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Theorem carsggect 31198
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 5107 . . . 4 ∅ ∈ V
32a1i 11 . . 3 (𝜑 → ∅ ∈ V)
4 carsggect.0 . . 3 (𝜑 → ¬ ∅ ∈ 𝐴)
5 padct 30148 . . 3 ((𝐴 ≼ ω ∧ ∅ ∈ V ∧ ¬ ∅ ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
61, 3, 4, 5syl3anc 1364 . 2 (𝜑 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
7 nfv 1892 . . . . 5 𝑧(𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
8 simpr1 1187 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝑓:ℕ⟶(𝐴 ∪ {∅}))
98feqmptd 6606 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝑓 = (𝑘 ∈ ℕ ↦ (𝑓𝑘)))
109rneqd 5695 . . . . 5 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran 𝑓 = ran (𝑘 ∈ ℕ ↦ (𝑓𝑘)))
117, 10esumeq1d 30916 . . . 4 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) = Σ*𝑧 ∈ ran (𝑘 ∈ ℕ ↦ (𝑓𝑘))(𝑀𝑧))
12 fvex 6556 . . . . . . . . . 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 31187 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ∅ ∈ (toCaraSiga‘𝑀))
2322snssd 4653 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → {∅} ⊆ (toCaraSiga‘𝑀))
2415, 23unssd 4087 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝐴 ∪ {∅}) ⊆ (toCaraSiga‘𝑀))
2513, 24ssexd 5124 . . . . . . . 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 31184 . . . . . . . . . . . . 13 (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂)
2814, 27sstrd 3903 . . . . . . . . . . . 12 (𝜑𝐴 ⊆ 𝒫 𝑂)
2928adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝐴 ⊆ 𝒫 𝑂)
30 0elpw 5152 . . . . . . . . . . . . 13 ∅ ∈ 𝒫 𝑂
3130a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ∅ ∈ 𝒫 𝑂)
3231snssd 4653 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → {∅} ⊆ 𝒫 𝑂)
3329, 32unssd 4087 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
3433sselda 3893 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → 𝑧 ∈ 𝒫 𝑂)
3526, 34ffvelrnd 6722 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → (𝑀𝑧) ∈ (0[,]+∞))
368frnd 6394 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran 𝑓 ⊆ (𝐴 ∪ {∅}))
377, 25, 35, 36esummono 30935 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) ≤ Σ*𝑧 ∈ (𝐴 ∪ {∅})(𝑀𝑧))
38 ctex 8377 . . . . . . . . . 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 5124 . . . . . . . 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 3893 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧𝐴) → 𝑧 ∈ 𝒫 𝑂)
4442, 43ffvelrnd 6722 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧𝐴) → (𝑀𝑧) ∈ (0[,]+∞))
45 elsni 4493 . . . . . . . . . . 11 (𝑧 ∈ {∅} → 𝑧 = ∅)
4645adantl 482 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ {∅}) → 𝑧 = ∅)
4746fveq2d 6547 . . . . . . . . 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 2831 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ {∅}) → (𝑀𝑧) = 0)
5040, 41, 44, 49esumpad 30936 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ (𝐴 ∪ {∅})(𝑀𝑧) = Σ*𝑧𝐴(𝑀𝑧))
5137, 50breqtrd 4992 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) ≤ Σ*𝑧𝐴(𝑀𝑧))
5236, 24sstrd 3903 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran 𝑓 ⊆ (toCaraSiga‘𝑀))
53 ssexg 5123 . . . . . . . 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 3903 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran 𝑓 ⊆ 𝒫 𝑂)
5756sselda 3893 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ ran 𝑓) → 𝑧 ∈ 𝒫 𝑂)
5855, 57ffvelrnd 6722 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ ran 𝑓) → (𝑀𝑧) ∈ (0[,]+∞))
59 simpr2 1188 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝐴 ⊆ ran 𝑓)
607, 54, 58, 59esummono 30935 . . . . . 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 12674 . . . . . . 7 (0[,]+∞) ⊆ ℝ*
6358ralrimiva 3149 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ∀𝑧 ∈ ran 𝑓(𝑀𝑧) ∈ (0[,]+∞))
64 nfcv 2949 . . . . . . . . 9 𝑧ran 𝑓
6564esumcl 30911 . . . . . . . 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, 66sseldi 3891 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) ∈ ℝ*)
6844ralrimiva 3149 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ∀𝑧𝐴 (𝑀𝑧) ∈ (0[,]+∞))
69 nfcv 2949 . . . . . . . . 9 𝑧𝐴
7069esumcl 30911 . . . . . . . 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, 71sseldi 3891 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧𝐴(𝑀𝑧) ∈ ℝ*)
73 xrletri3 12402 . . . . . 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 258 . . . 4 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) = Σ*𝑧𝐴(𝑀𝑧))
76 fveq2 6543 . . . . 5 (𝑧 = (𝑓𝑘) → (𝑀𝑧) = (𝑀‘(𝑓𝑘)))
77 nnex 11497 . . . . . 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, 82ffvelrnd 6722 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ (𝐴 ∪ {∅}))
8480, 83sseldd 3894 . . . . . 6 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ 𝒫 𝑂)
8579, 84ffvelrnd 6722 . . . . 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 6547 . . . . . 6 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓𝑘) = ∅) → (𝑀‘(𝑓𝑘)) = (𝑀‘∅))
8821ad2antrr 722 . . . . . 6 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓𝑘) = ∅) → (𝑀‘∅) = 0)
8987, 88eqtrd 2831 . . . . 5 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓𝑘) = ∅) → (𝑀‘(𝑓𝑘)) = 0)
90 cnvimass 5830 . . . . . . 7 (𝑓𝐴) ⊆ dom 𝑓
9190, 8fssdm 6403 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓𝐴) ⊆ ℕ)
92 ffun 6390 . . . . . . . . . . 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 6705 . . . . . . . . . . . . 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 6709 . . . . . . . . . . . . . 14 (𝑓:ℕ⟶(𝐴 ∪ {∅}) → (𝑓 “ (𝐴 ∪ {∅})) = ℕ)
988, 97syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓 “ (𝐴 ∪ {∅})) = ℕ)
9998difeq1d 4023 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ((𝑓 “ (𝐴 ∪ {∅})) ∖ (𝑓𝐴)) = (ℕ ∖ (𝑓𝐴)))
10096, 99eqtrd 2831 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) = (ℕ ∖ (𝑓𝐴)))
101 uncom 4054 . . . . . . . . . . . . . . . 16 ({∅} ∪ 𝐴) = (𝐴 ∪ {∅})
102101difeq1i 4020 . . . . . . . . . . . . . . 15 (({∅} ∪ 𝐴) ∖ 𝐴) = ((𝐴 ∪ {∅}) ∖ 𝐴)
103 difun2 4347 . . . . . . . . . . . . . . 15 (({∅} ∪ 𝐴) ∖ 𝐴) = ({∅} ∖ 𝐴)
104102, 103eqtr3i 2821 . . . . . . . . . . . . . 14 ((𝐴 ∪ {∅}) ∖ 𝐴) = ({∅} ∖ 𝐴)
105 difss 4033 . . . . . . . . . . . . . 14 ({∅} ∖ 𝐴) ⊆ {∅}
106104, 105eqsstri 3926 . . . . . . . . . . . . 13 ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅}
107106a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅})
108 sspreima 30087 . . . . . . . . . . . 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 3931 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (ℕ ∖ (𝑓𝐴)) ⊆ (𝑓 “ {∅}))
111110sselda 3893 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (𝑓𝐴))) → 𝑘 ∈ (𝑓 “ {∅}))
112 fvimacnvi 6692 . . . . . . . . 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 4493 . . . . . . . 8 ((𝑓𝑘) ∈ {∅} → (𝑓𝑘) = ∅)
115113, 114syl 17 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (𝑓𝐴))) → (𝑓𝑘) = ∅)
116115ralrimiva 3149 . . . . . 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 1189 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Fun (𝑓𝐴))
120 fresf1o 30071 . . . . . . . . . 10 ((Fun 𝑓𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto𝐴)
12193, 59, 119, 120syl3anc 1364 . . . . . . . . 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 30036 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (Disj 𝑘 ∈ (𝑓𝐴)((𝑓 ↾ (𝑓𝐴))‘𝑘) ↔ Disj 𝑦𝐴 𝑦))
124 fvres 6562 . . . . . . . . . 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 4939 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (Disj 𝑘 ∈ (𝑓𝐴)((𝑓 ↾ (𝑓𝐴))‘𝑘) ↔ Disj 𝑘 ∈ (𝑓𝐴)(𝑓𝑘)))
127123, 126bitr3d 282 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (Disj 𝑦𝐴 𝑦Disj 𝑘 ∈ (𝑓𝐴)(𝑓𝑘)))
128118, 127mpbid 233 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Disj 𝑘 ∈ (𝑓𝐴)(𝑓𝑘))
129 disjss3 4965 . . . . . . 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 834 . . . . 5 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Disj 𝑘 ∈ ℕ (𝑓𝑘))
13276, 78, 85, 84, 89, 131esumrnmpt2 30949 . . . 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 2840 . . 3 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑘 ∈ ℕ(𝑀‘(𝑓𝑘)) = Σ*𝑧𝐴(𝑀𝑧))
134 uniiun 4885 . . . . . . 7 𝐴 = 𝑥𝐴 𝑥
13528sselda 3893 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝑥 ∈ 𝒫 𝑂)
13639, 135elpwiuncl 29982 . . . . . . 7 (𝜑 𝑥𝐴 𝑥 ∈ 𝒫 𝑂)
137134, 136syl5eqel 2887 . . . . . 6 (𝜑 𝐴 ∈ 𝒫 𝑂)
138137adantr 481 . . . . 5 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝐴 ∈ 𝒫 𝑂)
13919, 138ffvelrnd 6722 . . . 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 1170 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))
142 fveq2 6543 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑀𝑦) = (𝑀𝑧))
143 nfcv 2949 . . . . . . . . . 10 𝑧𝑥
144 nfcv 2949 . . . . . . . . . 10 𝑦𝑥
145 nfcv 2949 . . . . . . . . . 10 𝑧(𝑀𝑦)
146 nfcv 2949 . . . . . . . . . 10 𝑦(𝑀𝑧)
147142, 143, 144, 145, 146cbvesum 30923 . . . . . . . . 9 Σ*𝑦𝑥(𝑀𝑦) = Σ*𝑧𝑥(𝑀𝑧)
148141, 147syl6breq 5007 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑧𝑥(𝑀𝑧))
149 ffn 6387 . . . . . . . . . 10 (𝑓:ℕ⟶(𝐴 ∪ {∅}) → 𝑓 Fn ℕ)
150 fz1ssnn 12793 . . . . . . . . . . 11 (1...𝑛) ⊆ ℕ
151 fnssres 6345 . . . . . . . . . . 11 ((𝑓 Fn ℕ ∧ (1...𝑛) ⊆ ℕ) → (𝑓 ↾ (1...𝑛)) Fn (1...𝑛))
152150, 151mpan2 687 . . . . . . . . . 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 13195 . . . . . . . . . 10 (1...𝑛) ∈ Fin
155 fnfi 8647 . . . . . . . . . 10 (((𝑓 ↾ (1...𝑛)) Fn (1...𝑛) ∧ (1...𝑛) ∈ Fin) → (𝑓 ↾ (1...𝑛)) ∈ Fin)
156154, 155mpan2 687 . . . . . . . . 9 ((𝑓 ↾ (1...𝑛)) Fn (1...𝑛) → (𝑓 ↾ (1...𝑛)) ∈ Fin)
157 rnfi 8658 . . . . . . . . 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 5764 . . . . . . . . . . 11 (𝑓 ↾ (1...𝑛)) ⊆ 𝑓
160 rnss 5696 . . . . . . . . . . 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 3903 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ (toCaraSiga‘𝑀))
164162, 36sstrd 3903 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ (𝐴 ∪ {∅}))
165 nfcv 2949 . . . . . . . . . . . . 13 𝑧𝑦
166 nfcv 2949 . . . . . . . . . . . . 13 𝑦𝑧
167 id 22 . . . . . . . . . . . . 13 (𝑦 = 𝑧𝑦 = 𝑧)
168165, 166, 167cbvdisj 4944 . . . . . . . . . . . 12 (Disj 𝑦𝐴 𝑦Disj 𝑧𝐴 𝑧)
169 disjun0 30040 . . . . . . . . . . . 12 (Disj 𝑧𝐴 𝑧Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
170168, 169sylbi 218 . . . . . . . . . . 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 4940 . . . . . . . . 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 4472 . . . . . . . . 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 31197 . . . . . . 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 4773 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ (𝐴 ∪ {∅}))
180 uniun 4768 . . . . . . . . . . . 12 (𝐴 ∪ {∅}) = ( 𝐴 {∅})
1812unisn 4765 . . . . . . . . . . . . 13 {∅} = ∅
182181uneq2i 4061 . . . . . . . . . . . 12 ( 𝐴 {∅}) = ( 𝐴 ∪ ∅)
183 un0 4268 . . . . . . . . . . . 12 ( 𝐴 ∪ ∅) = 𝐴
184180, 182, 1833eqtri 2823 . . . . . . . . . . 11 (𝐴 ∪ {∅}) = 𝐴
185179, 184syl6sseq 3942 . . . . . . . . . 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 4770 . . . . . . . . . . . 12 (𝐴 ⊆ 𝒫 𝑂 𝐴 𝒫 𝑂)
188 unipw 5239 . . . . . . . . . . . 12 𝒫 𝑂 = 𝑂
189187, 188syl6sseq 3942 . . . . . . . . . . 11 (𝐴 ⊆ 𝒫 𝑂 𝐴𝑂)
19028, 189syl 17 . . . . . . . . . 10 (𝜑 𝐴𝑂)
191190ad2antrr 722 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → 𝐴𝑂)
192186, 191sstrd 3903 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑓 ↾ (1...𝑛)) ⊆ 𝑂)
193 sseqin2 4116 . . . . . . . 8 ( ran (𝑓 ↾ (1...𝑛)) ⊆ 𝑂 ↔ (𝑂 ran (𝑓 ↾ (1...𝑛))) = ran (𝑓 ↾ (1...𝑛)))
194192, 193sylib 219 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑂 ran (𝑓 ↾ (1...𝑛))) = ran (𝑓 ↾ (1...𝑛)))
195194fveq2d 6547 . . . . . 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 1892 . . . . . . . 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 722 . . . . . . . . . . . . . . 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 4653 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → {∅} ⊆ 𝒫 𝑂)
201198, 200unssd 4087 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
202197, 201sstrd 3903 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑓 ↾ (1...𝑛)) ⊆ 𝒫 𝑂)
203202sselda 3893 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → 𝑧 ∈ 𝒫 𝑂)
204203elpwid 4469 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → 𝑧𝑂)
205 sseqin2 4116 . . . . . . . . . . 11 (𝑧𝑂 ↔ (𝑂𝑧) = 𝑧)
206204, 205sylib 219 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → (𝑂𝑧) = 𝑧)
207206fveq2d 6547 . . . . . . . . 9 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → (𝑀‘(𝑂𝑧)) = (𝑀𝑧))
208207ralrimiva 3149 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ∀𝑧 ∈ ran (𝑓 ↾ (1...𝑛))(𝑀‘(𝑂𝑧)) = (𝑀𝑧))
209196, 208esumeq2d 30918 . . . . . . 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 5738 . . . . . . . . . . . 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 5791 . . . . . . . . . . . 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, 213syl6eq 2847 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑓 ↾ (1...𝑛)) = (𝑘 ∈ (1...𝑛) ↦ (𝑓𝑘)))
215214eqcomd 2801 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ (1...𝑛) ↦ (𝑓𝑘)) = (𝑓 ↾ (1...𝑛)))
216215rneqd 5695 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑘 ∈ (1...𝑛) ↦ (𝑓𝑘)) = ran (𝑓 ↾ (1...𝑛)))
217196, 216esumeq1d 30916 . . . . . . 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 722 . . . . . . . . 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 3893 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
22284adantlr 711 . . . . . . . . . 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, 223ffvelrnd 6722 . . . . . . . 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 6547 . . . . . . . . 9 (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓𝑘) = ∅) → (𝑀‘(𝑓𝑘)) = (𝑀‘∅))
22721ad3antrrr 726 . . . . . . . . 9 (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓𝑘) = ∅) → (𝑀‘∅) = 0)
228226, 227eqtrd 2831 . . . . . . . 8 (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓𝑘) = ∅) → (𝑀‘(𝑓𝑘)) = 0)
229 disjss1 4940 . . . . . . . . . . 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 30949 . . . . . . 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 2837 . . . . . 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 2839 . . . . 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 1170 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))
23817, 19, 185, 138, 237carsgmon 31194 . . . . . 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 4990 . . . 4 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝑓𝑘)) ≤ (𝑀 𝐴))
241139, 85, 240esumgect 30971 . . 3 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑘 ∈ ℕ(𝑀‘(𝑓𝑘)) ≤ (𝑀 𝐴))
242133, 241eqbrtrrd 4990 . 2 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧𝐴(𝑀𝑧) ≤ (𝑀 𝐴))
2436, 242exlimddv 1913 1 (𝜑 → Σ*𝑧𝐴(𝑀𝑧) ≤ (𝑀 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wex 1761  wcel 2081  wral 3105  Vcvv 3437  cdif 3860  cun 3861  cin 3862  wss 3863  c0 4215  𝒫 cpw 4457  {csn 4476   cuni 4749   ciun 4829  Disj wdisj 4934   class class class wbr 4966  cmpt 5045  ccnv 5447  ran crn 5449  cres 5450  cima 5451  Fun wfun 6224   Fn wfn 6225  wf 6226  1-1-ontowf1o 6229  cfv 6230  (class class class)co 7021  ωcom 7441  cdom 8360  Fincfn 8362  0cc0 10388  1c1 10389  +∞cpnf 10523  *cxr 10525  cle 10527  cn 11491  [,]cicc 12596  ...cfz 12747  Σ*cesum 30908  toCaraSigaccarsg 31181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324  ax-inf2 8955  ax-cnex 10444  ax-resscn 10445  ax-1cn 10446  ax-icn 10447  ax-addcl 10448  ax-addrcl 10449  ax-mulcl 10450  ax-mulrcl 10451  ax-mulcom 10452  ax-addass 10453  ax-mulass 10454  ax-distr 10455  ax-i2m1 10456  ax-1ne0 10457  ax-1rid 10458  ax-rnegex 10459  ax-rrecex 10460  ax-cnre 10461  ax-pre-lttri 10462  ax-pre-lttrn 10463  ax-pre-ltadd 10464  ax-pre-mulgt0 10465  ax-pre-sup 10466  ax-addf 10467  ax-mulf 10468
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-int 4787  df-iun 4831  df-iin 4832  df-disj 4935  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-se 5408  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-pred 6028  df-ord 6074  df-on 6075  df-lim 6076  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-isom 6239  df-riota 6982  df-ov 7024  df-oprab 7025  df-mpo 7026  df-of 7272  df-om 7442  df-1st 7550  df-2nd 7551  df-supp 7687  df-wrecs 7803  df-recs 7865  df-rdg 7903  df-1o 7958  df-2o 7959  df-oadd 7962  df-er 8144  df-map 8263  df-pm 8264  df-ixp 8316  df-en 8363  df-dom 8364  df-sdom 8365  df-fin 8366  df-fsupp 8685  df-fi 8726  df-sup 8757  df-inf 8758  df-oi 8825  df-dju 9181  df-card 9219  df-pnf 10528  df-mnf 10529  df-xr 10530  df-ltxr 10531  df-le 10532  df-sub 10724  df-neg 10725  df-div 11151  df-nn 11492  df-2 11553  df-3 11554  df-4 11555  df-5 11556  df-6 11557  df-7 11558  df-8 11559  df-9 11560  df-n0 11751  df-z 11835  df-dec 11953  df-uz 12099  df-q 12203  df-rp 12245  df-xneg 12362  df-xadd 12363  df-xmul 12364  df-ioo 12597  df-ioc 12598  df-ico 12599  df-icc 12600  df-fz 12748  df-fzo 12889  df-fl 13017  df-mod 13093  df-seq 13225  df-exp 13285  df-fac 13489  df-bc 13518  df-hash 13546  df-shft 14265  df-cj 14297  df-re 14298  df-im 14299  df-sqrt 14433  df-abs 14434  df-limsup 14667  df-clim 14684  df-rlim 14685  df-sum 14882  df-ef 15259  df-sin 15261  df-cos 15262  df-pi 15264  df-struct 16319  df-ndx 16320  df-slot 16321  df-base 16323  df-sets 16324  df-ress 16325  df-plusg 16412  df-mulr 16413  df-starv 16414  df-sca 16415  df-vsca 16416  df-ip 16417  df-tset 16418  df-ple 16419  df-ds 16421  df-unif 16422  df-hom 16423  df-cco 16424  df-rest 16530  df-topn 16531  df-0g 16549  df-gsum 16550  df-topgen 16551  df-pt 16552  df-prds 16555  df-ordt 16608  df-xrs 16609  df-qtop 16614  df-imas 16615  df-xps 16617  df-mre 16691  df-mrc 16692  df-acs 16694  df-ps 17644  df-tsr 17645  df-plusf 17685  df-mgm 17686  df-sgrp 17728  df-mnd 17739  df-mhm 17779  df-submnd 17780  df-grp 17869  df-minusg 17870  df-sbg 17871  df-mulg 17987  df-subg 18035  df-cntz 18193  df-cmn 18640  df-abl 18641  df-mgp 18935  df-ur 18947  df-ring 18994  df-cring 18995  df-subrg 19228  df-abv 19283  df-lmod 19331  df-scaf 19332  df-sra 19639  df-rgmod 19640  df-psmet 20224  df-xmet 20225  df-met 20226  df-bl 20227  df-mopn 20228  df-fbas 20229  df-fg 20230  df-cnfld 20233  df-top 21191  df-topon 21208  df-topsp 21230  df-bases 21243  df-cld 21316  df-ntr 21317  df-cls 21318  df-nei 21395  df-lp 21433  df-perf 21434  df-cn 21524  df-cnp 21525  df-haus 21612  df-tx 21859  df-hmeo 22052  df-fil 22143  df-fm 22235  df-flim 22236  df-flf 22237  df-tmd 22369  df-tgp 22370  df-tsms 22423  df-trg 22456  df-xms 22618  df-ms 22619  df-tms 22620  df-nm 22880  df-ngp 22881  df-nrg 22883  df-nlm 22884  df-ii 23173  df-cncf 23174  df-limc 24152  df-dv 24153  df-log 24826  df-esum 30909  df-carsg 31182
This theorem is referenced by:  omsmeas  31203
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