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Theorem sge0fodjrnlem 39927
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 7085 . . . . 5 (𝐶𝑉 → (𝐹:𝐶onto𝐴𝐴 ∈ V))
52, 3, 4sylc 65 . . . 4 (𝜑𝐴 ∈ V)
6 difssd 3721 . . . 4 (𝜑 → (𝐴 ∖ {∅}) ⊆ 𝐴)
7 simpl 473 . . . . 5 ((𝜑𝑘 ∈ (𝐴 ∖ {∅})) → 𝜑)
86sselda 3588 . . . . 5 ((𝜑𝑘 ∈ (𝐴 ∖ {∅})) → 𝑘𝐴)
9 sge0fodjrnlem.b . . . . 5 ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))
107, 8, 9syl2anc 692 . . . 4 ((𝜑𝑘 ∈ (𝐴 ∖ {∅})) → 𝐵 ∈ (0[,]+∞))
11 simpl 473 . . . . 5 ((𝜑𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅}))) → 𝜑)
12 dfin4 3848 . . . . . . . . . 10 (𝐴 ∩ {∅}) = (𝐴 ∖ (𝐴 ∖ {∅}))
1312eqcomi 2635 . . . . . . . . 9 (𝐴 ∖ (𝐴 ∖ {∅})) = (𝐴 ∩ {∅})
14 inss2 3817 . . . . . . . . 9 (𝐴 ∩ {∅}) ⊆ {∅}
1513, 14eqsstri 3619 . . . . . . . 8 (𝐴 ∖ (𝐴 ∖ {∅})) ⊆ {∅}
16 id 22 . . . . . . . 8 (𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅})) → 𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅})))
1715, 16sseldi 3586 . . . . . . 7 (𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅})) → 𝑘 ∈ {∅})
18 elsni 4170 . . . . . . 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 692 . . . 4 ((𝜑𝑘 ∈ (𝐴 ∖ (𝐴 ∖ {∅}))) → 𝐵 = 0)
231, 5, 6, 10, 22sge0ss 39923 . . 3 (𝜑 → (Σ^‘(𝑘 ∈ (𝐴 ∖ {∅}) ↦ 𝐵)) = (Σ^‘(𝑘𝐴𝐵)))
2423eqcomd 2632 . 2 (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑘 ∈ (𝐴 ∖ {∅}) ↦ 𝐵)))
25 sge0fodjrnlem.n . . 3 𝑛𝜑
26 sge0fodjrnlem.bd . . 3 (𝑘 = 𝐺𝐵 = 𝐷)
27 difexg 4773 . . . 4 (𝐶𝑉 → (𝐶𝑍) ∈ V)
282, 27syl 17 . . 3 (𝜑 → (𝐶𝑍) ∈ V)
29 eqid 2626 . . . . 5 (𝑛𝐶 ↦ (𝐹𝑛)) = (𝑛𝐶 ↦ (𝐹𝑛))
30 fof 6074 . . . . . . 7 (𝐹:𝐶onto𝐴𝐹:𝐶𝐴)
313, 30syl 17 . . . . . 6 (𝜑𝐹:𝐶𝐴)
3231ffvelrnda 6316 . . . . 5 ((𝜑𝑛𝐶) → (𝐹𝑛) ∈ 𝐴)
33 sge0fodjrnlem.dj . . . . 5 (𝜑Disj 𝑛𝐶 (𝐹𝑛))
34 fveq2 6150 . . . . . . 7 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
3534neeq1d 2855 . . . . . 6 (𝑚 = 𝑛 → ((𝐹𝑚) ≠ ∅ ↔ (𝐹𝑛) ≠ ∅))
3635cbvrabv 3190 . . . . 5 {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅} = {𝑛𝐶 ∣ (𝐹𝑛) ≠ ∅}
3734cbvmptv 4715 . . . . . . 7 (𝑚𝐶 ↦ (𝐹𝑚)) = (𝑛𝐶 ↦ (𝐹𝑛))
3837rneqi 5316 . . . . . 6 ran (𝑚𝐶 ↦ (𝐹𝑚)) = ran (𝑛𝐶 ↦ (𝐹𝑛))
3938difeq1i 3707 . . . . 5 (ran (𝑚𝐶 ↦ (𝐹𝑚)) ∖ {∅}) = (ran (𝑛𝐶 ↦ (𝐹𝑛)) ∖ {∅})
4025, 29, 32, 33, 36, 39disjf1o 38838 . . . 4 (𝜑 → ((𝑛𝐶 ↦ (𝐹𝑛)) ↾ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}):{𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}–1-1-onto→(ran (𝑚𝐶 ↦ (𝐹𝑚)) ∖ {∅}))
4131feqmptd 6207 . . . . . 6 (𝜑𝐹 = (𝑛𝐶 ↦ (𝐹𝑛)))
42 difssd 3721 . . . . . . . . . . . . 13 (𝜑 → (𝐶𝑍) ⊆ 𝐶)
4342sselda 3588 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝐶𝑍)) → 𝑛𝐶)
44 eldifi 3715 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (𝐶𝑍) → 𝑛𝐶)
4544adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ (𝐶𝑍) ∧ (𝐹𝑛) = ∅) → 𝑛𝐶)
46 id 22 . . . . . . . . . . . . . . . . . . . 20 ((𝐹𝑛) = ∅ → (𝐹𝑛) = ∅)
47 fvex 6160 . . . . . . . . . . . . . . . . . . . . 21 (𝐹𝑛) ∈ V
4847elsn 4168 . . . . . . . . . . . . . . . . . . . 20 ((𝐹𝑛) ∈ {∅} ↔ (𝐹𝑛) = ∅)
4946, 48sylibr 224 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑛) = ∅ → (𝐹𝑛) ∈ {∅})
5049adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ (𝐶𝑍) ∧ (𝐹𝑛) = ∅) → (𝐹𝑛) ∈ {∅})
5145, 50jca 554 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (𝐶𝑍) ∧ (𝐹𝑛) = ∅) → (𝑛𝐶 ∧ (𝐹𝑛) ∈ {∅}))
5251adantll 749 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝐶𝑍)) ∧ (𝐹𝑛) = ∅) → (𝑛𝐶 ∧ (𝐹𝑛) ∈ {∅}))
5331ffnd 6005 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹 Fn 𝐶)
54 elpreima 6294 . . . . . . . . . . . . . . . . . 18 (𝐹 Fn 𝐶 → (𝑛 ∈ (𝐹 “ {∅}) ↔ (𝑛𝐶 ∧ (𝐹𝑛) ∈ {∅})))
5553, 54syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑛 ∈ (𝐹 “ {∅}) ↔ (𝑛𝐶 ∧ (𝐹𝑛) ∈ {∅})))
5655ad2antrr 761 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ (𝐶𝑍)) ∧ (𝐹𝑛) = ∅) → (𝑛 ∈ (𝐹 “ {∅}) ↔ (𝑛𝐶 ∧ (𝐹𝑛) ∈ {∅})))
5752, 56mpbird 247 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ (𝐶𝑍)) ∧ (𝐹𝑛) = ∅) → 𝑛 ∈ (𝐹 “ {∅}))
58 sge0fodjrnlem.z . . . . . . . . . . . . . . 15 𝑍 = (𝐹 “ {∅})
5957, 58syl6eleqr 2715 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (𝐶𝑍)) ∧ (𝐹𝑛) = ∅) → 𝑛𝑍)
60 eldifn 3716 . . . . . . . . . . . . . . 15 (𝑛 ∈ (𝐶𝑍) → ¬ 𝑛𝑍)
6160ad2antlr 762 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ (𝐶𝑍)) ∧ (𝐹𝑛) = ∅) → ¬ 𝑛𝑍)
6259, 61pm2.65da 599 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝐶𝑍)) → ¬ (𝐹𝑛) = ∅)
6362neqned 2803 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝐶𝑍)) → (𝐹𝑛) ≠ ∅)
6443, 63jca 554 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝐶𝑍)) → (𝑛𝐶 ∧ (𝐹𝑛) ≠ ∅))
6535elrab 3351 . . . . . . . . . . 11 (𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅} ↔ (𝑛𝐶 ∧ (𝐹𝑛) ≠ ∅))
6664, 65sylibr 224 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝐶𝑍)) → 𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅})
6766ex 450 . . . . . . . . 9 (𝜑 → (𝑛 ∈ (𝐶𝑍) → 𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}))
6865simplbi 476 . . . . . . . . . . . . . . 15 (𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅} → 𝑛𝐶)
6968adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}) → 𝑛𝐶)
7058eleq2i 2696 . . . . . . . . . . . . . . . . . . . . 21 (𝑛𝑍𝑛 ∈ (𝐹 “ {∅}))
7170biimpi 206 . . . . . . . . . . . . . . . . . . . 20 (𝑛𝑍𝑛 ∈ (𝐹 “ {∅}))
7271adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛𝑍) → 𝑛 ∈ (𝐹 “ {∅}))
7355adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛𝑍) → (𝑛 ∈ (𝐹 “ {∅}) ↔ (𝑛𝐶 ∧ (𝐹𝑛) ∈ {∅})))
7472, 73mpbid 222 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛𝑍) → (𝑛𝐶 ∧ (𝐹𝑛) ∈ {∅}))
7574simprd 479 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ {∅})
76 elsni 4170 . . . . . . . . . . . . . . . . 17 ((𝐹𝑛) ∈ {∅} → (𝐹𝑛) = ∅)
7775, 76syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑛𝑍) → (𝐹𝑛) = ∅)
7877adantlr 750 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}) ∧ 𝑛𝑍) → (𝐹𝑛) = ∅)
7965simprbi 480 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅} → (𝐹𝑛) ≠ ∅)
8079ad2antlr 762 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}) ∧ 𝑛𝑍) → (𝐹𝑛) ≠ ∅)
8180neneqd 2801 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}) ∧ 𝑛𝑍) → ¬ (𝐹𝑛) = ∅)
8278, 81pm2.65da 599 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}) → ¬ 𝑛𝑍)
8369, 82eldifd 3571 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}) → 𝑛 ∈ (𝐶𝑍))
8483ex 450 . . . . . . . . . . . 12 (𝜑 → (𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅} → 𝑛 ∈ (𝐶𝑍)))
8525, 84ralrimi 2956 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}𝑛 ∈ (𝐶𝑍))
86 dfss3 3578 . . . . . . . . . . 11 ({𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅} ⊆ (𝐶𝑍) ↔ ∀𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}𝑛 ∈ (𝐶𝑍))
8785, 86sylibr 224 . . . . . . . . . 10 (𝜑 → {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅} ⊆ (𝐶𝑍))
8887sseld 3587 . . . . . . . . 9 (𝜑 → (𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅} → 𝑛 ∈ (𝐶𝑍)))
8967, 88impbid 202 . . . . . . . 8 (𝜑 → (𝑛 ∈ (𝐶𝑍) ↔ 𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}))
9025, 89alrimi 2085 . . . . . . 7 (𝜑 → ∀𝑛(𝑛 ∈ (𝐶𝑍) ↔ 𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}))
91 dfcleq 2620 . . . . . . 7 ((𝐶𝑍) = {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅} ↔ ∀𝑛(𝑛 ∈ (𝐶𝑍) ↔ 𝑛 ∈ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}))
9290, 91sylibr 224 . . . . . 6 (𝜑 → (𝐶𝑍) = {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅})
9341, 92reseq12d 5361 . . . . 5 (𝜑 → (𝐹 ↾ (𝐶𝑍)) = ((𝑛𝐶 ↦ (𝐹𝑛)) ↾ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}))
9441, 37syl6eqr 2678 . . . . . . . . 9 (𝜑𝐹 = (𝑚𝐶 ↦ (𝐹𝑚)))
9594eqcomd 2632 . . . . . . . 8 (𝜑 → (𝑚𝐶 ↦ (𝐹𝑚)) = 𝐹)
9695rneqd 5317 . . . . . . 7 (𝜑 → ran (𝑚𝐶 ↦ (𝐹𝑚)) = ran 𝐹)
97 forn 6077 . . . . . . . 8 (𝐹:𝐶onto𝐴 → ran 𝐹 = 𝐴)
983, 97syl 17 . . . . . . 7 (𝜑 → ran 𝐹 = 𝐴)
9996, 98eqtr2d 2661 . . . . . 6 (𝜑𝐴 = ran (𝑚𝐶 ↦ (𝐹𝑚)))
10099difeq1d 3710 . . . . 5 (𝜑 → (𝐴 ∖ {∅}) = (ran (𝑚𝐶 ↦ (𝐹𝑚)) ∖ {∅}))
10193, 92, 100f1oeq123d 6092 . . . 4 (𝜑 → ((𝐹 ↾ (𝐶𝑍)):(𝐶𝑍)–1-1-onto→(𝐴 ∖ {∅}) ↔ ((𝑛𝐶 ↦ (𝐹𝑛)) ↾ {𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}):{𝑚𝐶 ∣ (𝐹𝑚) ≠ ∅}–1-1-onto→(ran (𝑚𝐶 ↦ (𝐹𝑚)) ∖ {∅})))
10240, 101mpbird 247 . . 3 (𝜑 → (𝐹 ↾ (𝐶𝑍)):(𝐶𝑍)–1-1-onto→(𝐴 ∖ {∅}))
103 fvres 6165 . . . . 5 (𝑛 ∈ (𝐶𝑍) → ((𝐹 ↾ (𝐶𝑍))‘𝑛) = (𝐹𝑛))
104103adantl 482 . . . 4 ((𝜑𝑛 ∈ (𝐶𝑍)) → ((𝐹 ↾ (𝐶𝑍))‘𝑛) = (𝐹𝑛))
105 simpl 473 . . . . 5 ((𝜑𝑛 ∈ (𝐶𝑍)) → 𝜑)
106 sge0fodjrnlem.fng . . . . 5 ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)
107105, 43, 106syl2anc 692 . . . 4 ((𝜑𝑛 ∈ (𝐶𝑍)) → (𝐹𝑛) = 𝐺)
108104, 107eqtrd 2660 . . 3 ((𝜑𝑛 ∈ (𝐶𝑍)) → ((𝐹 ↾ (𝐶𝑍))‘𝑛) = 𝐺)
1091, 25, 26, 28, 102, 108, 10sge0f1o 39893 . 2 (𝜑 → (Σ^‘(𝑘 ∈ (𝐴 ∖ {∅}) ↦ 𝐵)) = (Σ^‘(𝑛 ∈ (𝐶𝑍) ↦ 𝐷)))
110106eqcomd 2632 . . . . . 6 ((𝜑𝑛𝐶) → 𝐺 = (𝐹𝑛))
111110, 32eqeltrd 2704 . . . . 5 ((𝜑𝑛𝐶) → 𝐺𝐴)
112105, 43, 111syl2anc 692 . . . 4 ((𝜑𝑛 ∈ (𝐶𝑍)) → 𝐺𝐴)
113112ex 450 . . . . 5 (𝜑 → (𝑛 ∈ (𝐶𝑍) → 𝐺𝐴))
114113imdistani 725 . . . 4 ((𝜑𝑛 ∈ (𝐶𝑍)) → (𝜑𝐺𝐴))
115 nfcv 2767 . . . . 5 𝑘𝐺
116 nfv 1845 . . . . . . 7 𝑘 𝐺𝐴
1171, 116nfan 1830 . . . . . 6 𝑘(𝜑𝐺𝐴)
118 nfv 1845 . . . . . 6 𝑘 𝐷 ∈ (0[,]+∞)
119117, 118nfim 1827 . . . . 5 𝑘((𝜑𝐺𝐴) → 𝐷 ∈ (0[,]+∞))
120 eleq1 2692 . . . . . . 7 (𝑘 = 𝐺 → (𝑘𝐴𝐺𝐴))
121120anbi2d 739 . . . . . 6 (𝑘 = 𝐺 → ((𝜑𝑘𝐴) ↔ (𝜑𝐺𝐴)))
12226eleq1d 2688 . . . . . 6 (𝑘 = 𝐺 → (𝐵 ∈ (0[,]+∞) ↔ 𝐷 ∈ (0[,]+∞)))
123121, 122imbi12d 334 . . . . 5 (𝑘 = 𝐺 → (((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞)) ↔ ((𝜑𝐺𝐴) → 𝐷 ∈ (0[,]+∞))))
124115, 119, 123, 9vtoclgf 3255 . . . 4 (𝐺𝐴 → ((𝜑𝐺𝐴) → 𝐷 ∈ (0[,]+∞)))
125112, 114, 124sylc 65 . . 3 ((𝜑𝑛 ∈ (𝐶𝑍)) → 𝐷 ∈ (0[,]+∞))
126 simpl 473 . . . . 5 ((𝜑𝑛 ∈ (𝐶 ∖ (𝐶𝑍))) → 𝜑)
127 eldifi 3715 . . . . . 6 (𝑛 ∈ (𝐶 ∖ (𝐶𝑍)) → 𝑛𝐶)
128127adantl 482 . . . . 5 ((𝜑𝑛 ∈ (𝐶 ∖ (𝐶𝑍))) → 𝑛𝐶)
129126, 128, 111syl2anc 692 . . . 4 ((𝜑𝑛 ∈ (𝐶 ∖ (𝐶𝑍))) → 𝐺𝐴)
130 dfin4 3848 . . . . . . . . 9 (𝑍𝐶) = (𝑍 ∖ (𝑍𝐶))
131 difss 3720 . . . . . . . . 9 (𝑍 ∖ (𝑍𝐶)) ⊆ 𝑍
132130, 131eqsstri 3619 . . . . . . . 8 (𝑍𝐶) ⊆ 𝑍
133 inss2 3817 . . . . . . . . . 10 (𝐶𝑍) ⊆ 𝑍
134 id 22 . . . . . . . . . . 11 (𝑛 ∈ (𝐶 ∖ (𝐶𝑍)) → 𝑛 ∈ (𝐶 ∖ (𝐶𝑍)))
135 dfin4 3848 . . . . . . . . . . . 12 (𝐶𝑍) = (𝐶 ∖ (𝐶𝑍))
136135eqcomi 2635 . . . . . . . . . . 11 (𝐶 ∖ (𝐶𝑍)) = (𝐶𝑍)
137134, 136syl6eleq 2714 . . . . . . . . . 10 (𝑛 ∈ (𝐶 ∖ (𝐶𝑍)) → 𝑛 ∈ (𝐶𝑍))
138133, 137sseldi 3586 . . . . . . . . 9 (𝑛 ∈ (𝐶 ∖ (𝐶𝑍)) → 𝑛𝑍)
139138, 127elind 3781 . . . . . . . 8 (𝑛 ∈ (𝐶 ∖ (𝐶𝑍)) → 𝑛 ∈ (𝑍𝐶))
140132, 139sseldi 3586 . . . . . . 7 (𝑛 ∈ (𝐶 ∖ (𝐶𝑍)) → 𝑛𝑍)
141140adantl 482 . . . . . 6 ((𝜑𝑛 ∈ (𝐶 ∖ (𝐶𝑍))) → 𝑛𝑍)
14277eqcomd 2632 . . . . . . 7 ((𝜑𝑛𝑍) → ∅ = (𝐹𝑛))
143 simpl 473 . . . . . . . 8 ((𝜑𝑛𝑍) → 𝜑)
14474simpld 475 . . . . . . . 8 ((𝜑𝑛𝑍) → 𝑛𝐶)
145143, 144, 106syl2anc 692 . . . . . . 7 ((𝜑𝑛𝑍) → (𝐹𝑛) = 𝐺)
146142, 145eqtr2d 2661 . . . . . 6 ((𝜑𝑛𝑍) → 𝐺 = ∅)
147126, 141, 146syl2anc 692 . . . . 5 ((𝜑𝑛 ∈ (𝐶 ∖ (𝐶𝑍))) → 𝐺 = ∅)
148126, 147jca 554 . . . 4 ((𝜑𝑛 ∈ (𝐶 ∖ (𝐶𝑍))) → (𝜑𝐺 = ∅))
149 nfv 1845 . . . . . . 7 𝑘 𝐺 = ∅
1501, 149nfan 1830 . . . . . 6 𝑘(𝜑𝐺 = ∅)
151 nfv 1845 . . . . . 6 𝑘 𝐷 = 0
152150, 151nfim 1827 . . . . 5 𝑘((𝜑𝐺 = ∅) → 𝐷 = 0)
153 eqeq1 2630 . . . . . . 7 (𝑘 = 𝐺 → (𝑘 = ∅ ↔ 𝐺 = ∅))
154153anbi2d 739 . . . . . 6 (𝑘 = 𝐺 → ((𝜑𝑘 = ∅) ↔ (𝜑𝐺 = ∅)))
15526eqeq1d 2628 . . . . . 6 (𝑘 = 𝐺 → (𝐵 = 0 ↔ 𝐷 = 0))
156154, 155imbi12d 334 . . . . 5 (𝑘 = 𝐺 → (((𝜑𝑘 = ∅) → 𝐵 = 0) ↔ ((𝜑𝐺 = ∅) → 𝐷 = 0)))
157115, 152, 156, 21vtoclgf 3255 . . . 4 (𝐺𝐴 → ((𝜑𝐺 = ∅) → 𝐷 = 0))
158129, 148, 157sylc 65 . . 3 ((𝜑𝑛 ∈ (𝐶 ∖ (𝐶𝑍))) → 𝐷 = 0)
15925, 2, 42, 125, 158sge0ss 39923 . 2 (𝜑 → (Σ^‘(𝑛 ∈ (𝐶𝑍) ↦ 𝐷)) = (Σ^‘(𝑛𝐶𝐷)))
16024, 109, 1593eqtrd 2664 1 (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wnf 1705  wcel 1992  wne 2796  wral 2912  {crab 2916  Vcvv 3191  cdif 3557  cin 3559  wss 3560  c0 3896  {csn 4153  Disj wdisj 4588  cmpt 4678  ccnv 5078  ran crn 5080  cres 5081  cima 5082   Fn wfn 5845  wf 5846  ontowfo 5848  1-1-ontowf1o 5849  cfv 5850  (class class class)co 6605  0cc0 9881  +∞cpnf 10016  [,]cicc 12117  Σ^csumge0 39873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-disj 4589  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-sup 8293  df-oi 8360  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-2 11024  df-3 11025  df-n0 11238  df-z 11323  df-uz 11632  df-rp 11777  df-xadd 11891  df-ico 12120  df-icc 12121  df-fz 12266  df-fzo 12404  df-seq 12739  df-exp 12798  df-hash 13055  df-cj 13768  df-re 13769  df-im 13770  df-sqrt 13904  df-abs 13905  df-clim 14148  df-sum 14346  df-sumge0 39874
This theorem is referenced by:  sge0fodjrn  39928
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