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Theorem carsggect 32918
Description: The outer measure is countably superadditive on Caratheodory measurable sets. (Contributed by Thierry Arnoux, 31-May-2020.)
Hypotheses
Ref Expression
carsgval.1 (𝜑𝑂𝑉)
carsgval.2 (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))
carsgsiga.1 (𝜑 → (𝑀‘∅) = 0)
carsgsiga.2 ((𝜑𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))
carsggect.0 (𝜑 → ¬ ∅ ∈ 𝐴)
carsggect.1 (𝜑𝐴 ≼ ω)
carsggect.2 (𝜑𝐴 ⊆ (toCaraSiga‘𝑀))
carsggect.3 (𝜑Disj 𝑦𝐴 𝑦)
carsggect.4 ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))
Assertion
Ref Expression
carsggect (𝜑 → Σ*𝑧𝐴(𝑀𝑧) ≤ (𝑀 𝐴))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑀,𝑦   𝑥,𝑂,𝑦   𝜑,𝑥,𝑦   𝑧,𝐴   𝑧,𝑀   𝑧,𝑂,𝑥,𝑦   𝜑,𝑧
Allowed substitution hints:   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem carsggect
Dummy variables 𝑓 𝑘 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 carsggect.1 . . 3 (𝜑𝐴 ≼ ω)
2 0ex 5264 . . . 4 ∅ ∈ V
32a1i 11 . . 3 (𝜑 → ∅ ∈ V)
4 carsggect.0 . . 3 (𝜑 → ¬ ∅ ∈ 𝐴)
5 padct 31636 . . 3 ((𝐴 ≼ ω ∧ ∅ ∈ V ∧ ¬ ∅ ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
61, 3, 4, 5syl3anc 1371 . 2 (𝜑 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
7 nfv 1917 . . . . 5 𝑧(𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
8 simpr1 1194 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝑓:ℕ⟶(𝐴 ∪ {∅}))
98feqmptd 6910 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝑓 = (𝑘 ∈ ℕ ↦ (𝑓𝑘)))
109rneqd 5893 . . . . 5 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran 𝑓 = ran (𝑘 ∈ ℕ ↦ (𝑓𝑘)))
117, 10esumeq1d 32634 . . . 4 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) = Σ*𝑧 ∈ ran (𝑘 ∈ ℕ ↦ (𝑓𝑘))(𝑀𝑧))
12 fvex 6855 . . . . . . . . . 10 (toCaraSiga‘𝑀) ∈ V
1312a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (toCaraSiga‘𝑀) ∈ V)
14 carsggect.2 . . . . . . . . . . 11 (𝜑𝐴 ⊆ (toCaraSiga‘𝑀))
1514adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝐴 ⊆ (toCaraSiga‘𝑀))
16 carsgval.1 . . . . . . . . . . . . 13 (𝜑𝑂𝑉)
1716adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝑂𝑉)
18 carsgval.2 . . . . . . . . . . . . 13 (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))
1918adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
20 carsgsiga.1 . . . . . . . . . . . . 13 (𝜑 → (𝑀‘∅) = 0)
2120adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑀‘∅) = 0)
2217, 19, 210elcarsg 32907 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ∅ ∈ (toCaraSiga‘𝑀))
2322snssd 4769 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → {∅} ⊆ (toCaraSiga‘𝑀))
2415, 23unssd 4146 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝐴 ∪ {∅}) ⊆ (toCaraSiga‘𝑀))
2513, 24ssexd 5281 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝐴 ∪ {∅}) ∈ V)
2619adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
2716, 18carsgcl 32904 . . . . . . . . . . . . 13 (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂)
2814, 27sstrd 3954 . . . . . . . . . . . 12 (𝜑𝐴 ⊆ 𝒫 𝑂)
2928adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝐴 ⊆ 𝒫 𝑂)
30 0elpw 5311 . . . . . . . . . . . . 13 ∅ ∈ 𝒫 𝑂
3130a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ∅ ∈ 𝒫 𝑂)
3231snssd 4769 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → {∅} ⊆ 𝒫 𝑂)
3329, 32unssd 4146 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
3433sselda 3944 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → 𝑧 ∈ 𝒫 𝑂)
3526, 34ffvelcdmd 7036 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → (𝑀𝑧) ∈ (0[,]+∞))
368frnd 6676 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran 𝑓 ⊆ (𝐴 ∪ {∅}))
377, 25, 35, 36esummono 32653 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) ≤ Σ*𝑧 ∈ (𝐴 ∪ {∅})(𝑀𝑧))
38 ctex 8903 . . . . . . . . . 10 (𝐴 ≼ ω → 𝐴 ∈ V)
391, 38syl 17 . . . . . . . . 9 (𝜑𝐴 ∈ V)
4039adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝐴 ∈ V)
4113, 23ssexd 5281 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → {∅} ∈ V)
4219adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧𝐴) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
4329sselda 3944 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧𝐴) → 𝑧 ∈ 𝒫 𝑂)
4442, 43ffvelcdmd 7036 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧𝐴) → (𝑀𝑧) ∈ (0[,]+∞))
45 elsni 4603 . . . . . . . . . . 11 (𝑧 ∈ {∅} → 𝑧 = ∅)
4645adantl 482 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ {∅}) → 𝑧 = ∅)
4746fveq2d 6846 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ {∅}) → (𝑀𝑧) = (𝑀‘∅))
4821adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ {∅}) → (𝑀‘∅) = 0)
4947, 48eqtrd 2776 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ {∅}) → (𝑀𝑧) = 0)
5040, 41, 44, 49esumpad 32654 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ (𝐴 ∪ {∅})(𝑀𝑧) = Σ*𝑧𝐴(𝑀𝑧))
5137, 50breqtrd 5131 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) ≤ Σ*𝑧𝐴(𝑀𝑧))
5236, 24sstrd 3954 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran 𝑓 ⊆ (toCaraSiga‘𝑀))
53 ssexg 5280 . . . . . . . 8 ((ran 𝑓 ⊆ (toCaraSiga‘𝑀) ∧ (toCaraSiga‘𝑀) ∈ V) → ran 𝑓 ∈ V)
5452, 12, 53sylancl 586 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran 𝑓 ∈ V)
5519adantr 481 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ ran 𝑓) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
5636, 33sstrd 3954 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran 𝑓 ⊆ 𝒫 𝑂)
5756sselda 3944 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ ran 𝑓) → 𝑧 ∈ 𝒫 𝑂)
5855, 57ffvelcdmd 7036 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ ran 𝑓) → (𝑀𝑧) ∈ (0[,]+∞))
59 simpr2 1195 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝐴 ⊆ ran 𝑓)
607, 54, 58, 59esummono 32653 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧𝐴(𝑀𝑧) ≤ Σ*𝑧 ∈ ran 𝑓(𝑀𝑧))
6151, 60jca 512 . . . . 5 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) ≤ Σ*𝑧𝐴(𝑀𝑧) ∧ Σ*𝑧𝐴(𝑀𝑧) ≤ Σ*𝑧 ∈ ran 𝑓(𝑀𝑧)))
62 iccssxr 13347 . . . . . . 7 (0[,]+∞) ⊆ ℝ*
6358ralrimiva 3143 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ∀𝑧 ∈ ran 𝑓(𝑀𝑧) ∈ (0[,]+∞))
64 nfcv 2907 . . . . . . . . 9 𝑧ran 𝑓
6564esumcl 32629 . . . . . . . 8 ((ran 𝑓 ∈ V ∧ ∀𝑧 ∈ ran 𝑓(𝑀𝑧) ∈ (0[,]+∞)) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) ∈ (0[,]+∞))
6654, 63, 65syl2anc 584 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) ∈ (0[,]+∞))
6762, 66sselid 3942 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) ∈ ℝ*)
6844ralrimiva 3143 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ∀𝑧𝐴 (𝑀𝑧) ∈ (0[,]+∞))
69 nfcv 2907 . . . . . . . . 9 𝑧𝐴
7069esumcl 32629 . . . . . . . 8 ((𝐴 ∈ V ∧ ∀𝑧𝐴 (𝑀𝑧) ∈ (0[,]+∞)) → Σ*𝑧𝐴(𝑀𝑧) ∈ (0[,]+∞))
7140, 68, 70syl2anc 584 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧𝐴(𝑀𝑧) ∈ (0[,]+∞))
7262, 71sselid 3942 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧𝐴(𝑀𝑧) ∈ ℝ*)
73 xrletri3 13073 . . . . . 6 ((Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) ∈ ℝ* ∧ Σ*𝑧𝐴(𝑀𝑧) ∈ ℝ*) → (Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) = Σ*𝑧𝐴(𝑀𝑧) ↔ (Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) ≤ Σ*𝑧𝐴(𝑀𝑧) ∧ Σ*𝑧𝐴(𝑀𝑧) ≤ Σ*𝑧 ∈ ran 𝑓(𝑀𝑧))))
7467, 72, 73syl2anc 584 . . . . 5 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) = Σ*𝑧𝐴(𝑀𝑧) ↔ (Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) ≤ Σ*𝑧𝐴(𝑀𝑧) ∧ Σ*𝑧𝐴(𝑀𝑧) ≤ Σ*𝑧 ∈ ran 𝑓(𝑀𝑧))))
7561, 74mpbird 256 . . . 4 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) = Σ*𝑧𝐴(𝑀𝑧))
76 fveq2 6842 . . . . 5 (𝑧 = (𝑓𝑘) → (𝑀𝑧) = (𝑀‘(𝑓𝑘)))
77 nnex 12159 . . . . . 6 ℕ ∈ V
7877a1i 11 . . . . 5 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ℕ ∈ V)
7919adantr 481 . . . . . 6 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
8033adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
818adantr 481 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → 𝑓:ℕ⟶(𝐴 ∪ {∅}))
82 simpr 485 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
8381, 82ffvelcdmd 7036 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ (𝐴 ∪ {∅}))
8480, 83sseldd 3945 . . . . . 6 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ 𝒫 𝑂)
8579, 84ffvelcdmd 7036 . . . . 5 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑀‘(𝑓𝑘)) ∈ (0[,]+∞))
86 simpr 485 . . . . . . 7 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓𝑘) = ∅) → (𝑓𝑘) = ∅)
8786fveq2d 6846 . . . . . 6 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓𝑘) = ∅) → (𝑀‘(𝑓𝑘)) = (𝑀‘∅))
8821ad2antrr 724 . . . . . 6 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓𝑘) = ∅) → (𝑀‘∅) = 0)
8987, 88eqtrd 2776 . . . . 5 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓𝑘) = ∅) → (𝑀‘(𝑓𝑘)) = 0)
90 cnvimass 6033 . . . . . . 7 (𝑓𝐴) ⊆ dom 𝑓
9190, 8fssdm 6688 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓𝐴) ⊆ ℕ)
92 ffun 6671 . . . . . . . . . . 11 (𝑓:ℕ⟶(𝐴 ∪ {∅}) → Fun 𝑓)
938, 92syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Fun 𝑓)
9493adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (𝑓𝐴))) → Fun 𝑓)
95 difpreima 7015 . . . . . . . . . . . . 13 (Fun 𝑓 → (𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) = ((𝑓 “ (𝐴 ∪ {∅})) ∖ (𝑓𝐴)))
968, 92, 953syl 18 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) = ((𝑓 “ (𝐴 ∪ {∅})) ∖ (𝑓𝐴)))
97 fimacnv 6690 . . . . . . . . . . . . . 14 (𝑓:ℕ⟶(𝐴 ∪ {∅}) → (𝑓 “ (𝐴 ∪ {∅})) = ℕ)
988, 97syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓 “ (𝐴 ∪ {∅})) = ℕ)
9998difeq1d 4081 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ((𝑓 “ (𝐴 ∪ {∅})) ∖ (𝑓𝐴)) = (ℕ ∖ (𝑓𝐴)))
10096, 99eqtrd 2776 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) = (ℕ ∖ (𝑓𝐴)))
101 uncom 4113 . . . . . . . . . . . . . . . 16 ({∅} ∪ 𝐴) = (𝐴 ∪ {∅})
102101difeq1i 4078 . . . . . . . . . . . . . . 15 (({∅} ∪ 𝐴) ∖ 𝐴) = ((𝐴 ∪ {∅}) ∖ 𝐴)
103 difun2 4440 . . . . . . . . . . . . . . 15 (({∅} ∪ 𝐴) ∖ 𝐴) = ({∅} ∖ 𝐴)
104102, 103eqtr3i 2766 . . . . . . . . . . . . . 14 ((𝐴 ∪ {∅}) ∖ 𝐴) = ({∅} ∖ 𝐴)
105 difss 4091 . . . . . . . . . . . . . 14 ({∅} ∖ 𝐴) ⊆ {∅}
106104, 105eqsstri 3978 . . . . . . . . . . . . 13 ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅}
107106a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅})
108 sspreima 7018 . . . . . . . . . . . 12 ((Fun 𝑓 ∧ ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅}) → (𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) ⊆ (𝑓 “ {∅}))
10993, 107, 108syl2anc 584 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) ⊆ (𝑓 “ {∅}))
110100, 109eqsstrrd 3983 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (ℕ ∖ (𝑓𝐴)) ⊆ (𝑓 “ {∅}))
111110sselda 3944 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (𝑓𝐴))) → 𝑘 ∈ (𝑓 “ {∅}))
112 fvimacnvi 7002 . . . . . . . . 9 ((Fun 𝑓𝑘 ∈ (𝑓 “ {∅})) → (𝑓𝑘) ∈ {∅})
11394, 111, 112syl2anc 584 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (𝑓𝐴))) → (𝑓𝑘) ∈ {∅})
114 elsni 4603 . . . . . . . 8 ((𝑓𝑘) ∈ {∅} → (𝑓𝑘) = ∅)
115113, 114syl 17 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (𝑓𝐴))) → (𝑓𝑘) = ∅)
116115ralrimiva 3143 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ∀𝑘 ∈ (ℕ ∖ (𝑓𝐴))(𝑓𝑘) = ∅)
117 carsggect.3 . . . . . . . 8 (𝜑Disj 𝑦𝐴 𝑦)
118117adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Disj 𝑦𝐴 𝑦)
119 simpr3 1196 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Fun (𝑓𝐴))
120 fresf1o 31545 . . . . . . . . . 10 ((Fun 𝑓𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto𝐴)
12193, 59, 119, 120syl3anc 1371 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto𝐴)
122 simpr 485 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑦 = ((𝑓 ↾ (𝑓𝐴))‘𝑘)) → 𝑦 = ((𝑓 ↾ (𝑓𝐴))‘𝑘))
123121, 122disjrdx 31509 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (Disj 𝑘 ∈ (𝑓𝐴)((𝑓 ↾ (𝑓𝐴))‘𝑘) ↔ Disj 𝑦𝐴 𝑦))
124 fvres 6861 . . . . . . . . . 10 (𝑘 ∈ (𝑓𝐴) → ((𝑓 ↾ (𝑓𝐴))‘𝑘) = (𝑓𝑘))
125124adantl 482 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ (𝑓𝐴)) → ((𝑓 ↾ (𝑓𝐴))‘𝑘) = (𝑓𝑘))
126125disjeq2dv 5075 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (Disj 𝑘 ∈ (𝑓𝐴)((𝑓 ↾ (𝑓𝐴))‘𝑘) ↔ Disj 𝑘 ∈ (𝑓𝐴)(𝑓𝑘)))
127123, 126bitr3d 280 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (Disj 𝑦𝐴 𝑦Disj 𝑘 ∈ (𝑓𝐴)(𝑓𝑘)))
128118, 127mpbid 231 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Disj 𝑘 ∈ (𝑓𝐴)(𝑓𝑘))
129 disjss3 5104 . . . . . . 7 (((𝑓𝐴) ⊆ ℕ ∧ ∀𝑘 ∈ (ℕ ∖ (𝑓𝐴))(𝑓𝑘) = ∅) → (Disj 𝑘 ∈ (𝑓𝐴)(𝑓𝑘) ↔ Disj 𝑘 ∈ ℕ (𝑓𝑘)))
130129biimpa 477 . . . . . 6 ((((𝑓𝐴) ⊆ ℕ ∧ ∀𝑘 ∈ (ℕ ∖ (𝑓𝐴))(𝑓𝑘) = ∅) ∧ Disj 𝑘 ∈ (𝑓𝐴)(𝑓𝑘)) → Disj 𝑘 ∈ ℕ (𝑓𝑘))
13191, 116, 128, 130syl21anc 836 . . . . 5 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Disj 𝑘 ∈ ℕ (𝑓𝑘))
13276, 78, 85, 84, 89, 131esumrnmpt2 32667 . . . 4 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran (𝑘 ∈ ℕ ↦ (𝑓𝑘))(𝑀𝑧) = Σ*𝑘 ∈ ℕ(𝑀‘(𝑓𝑘)))
13311, 75, 1323eqtr3rd 2785 . . 3 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑘 ∈ ℕ(𝑀‘(𝑓𝑘)) = Σ*𝑧𝐴(𝑀𝑧))
134 uniiun 5018 . . . . . . 7 𝐴 = 𝑥𝐴 𝑥
13528sselda 3944 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝑥 ∈ 𝒫 𝑂)
13639, 135elpwiuncl 31455 . . . . . . 7 (𝜑 𝑥𝐴 𝑥 ∈ 𝒫 𝑂)
137134, 136eqeltrid 2842 . . . . . 6 (𝜑 𝐴 ∈ 𝒫 𝑂)
138137adantr 481 . . . . 5 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝐴 ∈ 𝒫 𝑂)
13919, 138ffvelcdmd 7036 . . . 4 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑀 𝐴) ∈ (0[,]+∞))
140 carsgsiga.2 . . . . . . . . . 10 ((𝜑𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))
1411403adant1r 1177 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))
142 fveq2 6842 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑀𝑦) = (𝑀𝑧))
143 nfcv 2907 . . . . . . . . . 10 𝑧𝑥
144 nfcv 2907 . . . . . . . . . 10 𝑦𝑥
145 nfcv 2907 . . . . . . . . . 10 𝑧(𝑀𝑦)
146 nfcv 2907 . . . . . . . . . 10 𝑦(𝑀𝑧)
147142, 143, 144, 145, 146cbvesum 32641 . . . . . . . . 9 Σ*𝑦𝑥(𝑀𝑦) = Σ*𝑧𝑥(𝑀𝑧)
148141, 147breqtrdi 5146 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑧𝑥(𝑀𝑧))
149 ffn 6668 . . . . . . . . . 10 (𝑓:ℕ⟶(𝐴 ∪ {∅}) → 𝑓 Fn ℕ)
150 fz1ssnn 13472 . . . . . . . . . . 11 (1...𝑛) ⊆ ℕ
151 fnssres 6624 . . . . . . . . . . 11 ((𝑓 Fn ℕ ∧ (1...𝑛) ⊆ ℕ) → (𝑓 ↾ (1...𝑛)) Fn (1...𝑛))
152150, 151mpan2 689 . . . . . . . . . 10 (𝑓 Fn ℕ → (𝑓 ↾ (1...𝑛)) Fn (1...𝑛))
1538, 149, 1523syl 18 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓 ↾ (1...𝑛)) Fn (1...𝑛))
154 fzfi 13877 . . . . . . . . . 10 (1...𝑛) ∈ Fin
155 fnfi 9125 . . . . . . . . . 10 (((𝑓 ↾ (1...𝑛)) Fn (1...𝑛) ∧ (1...𝑛) ∈ Fin) → (𝑓 ↾ (1...𝑛)) ∈ Fin)
156154, 155mpan2 689 . . . . . . . . 9 ((𝑓 ↾ (1...𝑛)) Fn (1...𝑛) → (𝑓 ↾ (1...𝑛)) ∈ Fin)
157 rnfi 9279 . . . . . . . . 9 ((𝑓 ↾ (1...𝑛)) ∈ Fin → ran (𝑓 ↾ (1...𝑛)) ∈ Fin)
158153, 156, 1573syl 18 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran (𝑓 ↾ (1...𝑛)) ∈ Fin)
159 resss 5962 . . . . . . . . . . 11 (𝑓 ↾ (1...𝑛)) ⊆ 𝑓
160 rnss 5894 . . . . . . . . . . 11 ((𝑓 ↾ (1...𝑛)) ⊆ 𝑓 → ran (𝑓 ↾ (1...𝑛)) ⊆ ran 𝑓)
161159, 160ax-mp 5 . . . . . . . . . 10 ran (𝑓 ↾ (1...𝑛)) ⊆ ran 𝑓
162161a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ ran 𝑓)
163162, 52sstrd 3954 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ (toCaraSiga‘𝑀))
164162, 36sstrd 3954 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ (𝐴 ∪ {∅}))
165 nfcv 2907 . . . . . . . . . . . . 13 𝑧𝑦
166 nfcv 2907 . . . . . . . . . . . . 13 𝑦𝑧
167 id 22 . . . . . . . . . . . . 13 (𝑦 = 𝑧𝑦 = 𝑧)
168165, 166, 167cbvdisj 5080 . . . . . . . . . . . 12 (Disj 𝑦𝐴 𝑦Disj 𝑧𝐴 𝑧)
169 disjun0 31513 . . . . . . . . . . . 12 (Disj 𝑧𝐴 𝑧Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
170168, 169sylbi 216 . . . . . . . . . . 11 (Disj 𝑦𝐴 𝑦Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
171117, 170syl 17 . . . . . . . . . 10 (𝜑Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
172171adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
173 disjss1 5076 . . . . . . . . 9 (ran (𝑓 ↾ (1...𝑛)) ⊆ (𝐴 ∪ {∅}) → (Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧Disj 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))𝑧))
174164, 172, 173sylc 65 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Disj 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))𝑧)
175 pwidg 4580 . . . . . . . . 9 (𝑂𝑉𝑂 ∈ 𝒫 𝑂)
17617, 175syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝑂 ∈ 𝒫 𝑂)
17717, 19, 21, 148, 158, 163, 174, 176carsgclctunlem1 32917 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑀‘(𝑂 ran (𝑓 ↾ (1...𝑛)))) = Σ*𝑧 ∈ ran (𝑓 ↾ (1...𝑛))(𝑀‘(𝑂𝑧)))
178177adantr 481 . . . . . 6 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝑂 ran (𝑓 ↾ (1...𝑛)))) = Σ*𝑧 ∈ ran (𝑓 ↾ (1...𝑛))(𝑀‘(𝑂𝑧)))
179164unissd 4875 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ (𝐴 ∪ {∅}))
180 uniun 4891 . . . . . . . . . . . 12 (𝐴 ∪ {∅}) = ( 𝐴 {∅})
1812unisn 4887 . . . . . . . . . . . . 13 {∅} = ∅
182181uneq2i 4120 . . . . . . . . . . . 12 ( 𝐴 {∅}) = ( 𝐴 ∪ ∅)
183 un0 4350 . . . . . . . . . . . 12 ( 𝐴 ∪ ∅) = 𝐴
184180, 182, 1833eqtri 2768 . . . . . . . . . . 11 (𝐴 ∪ {∅}) = 𝐴
185179, 184sseqtrdi 3994 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ 𝐴)
186185adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑓 ↾ (1...𝑛)) ⊆ 𝐴)
187 uniss 4873 . . . . . . . . . . . 12 (𝐴 ⊆ 𝒫 𝑂 𝐴 𝒫 𝑂)
188 unipw 5407 . . . . . . . . . . . 12 𝒫 𝑂 = 𝑂
189187, 188sseqtrdi 3994 . . . . . . . . . . 11 (𝐴 ⊆ 𝒫 𝑂 𝐴𝑂)
19028, 189syl 17 . . . . . . . . . 10 (𝜑 𝐴𝑂)
191190ad2antrr 724 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → 𝐴𝑂)
192186, 191sstrd 3954 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑓 ↾ (1...𝑛)) ⊆ 𝑂)
193 sseqin2 4175 . . . . . . . 8 ( ran (𝑓 ↾ (1...𝑛)) ⊆ 𝑂 ↔ (𝑂 ran (𝑓 ↾ (1...𝑛))) = ran (𝑓 ↾ (1...𝑛)))
194192, 193sylib 217 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑂 ran (𝑓 ↾ (1...𝑛))) = ran (𝑓 ↾ (1...𝑛)))
195194fveq2d 6846 . . . . . 6 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝑂 ran (𝑓 ↾ (1...𝑛)))) = (𝑀 ran (𝑓 ↾ (1...𝑛))))
196 nfv 1917 . . . . . . . 8 𝑧((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ)
197164adantr 481 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑓 ↾ (1...𝑛)) ⊆ (𝐴 ∪ {∅}))
19828ad2antrr 724 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ 𝒫 𝑂)
19930a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ∅ ∈ 𝒫 𝑂)
200199snssd 4769 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → {∅} ⊆ 𝒫 𝑂)
201198, 200unssd 4146 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
202197, 201sstrd 3954 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑓 ↾ (1...𝑛)) ⊆ 𝒫 𝑂)
203202sselda 3944 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → 𝑧 ∈ 𝒫 𝑂)
204203elpwid 4569 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → 𝑧𝑂)
205 sseqin2 4175 . . . . . . . . . . 11 (𝑧𝑂 ↔ (𝑂𝑧) = 𝑧)
206204, 205sylib 217 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → (𝑂𝑧) = 𝑧)
207206fveq2d 6846 . . . . . . . . 9 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → (𝑀‘(𝑂𝑧)) = (𝑀𝑧))
208207ralrimiva 3143 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ∀𝑧 ∈ ran (𝑓 ↾ (1...𝑛))(𝑀‘(𝑂𝑧)) = (𝑀𝑧))
209196, 208esumeq2d 32636 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → Σ*𝑧 ∈ ran (𝑓 ↾ (1...𝑛))(𝑀‘(𝑂𝑧)) = Σ*𝑧 ∈ ran (𝑓 ↾ (1...𝑛))(𝑀𝑧))
2109reseq1d 5936 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓 ↾ (1...𝑛)) = ((𝑘 ∈ ℕ ↦ (𝑓𝑘)) ↾ (1...𝑛)))
211210adantr 481 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑓 ↾ (1...𝑛)) = ((𝑘 ∈ ℕ ↦ (𝑓𝑘)) ↾ (1...𝑛)))
212 resmpt 5991 . . . . . . . . . . . 12 ((1...𝑛) ⊆ ℕ → ((𝑘 ∈ ℕ ↦ (𝑓𝑘)) ↾ (1...𝑛)) = (𝑘 ∈ (1...𝑛) ↦ (𝑓𝑘)))
213150, 212ax-mp 5 . . . . . . . . . . 11 ((𝑘 ∈ ℕ ↦ (𝑓𝑘)) ↾ (1...𝑛)) = (𝑘 ∈ (1...𝑛) ↦ (𝑓𝑘))
214211, 213eqtrdi 2792 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑓 ↾ (1...𝑛)) = (𝑘 ∈ (1...𝑛) ↦ (𝑓𝑘)))
215214eqcomd 2742 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ (1...𝑛) ↦ (𝑓𝑘)) = (𝑓 ↾ (1...𝑛)))
216215rneqd 5893 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑘 ∈ (1...𝑛) ↦ (𝑓𝑘)) = ran (𝑓 ↾ (1...𝑛)))
217196, 216esumeq1d 32634 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → Σ*𝑧 ∈ ran (𝑘 ∈ (1...𝑛) ↦ (𝑓𝑘))(𝑀𝑧) = Σ*𝑧 ∈ ran (𝑓 ↾ (1...𝑛))(𝑀𝑧))
218154a1i 11 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
21919ad2antrr 724 . . . . . . . . 9 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
220150a1i 11 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
221220sselda 3944 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
22284adantlr 713 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ 𝒫 𝑂)
223221, 222syldan 591 . . . . . . . . 9 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑓𝑘) ∈ 𝒫 𝑂)
224219, 223ffvelcdmd 7036 . . . . . . . 8 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑀‘(𝑓𝑘)) ∈ (0[,]+∞))
225 simpr 485 . . . . . . . . . 10 (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓𝑘) = ∅) → (𝑓𝑘) = ∅)
226225fveq2d 6846 . . . . . . . . 9 (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓𝑘) = ∅) → (𝑀‘(𝑓𝑘)) = (𝑀‘∅))
22721ad3antrrr 728 . . . . . . . . 9 (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓𝑘) = ∅) → (𝑀‘∅) = 0)
228226, 227eqtrd 2776 . . . . . . . 8 (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓𝑘) = ∅) → (𝑀‘(𝑓𝑘)) = 0)
229 disjss1 5076 . . . . . . . . . . 11 ((1...𝑛) ⊆ ℕ → (Disj 𝑘 ∈ ℕ (𝑓𝑘) → Disj 𝑘 ∈ (1...𝑛)(𝑓𝑘)))
230150, 229ax-mp 5 . . . . . . . . . 10 (Disj 𝑘 ∈ ℕ (𝑓𝑘) → Disj 𝑘 ∈ (1...𝑛)(𝑓𝑘))
231131, 230syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Disj 𝑘 ∈ (1...𝑛)(𝑓𝑘))
232231adantr 481 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → Disj 𝑘 ∈ (1...𝑛)(𝑓𝑘))
23376, 218, 224, 223, 228, 232esumrnmpt2 32667 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → Σ*𝑧 ∈ ran (𝑘 ∈ (1...𝑛) ↦ (𝑓𝑘))(𝑀𝑧) = Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝑓𝑘)))
234209, 217, 2333eqtr2d 2782 . . . . . 6 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → Σ*𝑧 ∈ ran (𝑓 ↾ (1...𝑛))(𝑀‘(𝑂𝑧)) = Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝑓𝑘)))
235178, 195, 2343eqtr3d 2784 . . . . 5 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑀 ran (𝑓 ↾ (1...𝑛))) = Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝑓𝑘)))
236 carsggect.4 . . . . . . . 8 ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))
2372363adant1r 1177 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))
23817, 19, 185, 138, 237carsgmon 32914 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑀 ran (𝑓 ↾ (1...𝑛))) ≤ (𝑀 𝐴))
239238adantr 481 . . . . 5 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑀 ran (𝑓 ↾ (1...𝑛))) ≤ (𝑀 𝐴))
240235, 239eqbrtrrd 5129 . . . 4 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝑓𝑘)) ≤ (𝑀 𝐴))
241139, 85, 240esumgect 32689 . . 3 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑘 ∈ ℕ(𝑀‘(𝑓𝑘)) ≤ (𝑀 𝐴))
242133, 241eqbrtrrd 5129 . 2 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧𝐴(𝑀𝑧) ≤ (𝑀 𝐴))
2436, 242exlimddv 1938 1 (𝜑 → Σ*𝑧𝐴(𝑀𝑧) ≤ (𝑀 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wral 3064  Vcvv 3445  cdif 3907  cun 3908  cin 3909  wss 3910  c0 4282  𝒫 cpw 4560  {csn 4586   cuni 4865   ciun 4954  Disj wdisj 5070   class class class wbr 5105  cmpt 5188  ccnv 5632  ran crn 5634  cres 5635  cima 5636  Fun wfun 6490   Fn wfn 6491  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  ωcom 7802  cdom 8881  Fincfn 8883  0cc0 11051  1c1 11052  +∞cpnf 11186  *cxr 11188  cle 11190  cn 12153  [,]cicc 13267  ...cfz 13424  Σ*cesum 32626  toCaraSigaccarsg 32901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-shft 14952  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-sum 15571  df-ef 15950  df-sin 15952  df-cos 15953  df-pi 15955  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-ordt 17383  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-ps 18455  df-tsr 18456  df-plusf 18496  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-subrg 20220  df-abv 20276  df-lmod 20324  df-scaf 20325  df-sra 20633  df-rgmod 20634  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lp 22487  df-perf 22488  df-cn 22578  df-cnp 22579  df-haus 22666  df-tx 22913  df-hmeo 23106  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-tmd 23423  df-tgp 23424  df-tsms 23478  df-trg 23511  df-xms 23673  df-ms 23674  df-tms 23675  df-nm 23938  df-ngp 23939  df-nrg 23941  df-nlm 23942  df-ii 24240  df-cncf 24241  df-limc 25230  df-dv 25231  df-log 25912  df-esum 32627  df-carsg 32902
This theorem is referenced by:  omsmeas  32923
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