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Theorem sge0fodjrnlem 42566
Description: Re-index a nonnegative extended sum using an onto function with disjoint range, when the empty set is assigned 0 in the sum (this is true, for example, both for measures and outer measures). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
sge0fodjrnlem.k 𝑘𝜑
sge0fodjrnlem.n 𝑛𝜑
sge0fodjrnlem.bd (𝑘 = 𝐺𝐵 = 𝐷)
sge0fodjrnlem.c (𝜑𝐶𝑉)
sge0fodjrnlem.f (𝜑𝐹:𝐶onto𝐴)
sge0fodjrnlem.dj (𝜑Disj 𝑛𝐶 (𝐹𝑛))
sge0fodjrnlem.fng ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)
sge0fodjrnlem.b ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))
sge0fodjrnlem.b0 ((𝜑𝑘 = ∅) → 𝐵 = 0)
sge0fodjrnlem.z 𝑍 = (𝐹 “ {∅})
Assertion
Ref Expression
sge0fodjrnlem (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))
Distinct variable groups:   𝐴,𝑘,𝑛   𝐵,𝑛   𝐶,𝑘,𝑛   𝐷,𝑘   𝑘,𝐹,𝑛   𝑘,𝐺   𝑘,𝑍,𝑛
Allowed substitution hints:   𝜑(𝑘,𝑛)   𝐵(𝑘)   𝐷(𝑛)   𝐺(𝑛)   𝑉(𝑘,𝑛)

Proof of Theorem sge0fodjrnlem
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 sge0fodjrnlem.k . . . 4 𝑘𝜑
2 sge0fodjrnlem.c . . . . 5 (𝜑𝐶𝑉)
3 sge0fodjrnlem.f . . . . 5 (𝜑𝐹:𝐶onto𝐴)
4 fornex 7651 . . . . 5 (𝐶𝑉 → (𝐹:𝐶onto𝐴𝐴 ∈ V))
52, 3, 4sylc 65 . . . 4 (𝜑𝐴 ∈ V)
6 difssd 4112 . . . 4 (𝜑 → (𝐴 ∖ {∅}) ⊆ 𝐴)
7 simpl 483 . . . . 5 ((𝜑𝑘 ∈ (𝐴 ∖ {∅})) → 𝜑)
86sselda 3970 . . . . 5 ((𝜑𝑘 ∈ (𝐴 ∖ {∅})) → 𝑘𝐴)
9 sge0fodjrnlem.b . . . . 5 ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))
107, 8, 9syl2anc 584 . . . 4 ((𝜑𝑘 ∈ (𝐴 ∖ {∅})) → 𝐵 ∈ (0[,]+∞))
11 simpl 483 . . . . 5 ((𝜑𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅}))) → 𝜑)
12 dfin4 4247 . . . . . . . . . 10 (𝐴 ∩ {∅}) = (𝐴 ∖ (𝐴 ∖ {∅}))
1312eqcomi 2834 . . . . . . . . 9 (𝐴 ∖ (𝐴 ∖ {∅})) = (𝐴 ∩ {∅})
14 inss2 4209 . . . . . . . . 9 (𝐴 ∩ {∅}) ⊆ {∅}
1513, 14eqsstri 4004 . . . . . . . 8 (𝐴 ∖ (𝐴 ∖ {∅})) ⊆ {∅}
16 id 22 . . . . . . . 8 (𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅})) → 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅})))
1715, 16sseldi 3968 . . . . . . 7 (𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅})) → 𝑘 ∈ {∅})
18 elsni 4580 . . . . . . 7 (𝑘 ∈ {∅} → 𝑘 = ∅)
1917, 18syl 17 . . . . . 6 (𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅})) → 𝑘 = ∅)
2019adantl 482 . . . . 5 ((𝜑𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅}))) → 𝑘 = ∅)
21 sge0fodjrnlem.b0 . . . . 5 ((𝜑𝑘 = ∅) → 𝐵 = 0)
2211, 20, 21syl2anc 584 . . . 4 ((𝜑𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅}))) → 𝐵 = 0)
231, 5, 6, 10, 22sge0ss 42562 . . 3 (𝜑 → (Σ^‘(𝑘 ∈ (𝐴 ∖ {∅}) ↦ 𝐵)) = (Σ^‘(𝑘𝐴𝐵)))
2423eqcomd 2831 . 2 (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑘 ∈ (𝐴 ∖ {∅}) ↦ 𝐵)))
25 sge0fodjrnlem.n . . 3 𝑛𝜑
26 sge0fodjrnlem.bd . . 3 (𝑘 = 𝐺𝐵 = 𝐷)
27 difexg 5227 . . . 4 (𝐶𝑉 → (𝐶𝑍) ∈ V)
282, 27syl 17 . . 3 (𝜑 → (𝐶𝑍) ∈ V)
29 eqid 2825 . . . . 5 (𝑛𝐶 ↦ (𝐹𝑛)) = (𝑛𝐶 ↦ (𝐹𝑛))
30 fof 6586 . . . . . . 7 (𝐹:𝐶onto𝐴𝐹:𝐶𝐴)
313, 30syl 17 . . . . . 6 (𝜑𝐹:𝐶𝐴)
3231ffvelrnda 6846 . . . . 5 ((𝜑𝑛𝐶) → (𝐹𝑛) ∈ 𝐴)
33 sge0fodjrnlem.dj . . . . 5 (𝜑Disj 𝑛𝐶 (𝐹𝑛))
34 fveq2 6666 . . . . . . 7 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
3534neeq1d 3079 . . . . . 6 (𝑚 = 𝑛 → ((𝐹𝑚) ≠ ∅ ↔ (𝐹𝑛) ≠ ∅))
3635cbvrabv 3496 . . . . 5 {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅} = {𝑛𝐶 ∣ (𝐹𝑛) ≠ ∅}
3734cbvmptv 5165 . . . . . . 7 (𝑚𝐶 ↦ (𝐹𝑚)) = (𝑛𝐶 ↦ (𝐹𝑛))
3837rneqi 5805 . . . . . 6 ran (𝑚𝐶 ↦ (𝐹𝑚)) = ran (𝑛𝐶 ↦ (𝐹𝑛))
3938difeq1i 4098 . . . . 5 (ran (𝑚𝐶 ↦ (𝐹𝑚)) ∖ {∅}) = (ran (𝑛𝐶 ↦ (𝐹𝑛)) ∖ {∅})
4025, 29, 32, 33, 36, 39disjf1o 41319 . . . 4 (𝜑 → ((𝑛𝐶 ↦ (𝐹𝑛)) ↾ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}):{𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}–1-1-onto→(ran (𝑚𝐶 ↦ (𝐹𝑚)) ∖ {∅}))
4131feqmptd 6729 . . . . . 6 (𝜑𝐹 = (𝑛𝐶 ↦ (𝐹𝑛)))
42 difssd 4112 . . . . . . . . . . . . 13 (𝜑 → (𝐶𝑍) ⊆ 𝐶)
4342sselda 3970 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝐶𝑍)) → 𝑛𝐶)
44 eldifi 4106 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (𝐶𝑍) → 𝑛𝐶)
4544adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ (𝐶𝑍) ∧ (𝐹𝑛) = ∅) → 𝑛𝐶)
46 id 22 . . . . . . . . . . . . . . . . . . . 20 ((𝐹𝑛) = ∅ → (𝐹𝑛) = ∅)
47 fvex 6679 . . . . . . . . . . . . . . . . . . . . 21 (𝐹𝑛) ∈ V
4847elsn 4578 . . . . . . . . . . . . . . . . . . . 20 ((𝐹𝑛) ∈ {∅} ↔ (𝐹𝑛) = ∅)
4946, 48sylibr 235 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑛) = ∅ → (𝐹𝑛) ∈ {∅})
5049adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ (𝐶𝑍) ∧ (𝐹𝑛) = ∅) → (𝐹𝑛) ∈ {∅})
5145, 50jca 512 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (𝐶𝑍) ∧ (𝐹𝑛) = ∅) → (𝑛𝐶 ∧ (𝐹𝑛) ∈ {∅}))
5251adantll 710 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝐶𝑍)) ∧ (𝐹𝑛) = ∅) → (𝑛𝐶 ∧ (𝐹𝑛) ∈ {∅}))
5331ffnd 6511 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹 Fn 𝐶)
54 elpreima 6823 . . . . . . . . . . . . . . . . . 18 (𝐹 Fn 𝐶 → (𝑛 ∈ (𝐹 “ {∅}) ↔ (𝑛𝐶 ∧ (𝐹𝑛) ∈ {∅})))
5553, 54syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑛 ∈ (𝐹 “ {∅}) ↔ (𝑛𝐶 ∧ (𝐹𝑛) ∈ {∅})))
5655ad2antrr 722 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝐶𝑍)) ∧ (𝐹𝑛) = ∅) → (𝑛 ∈ (𝐹 “ {∅}) ↔ (𝑛𝐶 ∧ (𝐹𝑛) ∈ {∅})))
5752, 56mpbird 258 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (𝐶𝑍)) ∧ (𝐹𝑛) = ∅) → 𝑛 ∈ (𝐹 “ {∅}))
58 sge0fodjrnlem.z . . . . . . . . . . . . . . 15 𝑍 = (𝐹 “ {∅})
5957, 58syl6eleqr 2928 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (𝐶𝑍)) ∧ (𝐹𝑛) = ∅) → 𝑛𝑍)
60 eldifn 4107 . . . . . . . . . . . . . . 15 (𝑛 ∈ (𝐶𝑍) → ¬ 𝑛𝑍)
6160ad2antlr 723 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (𝐶𝑍)) ∧ (𝐹𝑛) = ∅) → ¬ 𝑛𝑍)
6259, 61pm2.65da 813 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝐶𝑍)) → ¬ (𝐹𝑛) = ∅)
6362neqned 3027 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝐶𝑍)) → (𝐹𝑛) ≠ ∅)
6443, 63jca 512 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝐶𝑍)) → (𝑛𝐶 ∧ (𝐹𝑛) ≠ ∅))
6535elrab 3683 . . . . . . . . . . 11 (𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅} ↔ (𝑛𝐶 ∧ (𝐹𝑛) ≠ ∅))
6664, 65sylibr 235 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝐶𝑍)) → 𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅})
6766ex 413 . . . . . . . . 9 (𝜑 → (𝑛 ∈ (𝐶𝑍) → 𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}))
6865simplbi 498 . . . . . . . . . . . . . . 15 (𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅} → 𝑛𝐶)
6968adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}) → 𝑛𝐶)
7058eleq2i 2908 . . . . . . . . . . . . . . . . . . . . 21 (𝑛𝑍𝑛 ∈ (𝐹 “ {∅}))
7170biimpi 217 . . . . . . . . . . . . . . . . . . . 20 (𝑛𝑍𝑛 ∈ (𝐹 “ {∅}))
7271adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛𝑍) → 𝑛 ∈ (𝐹 “ {∅}))
7355adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛𝑍) → (𝑛 ∈ (𝐹 “ {∅}) ↔ (𝑛𝐶 ∧ (𝐹𝑛) ∈ {∅})))
7472, 73mpbid 233 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛𝑍) → (𝑛𝐶 ∧ (𝐹𝑛) ∈ {∅}))
7574simprd 496 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ {∅})
76 elsni 4580 . . . . . . . . . . . . . . . . 17 ((𝐹𝑛) ∈ {∅} → (𝐹𝑛) = ∅)
7775, 76syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑛𝑍) → (𝐹𝑛) = ∅)
7877adantlr 711 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}) ∧ 𝑛𝑍) → (𝐹𝑛) = ∅)
7965simprbi 497 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅} → (𝐹𝑛) ≠ ∅)
8079ad2antlr 723 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}) ∧ 𝑛𝑍) → (𝐹𝑛) ≠ ∅)
8180neneqd 3025 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}) ∧ 𝑛𝑍) → ¬ (𝐹𝑛) = ∅)
8278, 81pm2.65da 813 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}) → ¬ 𝑛𝑍)
8369, 82eldifd 3950 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}) → 𝑛 ∈ (𝐶𝑍))
8483ex 413 . . . . . . . . . . . 12 (𝜑 → (𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅} → 𝑛 ∈ (𝐶𝑍)))
8525, 84ralrimi 3220 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}𝑛 ∈ (𝐶𝑍))
86 dfss3 3959 . . . . . . . . . . 11 ({𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅} ⊆ (𝐶𝑍) ↔ ∀𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}𝑛 ∈ (𝐶𝑍))
8785, 86sylibr 235 . . . . . . . . . 10 (𝜑 → {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅} ⊆ (𝐶𝑍))
8887sseld 3969 . . . . . . . . 9 (𝜑 → (𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅} → 𝑛 ∈ (𝐶𝑍)))
8967, 88impbid 213 . . . . . . . 8 (𝜑 → (𝑛 ∈ (𝐶𝑍) ↔ 𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}))
9025, 89alrimi 2206 . . . . . . 7 (𝜑 → ∀𝑛(𝑛 ∈ (𝐶𝑍) ↔ 𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}))
91 dfcleq 2819 . . . . . . 7 ((𝐶𝑍) = {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅} ↔ ∀𝑛(𝑛 ∈ (𝐶𝑍) ↔ 𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}))
9290, 91sylibr 235 . . . . . 6 (𝜑 → (𝐶𝑍) = {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅})
9341, 92reseq12d 5852 . . . . 5 (𝜑 → (𝐹 ↾ (𝐶𝑍)) = ((𝑛𝐶 ↦ (𝐹𝑛)) ↾ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}))
9441, 37syl6eqr 2878 . . . . . . . . 9 (𝜑𝐹 = (𝑚𝐶 ↦ (𝐹𝑚)))
9594eqcomd 2831 . . . . . . . 8 (𝜑 → (𝑚𝐶 ↦ (𝐹𝑚)) = 𝐹)
9695rneqd 5806 . . . . . . 7 (𝜑 → ran (𝑚𝐶 ↦ (𝐹𝑚)) = ran 𝐹)
97 forn 6589 . . . . . . . 8 (𝐹:𝐶onto𝐴 → ran 𝐹 = 𝐴)
983, 97syl 17 . . . . . . 7 (𝜑 → ran 𝐹 = 𝐴)
9996, 98eqtr2d 2861 . . . . . 6 (𝜑𝐴 = ran (𝑚𝐶 ↦ (𝐹𝑚)))
10099difeq1d 4101 . . . . 5 (𝜑 → (𝐴 ∖ {∅}) = (ran (𝑚𝐶 ↦ (𝐹𝑚)) ∖ {∅}))
10193, 92, 100f1oeq123d 6606 . . . 4 (𝜑 → ((𝐹 ↾ (𝐶𝑍)):(𝐶𝑍)–1-1-onto→(𝐴 ∖ {∅}) ↔ ((𝑛𝐶 ↦ (𝐹𝑛)) ↾ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}):{𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}–1-1-onto→(ran (𝑚𝐶 ↦ (𝐹𝑚)) ∖ {∅})))
10240, 101mpbird 258 . . 3 (𝜑 → (𝐹 ↾ (𝐶𝑍)):(𝐶𝑍)–1-1-onto→(𝐴 ∖ {∅}))
103 fvres 6685 . . . . 5 (𝑛 ∈ (𝐶𝑍) → ((𝐹 ↾ (𝐶𝑍))‘𝑛) = (𝐹𝑛))
104103adantl 482 . . . 4 ((𝜑𝑛 ∈ (𝐶𝑍)) → ((𝐹 ↾ (𝐶𝑍))‘𝑛) = (𝐹𝑛))
105 simpl 483 . . . . 5 ((𝜑𝑛 ∈ (𝐶𝑍)) → 𝜑)
106 sge0fodjrnlem.fng . . . . 5 ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)
107105, 43, 106syl2anc 584 . . . 4 ((𝜑𝑛 ∈ (𝐶𝑍)) → (𝐹𝑛) = 𝐺)
108104, 107eqtrd 2860 . . 3 ((𝜑𝑛 ∈ (𝐶𝑍)) → ((𝐹 ↾ (𝐶𝑍))‘𝑛) = 𝐺)
1091, 25, 26, 28, 102, 108, 10sge0f1o 42532 . 2 (𝜑 → (Σ^‘(𝑘 ∈ (𝐴 ∖ {∅}) ↦ 𝐵)) = (Σ^‘(𝑛 ∈ (𝐶𝑍) ↦ 𝐷)))
110106eqcomd 2831 . . . . . 6 ((𝜑𝑛𝐶) → 𝐺 = (𝐹𝑛))
111110, 32eqeltrd 2917 . . . . 5 ((𝜑𝑛𝐶) → 𝐺𝐴)
112105, 43, 111syl2anc 584 . . . 4 ((𝜑𝑛 ∈ (𝐶𝑍)) → 𝐺𝐴)
113112ex 413 . . . . 5 (𝜑 → (𝑛 ∈ (𝐶𝑍) → 𝐺𝐴))
114113imdistani 569 . . . 4 ((𝜑𝑛 ∈ (𝐶𝑍)) → (𝜑𝐺𝐴))
115 nfcv 2981 . . . . 5 𝑘𝐺
116 nfv 1908 . . . . . . 7 𝑘 𝐺𝐴
1171, 116nfan 1893 . . . . . 6 𝑘(𝜑𝐺𝐴)
118 nfv 1908 . . . . . 6 𝑘 𝐷 ∈ (0[,]+∞)
119117, 118nfim 1890 . . . . 5 𝑘((𝜑𝐺𝐴) → 𝐷 ∈ (0[,]+∞))
120 eleq1 2904 . . . . . . 7 (𝑘 = 𝐺 → (𝑘𝐴𝐺𝐴))
121120anbi2d 628 . . . . . 6 (𝑘 = 𝐺 → ((𝜑𝑘𝐴) ↔ (𝜑𝐺𝐴)))
12226eleq1d 2901 . . . . . 6 (𝑘 = 𝐺 → (𝐵 ∈ (0[,]+∞) ↔ 𝐷 ∈ (0[,]+∞)))
123121, 122imbi12d 346 . . . . 5 (𝑘 = 𝐺 → (((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞)) ↔ ((𝜑𝐺𝐴) → 𝐷 ∈ (0[,]+∞))))
124115, 119, 123, 9vtoclgf 3570 . . . 4 (𝐺𝐴 → ((𝜑𝐺𝐴) → 𝐷 ∈ (0[,]+∞)))
125112, 114, 124sylc 65 . . 3 ((𝜑𝑛 ∈ (𝐶𝑍)) → 𝐷 ∈ (0[,]+∞))
126 simpl 483 . . . . 5 ((𝜑𝑛 ∈ (𝐶 ∖ (𝐶𝑍))) → 𝜑)
127 eldifi 4106 . . . . . 6 (𝑛 ∈ (𝐶 ∖ (𝐶𝑍)) → 𝑛𝐶)
128127adantl 482 . . . . 5 ((𝜑𝑛 ∈ (𝐶 ∖ (𝐶𝑍))) → 𝑛𝐶)
129126, 128, 111syl2anc 584 . . . 4 ((𝜑𝑛 ∈ (𝐶 ∖ (𝐶𝑍))) → 𝐺𝐴)
130 dfin4 4247 . . . . . . . . 9 (𝑍𝐶) = (𝑍 ∖ (𝑍𝐶))
131 difss 4111 . . . . . . . . 9 (𝑍 ∖ (𝑍𝐶)) ⊆ 𝑍
132130, 131eqsstri 4004 . . . . . . . 8 (𝑍𝐶) ⊆ 𝑍
133 inss2 4209 . . . . . . . . . 10 (𝐶𝑍) ⊆ 𝑍
134 id 22 . . . . . . . . . . 11 (𝑛 ∈ (𝐶 ∖ (𝐶𝑍)) → 𝑛 ∈ (𝐶 ∖ (𝐶𝑍)))
135 dfin4 4247 . . . . . . . . . . . 12 (𝐶𝑍) = (𝐶 ∖ (𝐶𝑍))
136135eqcomi 2834 . . . . . . . . . . 11 (𝐶 ∖ (𝐶𝑍)) = (𝐶𝑍)
137134, 136syl6eleq 2927 . . . . . . . . . 10 (𝑛 ∈ (𝐶 ∖ (𝐶𝑍)) → 𝑛 ∈ (𝐶𝑍))
138133, 137sseldi 3968 . . . . . . . . 9 (𝑛 ∈ (𝐶 ∖ (𝐶𝑍)) → 𝑛𝑍)
139138, 127elind 4174 . . . . . . . 8 (𝑛 ∈ (𝐶 ∖ (𝐶𝑍)) → 𝑛 ∈ (𝑍𝐶))
140132, 139sseldi 3968 . . . . . . 7 (𝑛 ∈ (𝐶 ∖ (𝐶𝑍)) → 𝑛𝑍)
141140adantl 482 . . . . . 6 ((𝜑𝑛 ∈ (𝐶 ∖ (𝐶𝑍))) → 𝑛𝑍)
14277eqcomd 2831 . . . . . . 7 ((𝜑𝑛𝑍) → ∅ = (𝐹𝑛))
143 simpl 483 . . . . . . . 8 ((𝜑𝑛𝑍) → 𝜑)
14474simpld 495 . . . . . . . 8 ((𝜑𝑛𝑍) → 𝑛𝐶)
145143, 144, 106syl2anc 584 . . . . . . 7 ((𝜑𝑛𝑍) → (𝐹𝑛) = 𝐺)
146142, 145eqtr2d 2861 . . . . . 6 ((𝜑𝑛𝑍) → 𝐺 = ∅)
147126, 141, 146syl2anc 584 . . . . 5 ((𝜑𝑛 ∈ (𝐶 ∖ (𝐶𝑍))) → 𝐺 = ∅)
148126, 147jca 512 . . . 4 ((𝜑𝑛 ∈ (𝐶 ∖ (𝐶𝑍))) → (𝜑𝐺 = ∅))
149 nfv 1908 . . . . . . 7 𝑘 𝐺 = ∅
1501, 149nfan 1893 . . . . . 6 𝑘(𝜑𝐺 = ∅)
151 nfv 1908 . . . . . 6 𝑘 𝐷 = 0
152150, 151nfim 1890 . . . . 5 𝑘((𝜑𝐺 = ∅) → 𝐷 = 0)
153 eqeq1 2829 . . . . . . 7 (𝑘 = 𝐺 → (𝑘 = ∅ ↔ 𝐺 = ∅))
154153anbi2d 628 . . . . . 6 (𝑘 = 𝐺 → ((𝜑𝑘 = ∅) ↔ (𝜑𝐺 = ∅)))
15526eqeq1d 2827 . . . . . 6 (𝑘 = 𝐺 → (𝐵 = 0 ↔ 𝐷 = 0))
156154, 155imbi12d 346 . . . . 5 (𝑘 = 𝐺 → (((𝜑𝑘 = ∅) → 𝐵 = 0) ↔ ((𝜑𝐺 = ∅) → 𝐷 = 0)))
157115, 152, 156, 21vtoclgf 3570 . . . 4 (𝐺𝐴 → ((𝜑𝐺 = ∅) → 𝐷 = 0))
158129, 148, 157sylc 65 . . 3 ((𝜑𝑛 ∈ (𝐶 ∖ (𝐶𝑍))) → 𝐷 = 0)
15925, 2, 42, 125, 158sge0ss 42562 . 2 (𝜑 → (Σ^‘(𝑛 ∈ (𝐶𝑍) ↦ 𝐷)) = (Σ^‘(𝑛𝐶𝐷)))
16024, 109, 1593eqtrd 2864 1 (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wal 1528   = wceq 1530  wnf 1777  wcel 2107  wne 3020  wral 3142  {crab 3146  Vcvv 3499  cdif 3936  cin 3938  wss 3939  c0 4294  {csn 4563  Disj wdisj 5027  cmpt 5142  ccnv 5552  ran crn 5554  cres 5555  cima 5556   Fn wfn 6346  wf 6347  ontowfo 6349  1-1-ontowf1o 6350  cfv 6351  (class class class)co 7151  0cc0 10529  +∞cpnf 10664  [,]cicc 12734  Σ^csumge0 42512
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-13 2385  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-disj 5028  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-xadd 12501  df-ico 12737  df-icc 12738  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  df-sumge0 42513
This theorem is referenced by:  sge0fodjrn  42567
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