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Theorem carsggect 31690
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 5178 . . . 4 ∅ ∈ V
32a1i 11 . . 3 (𝜑 → ∅ ∈ V)
4 carsggect.0 . . 3 (𝜑 → ¬ ∅ ∈ 𝐴)
5 padct 30485 . . 3 ((𝐴 ≼ ω ∧ ∅ ∈ V ∧ ¬ ∅ ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
61, 3, 4, 5syl3anc 1368 . 2 (𝜑 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
7 nfv 1915 . . . . 5 𝑧(𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
8 simpr1 1191 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝑓:ℕ⟶(𝐴 ∪ {∅}))
98feqmptd 6712 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝑓 = (𝑘 ∈ ℕ ↦ (𝑓𝑘)))
109rneqd 5776 . . . . 5 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran 𝑓 = ran (𝑘 ∈ ℕ ↦ (𝑓𝑘)))
117, 10esumeq1d 31408 . . . 4 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) = Σ*𝑧 ∈ ran (𝑘 ∈ ℕ ↦ (𝑓𝑘))(𝑀𝑧))
12 fvex 6662 . . . . . . . . . 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 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝐴 ⊆ (toCaraSiga‘𝑀))
16 carsgval.1 . . . . . . . . . . . . 13 (𝜑𝑂𝑉)
1716adantr 484 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝑂𝑉)
18 carsgval.2 . . . . . . . . . . . . 13 (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))
1918adantr 484 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
20 carsgsiga.1 . . . . . . . . . . . . 13 (𝜑 → (𝑀‘∅) = 0)
2120adantr 484 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑀‘∅) = 0)
2217, 19, 210elcarsg 31679 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ∅ ∈ (toCaraSiga‘𝑀))
2322snssd 4705 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → {∅} ⊆ (toCaraSiga‘𝑀))
2415, 23unssd 4116 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝐴 ∪ {∅}) ⊆ (toCaraSiga‘𝑀))
2513, 24ssexd 5195 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝐴 ∪ {∅}) ∈ V)
2619adantr 484 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
2716, 18carsgcl 31676 . . . . . . . . . . . . 13 (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂)
2814, 27sstrd 3928 . . . . . . . . . . . 12 (𝜑𝐴 ⊆ 𝒫 𝑂)
2928adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝐴 ⊆ 𝒫 𝑂)
30 0elpw 5224 . . . . . . . . . . . . 13 ∅ ∈ 𝒫 𝑂
3130a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ∅ ∈ 𝒫 𝑂)
3231snssd 4705 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → {∅} ⊆ 𝒫 𝑂)
3329, 32unssd 4116 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
3433sselda 3918 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → 𝑧 ∈ 𝒫 𝑂)
3526, 34ffvelrnd 6833 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → (𝑀𝑧) ∈ (0[,]+∞))
368frnd 6498 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran 𝑓 ⊆ (𝐴 ∪ {∅}))
377, 25, 35, 36esummono 31427 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) ≤ Σ*𝑧 ∈ (𝐴 ∪ {∅})(𝑀𝑧))
38 ctex 8511 . . . . . . . . . 10 (𝐴 ≼ ω → 𝐴 ∈ V)
391, 38syl 17 . . . . . . . . 9 (𝜑𝐴 ∈ V)
4039adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝐴 ∈ V)
4113, 23ssexd 5195 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → {∅} ∈ V)
4219adantr 484 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧𝐴) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
4329sselda 3918 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧𝐴) → 𝑧 ∈ 𝒫 𝑂)
4442, 43ffvelrnd 6833 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧𝐴) → (𝑀𝑧) ∈ (0[,]+∞))
45 elsni 4545 . . . . . . . . . . 11 (𝑧 ∈ {∅} → 𝑧 = ∅)
4645adantl 485 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ {∅}) → 𝑧 = ∅)
4746fveq2d 6653 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ {∅}) → (𝑀𝑧) = (𝑀‘∅))
4821adantr 484 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ {∅}) → (𝑀‘∅) = 0)
4947, 48eqtrd 2836 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ {∅}) → (𝑀𝑧) = 0)
5040, 41, 44, 49esumpad 31428 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ (𝐴 ∪ {∅})(𝑀𝑧) = Σ*𝑧𝐴(𝑀𝑧))
5137, 50breqtrd 5059 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) ≤ Σ*𝑧𝐴(𝑀𝑧))
5236, 24sstrd 3928 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran 𝑓 ⊆ (toCaraSiga‘𝑀))
53 ssexg 5194 . . . . . . . 8 ((ran 𝑓 ⊆ (toCaraSiga‘𝑀) ∧ (toCaraSiga‘𝑀) ∈ V) → ran 𝑓 ∈ V)
5452, 12, 53sylancl 589 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran 𝑓 ∈ V)
5519adantr 484 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ ran 𝑓) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
5636, 33sstrd 3928 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran 𝑓 ⊆ 𝒫 𝑂)
5756sselda 3918 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ ran 𝑓) → 𝑧 ∈ 𝒫 𝑂)
5855, 57ffvelrnd 6833 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ ran 𝑓) → (𝑀𝑧) ∈ (0[,]+∞))
59 simpr2 1192 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝐴 ⊆ ran 𝑓)
607, 54, 58, 59esummono 31427 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧𝐴(𝑀𝑧) ≤ Σ*𝑧 ∈ ran 𝑓(𝑀𝑧))
6151, 60jca 515 . . . . 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 12812 . . . . . . 7 (0[,]+∞) ⊆ ℝ*
6358ralrimiva 3152 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ∀𝑧 ∈ ran 𝑓(𝑀𝑧) ∈ (0[,]+∞))
64 nfcv 2958 . . . . . . . . 9 𝑧ran 𝑓
6564esumcl 31403 . . . . . . . 8 ((ran 𝑓 ∈ V ∧ ∀𝑧 ∈ ran 𝑓(𝑀𝑧) ∈ (0[,]+∞)) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) ∈ (0[,]+∞))
6654, 63, 65syl2anc 587 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) ∈ (0[,]+∞))
6762, 66sseldi 3916 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) ∈ ℝ*)
6844ralrimiva 3152 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ∀𝑧𝐴 (𝑀𝑧) ∈ (0[,]+∞))
69 nfcv 2958 . . . . . . . . 9 𝑧𝐴
7069esumcl 31403 . . . . . . . 8 ((𝐴 ∈ V ∧ ∀𝑧𝐴 (𝑀𝑧) ∈ (0[,]+∞)) → Σ*𝑧𝐴(𝑀𝑧) ∈ (0[,]+∞))
7140, 68, 70syl2anc 587 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧𝐴(𝑀𝑧) ∈ (0[,]+∞))
7262, 71sseldi 3916 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧𝐴(𝑀𝑧) ∈ ℝ*)
73 xrletri3 12539 . . . . . 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 587 . . . . 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 260 . . . 4 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) = Σ*𝑧𝐴(𝑀𝑧))
76 fveq2 6649 . . . . 5 (𝑧 = (𝑓𝑘) → (𝑀𝑧) = (𝑀‘(𝑓𝑘)))
77 nnex 11635 . . . . . 6 ℕ ∈ V
7877a1i 11 . . . . 5 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ℕ ∈ V)
7919adantr 484 . . . . . 6 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
8033adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
818adantr 484 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → 𝑓:ℕ⟶(𝐴 ∪ {∅}))
82 simpr 488 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
8381, 82ffvelrnd 6833 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ (𝐴 ∪ {∅}))
8480, 83sseldd 3919 . . . . . 6 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ 𝒫 𝑂)
8579, 84ffvelrnd 6833 . . . . 5 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑀‘(𝑓𝑘)) ∈ (0[,]+∞))
86 simpr 488 . . . . . . 7 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓𝑘) = ∅) → (𝑓𝑘) = ∅)
8786fveq2d 6653 . . . . . 6 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓𝑘) = ∅) → (𝑀‘(𝑓𝑘)) = (𝑀‘∅))
8821ad2antrr 725 . . . . . 6 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓𝑘) = ∅) → (𝑀‘∅) = 0)
8987, 88eqtrd 2836 . . . . 5 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓𝑘) = ∅) → (𝑀‘(𝑓𝑘)) = 0)
90 cnvimass 5920 . . . . . . 7 (𝑓𝐴) ⊆ dom 𝑓
9190, 8fssdm 6508 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓𝐴) ⊆ ℕ)
92 ffun 6494 . . . . . . . . . . 11 (𝑓:ℕ⟶(𝐴 ∪ {∅}) → Fun 𝑓)
938, 92syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Fun 𝑓)
9493adantr 484 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (𝑓𝐴))) → Fun 𝑓)
95 difpreima 6816 . . . . . . . . . . . . 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 6820 . . . . . . . . . . . . . 14 (𝑓:ℕ⟶(𝐴 ∪ {∅}) → (𝑓 “ (𝐴 ∪ {∅})) = ℕ)
988, 97syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓 “ (𝐴 ∪ {∅})) = ℕ)
9998difeq1d 4052 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ((𝑓 “ (𝐴 ∪ {∅})) ∖ (𝑓𝐴)) = (ℕ ∖ (𝑓𝐴)))
10096, 99eqtrd 2836 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) = (ℕ ∖ (𝑓𝐴)))
101 uncom 4083 . . . . . . . . . . . . . . . 16 ({∅} ∪ 𝐴) = (𝐴 ∪ {∅})
102101difeq1i 4049 . . . . . . . . . . . . . . 15 (({∅} ∪ 𝐴) ∖ 𝐴) = ((𝐴 ∪ {∅}) ∖ 𝐴)
103 difun2 4390 . . . . . . . . . . . . . . 15 (({∅} ∪ 𝐴) ∖ 𝐴) = ({∅} ∖ 𝐴)
104102, 103eqtr3i 2826 . . . . . . . . . . . . . 14 ((𝐴 ∪ {∅}) ∖ 𝐴) = ({∅} ∖ 𝐴)
105 difss 4062 . . . . . . . . . . . . . 14 ({∅} ∖ 𝐴) ⊆ {∅}
106104, 105eqsstri 3952 . . . . . . . . . . . . 13 ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅}
107106a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅})
108 sspreima 30410 . . . . . . . . . . . 12 ((Fun 𝑓 ∧ ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅}) → (𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) ⊆ (𝑓 “ {∅}))
10993, 107, 108syl2anc 587 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) ⊆ (𝑓 “ {∅}))
110100, 109eqsstrrd 3957 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (ℕ ∖ (𝑓𝐴)) ⊆ (𝑓 “ {∅}))
111110sselda 3918 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (𝑓𝐴))) → 𝑘 ∈ (𝑓 “ {∅}))
112 fvimacnvi 6803 . . . . . . . . 9 ((Fun 𝑓𝑘 ∈ (𝑓 “ {∅})) → (𝑓𝑘) ∈ {∅})
11394, 111, 112syl2anc 587 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (𝑓𝐴))) → (𝑓𝑘) ∈ {∅})
114 elsni 4545 . . . . . . . 8 ((𝑓𝑘) ∈ {∅} → (𝑓𝑘) = ∅)
115113, 114syl 17 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (𝑓𝐴))) → (𝑓𝑘) = ∅)
116115ralrimiva 3152 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ∀𝑘 ∈ (ℕ ∖ (𝑓𝐴))(𝑓𝑘) = ∅)
117 carsggect.3 . . . . . . . 8 (𝜑Disj 𝑦𝐴 𝑦)
118117adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Disj 𝑦𝐴 𝑦)
119 simpr3 1193 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Fun (𝑓𝐴))
120 fresf1o 30394 . . . . . . . . . 10 ((Fun 𝑓𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto𝐴)
12193, 59, 119, 120syl3anc 1368 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto𝐴)
122 simpr 488 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑦 = ((𝑓 ↾ (𝑓𝐴))‘𝑘)) → 𝑦 = ((𝑓 ↾ (𝑓𝐴))‘𝑘))
123121, 122disjrdx 30358 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (Disj 𝑘 ∈ (𝑓𝐴)((𝑓 ↾ (𝑓𝐴))‘𝑘) ↔ Disj 𝑦𝐴 𝑦))
124 fvres 6668 . . . . . . . . . 10 (𝑘 ∈ (𝑓𝐴) → ((𝑓 ↾ (𝑓𝐴))‘𝑘) = (𝑓𝑘))
125124adantl 485 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ (𝑓𝐴)) → ((𝑓 ↾ (𝑓𝐴))‘𝑘) = (𝑓𝑘))
126125disjeq2dv 5003 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (Disj 𝑘 ∈ (𝑓𝐴)((𝑓 ↾ (𝑓𝐴))‘𝑘) ↔ Disj 𝑘 ∈ (𝑓𝐴)(𝑓𝑘)))
127123, 126bitr3d 284 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (Disj 𝑦𝐴 𝑦Disj 𝑘 ∈ (𝑓𝐴)(𝑓𝑘)))
128118, 127mpbid 235 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Disj 𝑘 ∈ (𝑓𝐴)(𝑓𝑘))
129 disjss3 5032 . . . . . . 7 (((𝑓𝐴) ⊆ ℕ ∧ ∀𝑘 ∈ (ℕ ∖ (𝑓𝐴))(𝑓𝑘) = ∅) → (Disj 𝑘 ∈ (𝑓𝐴)(𝑓𝑘) ↔ Disj 𝑘 ∈ ℕ (𝑓𝑘)))
130129biimpa 480 . . . . . 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 31441 . . . 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 2845 . . 3 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑘 ∈ ℕ(𝑀‘(𝑓𝑘)) = Σ*𝑧𝐴(𝑀𝑧))
134 uniiun 4948 . . . . . . 7 𝐴 = 𝑥𝐴 𝑥
13528sselda 3918 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝑥 ∈ 𝒫 𝑂)
13639, 135elpwiuncl 30304 . . . . . . 7 (𝜑 𝑥𝐴 𝑥 ∈ 𝒫 𝑂)
137134, 136eqeltrid 2897 . . . . . 6 (𝜑 𝐴 ∈ 𝒫 𝑂)
138137adantr 484 . . . . 5 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝐴 ∈ 𝒫 𝑂)
13919, 138ffvelrnd 6833 . . . 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 1174 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))
142 fveq2 6649 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑀𝑦) = (𝑀𝑧))
143 nfcv 2958 . . . . . . . . . 10 𝑧𝑥
144 nfcv 2958 . . . . . . . . . 10 𝑦𝑥
145 nfcv 2958 . . . . . . . . . 10 𝑧(𝑀𝑦)
146 nfcv 2958 . . . . . . . . . 10 𝑦(𝑀𝑧)
147142, 143, 144, 145, 146cbvesum 31415 . . . . . . . . 9 Σ*𝑦𝑥(𝑀𝑦) = Σ*𝑧𝑥(𝑀𝑧)
148141, 147breqtrdi 5074 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑧𝑥(𝑀𝑧))
149 ffn 6491 . . . . . . . . . 10 (𝑓:ℕ⟶(𝐴 ∪ {∅}) → 𝑓 Fn ℕ)
150 fz1ssnn 12937 . . . . . . . . . . 11 (1...𝑛) ⊆ ℕ
151 fnssres 6446 . . . . . . . . . . 11 ((𝑓 Fn ℕ ∧ (1...𝑛) ⊆ ℕ) → (𝑓 ↾ (1...𝑛)) Fn (1...𝑛))
152150, 151mpan2 690 . . . . . . . . . 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 13339 . . . . . . . . . 10 (1...𝑛) ∈ Fin
155 fnfi 8784 . . . . . . . . . 10 (((𝑓 ↾ (1...𝑛)) Fn (1...𝑛) ∧ (1...𝑛) ∈ Fin) → (𝑓 ↾ (1...𝑛)) ∈ Fin)
156154, 155mpan2 690 . . . . . . . . 9 ((𝑓 ↾ (1...𝑛)) Fn (1...𝑛) → (𝑓 ↾ (1...𝑛)) ∈ Fin)
157 rnfi 8795 . . . . . . . . 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 5847 . . . . . . . . . . 11 (𝑓 ↾ (1...𝑛)) ⊆ 𝑓
160 rnss 5777 . . . . . . . . . . 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 3928 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ (toCaraSiga‘𝑀))
164162, 36sstrd 3928 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ (𝐴 ∪ {∅}))
165 nfcv 2958 . . . . . . . . . . . . 13 𝑧𝑦
166 nfcv 2958 . . . . . . . . . . . . 13 𝑦𝑧
167 id 22 . . . . . . . . . . . . 13 (𝑦 = 𝑧𝑦 = 𝑧)
168165, 166, 167cbvdisj 5008 . . . . . . . . . . . 12 (Disj 𝑦𝐴 𝑦Disj 𝑧𝐴 𝑧)
169 disjun0 30362 . . . . . . . . . . . 12 (Disj 𝑧𝐴 𝑧Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
170168, 169sylbi 220 . . . . . . . . . . 11 (Disj 𝑦𝐴 𝑦Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
171117, 170syl 17 . . . . . . . . . 10 (𝜑Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
172171adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
173 disjss1 5004 . . . . . . . . 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 4522 . . . . . . . . 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 31689 . . . . . . 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 484 . . . . . 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 4813 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ (𝐴 ∪ {∅}))
180 uniun 4826 . . . . . . . . . . . 12 (𝐴 ∪ {∅}) = ( 𝐴 {∅})
1812unisn 4823 . . . . . . . . . . . . 13 {∅} = ∅
182181uneq2i 4090 . . . . . . . . . . . 12 ( 𝐴 {∅}) = ( 𝐴 ∪ ∅)
183 un0 4301 . . . . . . . . . . . 12 ( 𝐴 ∪ ∅) = 𝐴
184180, 182, 1833eqtri 2828 . . . . . . . . . . 11 (𝐴 ∪ {∅}) = 𝐴
185179, 184sseqtrdi 3968 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ 𝐴)
186185adantr 484 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑓 ↾ (1...𝑛)) ⊆ 𝐴)
187 uniss 4811 . . . . . . . . . . . 12 (𝐴 ⊆ 𝒫 𝑂 𝐴 𝒫 𝑂)
188 unipw 5311 . . . . . . . . . . . 12 𝒫 𝑂 = 𝑂
189187, 188sseqtrdi 3968 . . . . . . . . . . 11 (𝐴 ⊆ 𝒫 𝑂 𝐴𝑂)
19028, 189syl 17 . . . . . . . . . 10 (𝜑 𝐴𝑂)
191190ad2antrr 725 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → 𝐴𝑂)
192186, 191sstrd 3928 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑓 ↾ (1...𝑛)) ⊆ 𝑂)
193 sseqin2 4145 . . . . . . . 8 ( ran (𝑓 ↾ (1...𝑛)) ⊆ 𝑂 ↔ (𝑂 ran (𝑓 ↾ (1...𝑛))) = ran (𝑓 ↾ (1...𝑛)))
194192, 193sylib 221 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑂 ran (𝑓 ↾ (1...𝑛))) = ran (𝑓 ↾ (1...𝑛)))
195194fveq2d 6653 . . . . . 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 1915 . . . . . . . 8 𝑧((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ)
197164adantr 484 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑓 ↾ (1...𝑛)) ⊆ (𝐴 ∪ {∅}))
19828ad2antrr 725 . . . . . . . . . . . . . . 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 4705 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → {∅} ⊆ 𝒫 𝑂)
201198, 200unssd 4116 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
202197, 201sstrd 3928 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑓 ↾ (1...𝑛)) ⊆ 𝒫 𝑂)
203202sselda 3918 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → 𝑧 ∈ 𝒫 𝑂)
204203elpwid 4511 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → 𝑧𝑂)
205 sseqin2 4145 . . . . . . . . . . 11 (𝑧𝑂 ↔ (𝑂𝑧) = 𝑧)
206204, 205sylib 221 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → (𝑂𝑧) = 𝑧)
207206fveq2d 6653 . . . . . . . . 9 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → (𝑀‘(𝑂𝑧)) = (𝑀𝑧))
208207ralrimiva 3152 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ∀𝑧 ∈ ran (𝑓 ↾ (1...𝑛))(𝑀‘(𝑂𝑧)) = (𝑀𝑧))
209196, 208esumeq2d 31410 . . . . . . 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 5821 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓 ↾ (1...𝑛)) = ((𝑘 ∈ ℕ ↦ (𝑓𝑘)) ↾ (1...𝑛)))
211210adantr 484 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑓 ↾ (1...𝑛)) = ((𝑘 ∈ ℕ ↦ (𝑓𝑘)) ↾ (1...𝑛)))
212 resmpt 5876 . . . . . . . . . . . 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 2852 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑓 ↾ (1...𝑛)) = (𝑘 ∈ (1...𝑛) ↦ (𝑓𝑘)))
215214eqcomd 2807 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ (1...𝑛) ↦ (𝑓𝑘)) = (𝑓 ↾ (1...𝑛)))
216215rneqd 5776 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑘 ∈ (1...𝑛) ↦ (𝑓𝑘)) = ran (𝑓 ↾ (1...𝑛)))
217196, 216esumeq1d 31408 . . . . . . 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 725 . . . . . . . . 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 3918 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
22284adantlr 714 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ 𝒫 𝑂)
223221, 222syldan 594 . . . . . . . . 9 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑓𝑘) ∈ 𝒫 𝑂)
224219, 223ffvelrnd 6833 . . . . . . . 8 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑀‘(𝑓𝑘)) ∈ (0[,]+∞))
225 simpr 488 . . . . . . . . . 10 (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓𝑘) = ∅) → (𝑓𝑘) = ∅)
226225fveq2d 6653 . . . . . . . . 9 (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓𝑘) = ∅) → (𝑀‘(𝑓𝑘)) = (𝑀‘∅))
22721ad3antrrr 729 . . . . . . . . 9 (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓𝑘) = ∅) → (𝑀‘∅) = 0)
228226, 227eqtrd 2836 . . . . . . . 8 (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓𝑘) = ∅) → (𝑀‘(𝑓𝑘)) = 0)
229 disjss1 5004 . . . . . . . . . . 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 484 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → Disj 𝑘 ∈ (1...𝑛)(𝑓𝑘))
23376, 218, 224, 223, 228, 232esumrnmpt2 31441 . . . . . . 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 2842 . . . . . 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 2844 . . . . 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 1174 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))
23817, 19, 185, 138, 237carsgmon 31686 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑀 ran (𝑓 ↾ (1...𝑛))) ≤ (𝑀 𝐴))
239238adantr 484 . . . . 5 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑀 ran (𝑓 ↾ (1...𝑛))) ≤ (𝑀 𝐴))
240235, 239eqbrtrrd 5057 . . . 4 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝑓𝑘)) ≤ (𝑀 𝐴))
241139, 85, 240esumgect 31463 . . 3 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑘 ∈ ℕ(𝑀‘(𝑓𝑘)) ≤ (𝑀 𝐴))
242133, 241eqbrtrrd 5057 . 2 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧𝐴(𝑀𝑧) ≤ (𝑀 𝐴))
2436, 242exlimddv 1936 1 (𝜑 → Σ*𝑧𝐴(𝑀𝑧) ≤ (𝑀 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2112  wral 3109  Vcvv 3444  cdif 3881  cun 3882  cin 3883  wss 3884  c0 4246  𝒫 cpw 4500  {csn 4528   cuni 4803   ciun 4884  Disj wdisj 4998   class class class wbr 5033  cmpt 5113  ccnv 5522  ran crn 5524  cres 5525  cima 5526  Fun wfun 6322   Fn wfn 6323  wf 6324  1-1-ontowf1o 6327  cfv 6328  (class class class)co 7139  ωcom 7564  cdom 8494  Fincfn 8496  0cc0 10530  1c1 10531  +∞cpnf 10665  *cxr 10667  cle 10669  cn 11629  [,]cicc 12733  ...cfz 12889  Σ*cesum 31400  toCaraSigaccarsg 31673
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608  ax-addf 10609  ax-mulf 10610
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-disj 4999  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-q 12341  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-ioo 12734  df-ioc 12735  df-ico 12736  df-icc 12737  df-fz 12890  df-fzo 13033  df-fl 13161  df-mod 13237  df-seq 13369  df-exp 13430  df-fac 13634  df-bc 13663  df-hash 13691  df-shft 14422  df-cj 14454  df-re 14455  df-im 14456  df-sqrt 14590  df-abs 14591  df-limsup 14824  df-clim 14841  df-rlim 14842  df-sum 15039  df-ef 15417  df-sin 15419  df-cos 15420  df-pi 15422  df-struct 16481  df-ndx 16482  df-slot 16483  df-base 16485  df-sets 16486  df-ress 16487  df-plusg 16574  df-mulr 16575  df-starv 16576  df-sca 16577  df-vsca 16578  df-ip 16579  df-tset 16580  df-ple 16581  df-ds 16583  df-unif 16584  df-hom 16585  df-cco 16586  df-rest 16692  df-topn 16693  df-0g 16711  df-gsum 16712  df-topgen 16713  df-pt 16714  df-prds 16717  df-ordt 16770  df-xrs 16771  df-qtop 16776  df-imas 16777  df-xps 16779  df-mre 16853  df-mrc 16854  df-acs 16856  df-ps 17806  df-tsr 17807  df-plusf 17847  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-mhm 17952  df-submnd 17953  df-grp 18102  df-minusg 18103  df-sbg 18104  df-mulg 18221  df-subg 18272  df-cntz 18443  df-cmn 18904  df-abl 18905  df-mgp 19237  df-ur 19249  df-ring 19296  df-cring 19297  df-subrg 19530  df-abv 19585  df-lmod 19633  df-scaf 19634  df-sra 19941  df-rgmod 19942  df-psmet 20087  df-xmet 20088  df-met 20089  df-bl 20090  df-mopn 20091  df-fbas 20092  df-fg 20093  df-cnfld 20096  df-top 21503  df-topon 21520  df-topsp 21542  df-bases 21555  df-cld 21628  df-ntr 21629  df-cls 21630  df-nei 21707  df-lp 21745  df-perf 21746  df-cn 21836  df-cnp 21837  df-haus 21924  df-tx 22171  df-hmeo 22364  df-fil 22455  df-fm 22547  df-flim 22548  df-flf 22549  df-tmd 22681  df-tgp 22682  df-tsms 22736  df-trg 22769  df-xms 22931  df-ms 22932  df-tms 22933  df-nm 23193  df-ngp 23194  df-nrg 23196  df-nlm 23197  df-ii 23486  df-cncf 23487  df-limc 24473  df-dv 24474  df-log 25152  df-esum 31401  df-carsg 31674
This theorem is referenced by:  omsmeas  31695
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