1f1o 10728 . . 3 (𝐵 ∈ Fin → (𝐵 = ∅ ∨ ((♯‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)))
173122, 172syl 14 . 2 (𝜑 → (𝐵 = ∅ ∨ ((♯‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto𝐵)))
1748, 171, 173mpjaod 673 1 (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 664  DECID wdc 780   = wceq 1289  wex 1426  wcel 1438  wral 2359  cdif 2994  wss 2997  c0 3284  {csn 3441  cmpt 3891  ccnv 4427  dom cdm 4428  cres 4430  cima 4431  Fun wfun 4996   Fn wfn 4997  wf 4998  1-1wf1 4999  ontowfo 5000  1-1-ontowf1o 5001  cfv 5002  (class class class)co 5634  Fincfn 6437  cc 7327  0cc0 7329  1c1 7330  cn 8394  cz 8720  cuz 8988  ...cfz 9393  chash 10148  Σcsu 10706
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443  ax-caucvg 7444
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-isom 5011  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-frec 6138  df-1o 6163  df-oadd 6167  df-er 6272  df-en 6438  df-dom 6439  df-fin 6440  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-3 8453  df-4 8454  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-fz 9394  df-fzo 9519  df-iseq 9818  df-seq3 9819  df-exp 9920  df-ihash 10149  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397  df-clim 10631  df-isum 10707
This theorem is referenced by:  isumss2  10749
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