Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  carsggect Structured version   Visualization version   GIF version

Theorem carsggect 30508
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 4823 . . . 4 ∅ ∈ V
32a1i 11 . . 3 (𝜑 → ∅ ∈ V)
4 carsggect.0 . . 3 (𝜑 → ¬ ∅ ∈ 𝐴)
5 padct 29625 . . 3 ((𝐴 ≼ ω ∧ ∅ ∈ V ∧ ¬ ∅ ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
61, 3, 4, 5syl3anc 1366 . 2 (𝜑 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
7 nfv 1883 . . . . 5 𝑧(𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
8 simpr1 1087 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝑓:ℕ⟶(𝐴 ∪ {∅}))
98feqmptd 6288 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝑓 = (𝑘 ∈ ℕ ↦ (𝑓𝑘)))
109rneqd 5385 . . . . 5 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran 𝑓 = ran (𝑘 ∈ ℕ ↦ (𝑓𝑘)))
117, 10esumeq1d 30225 . . . 4 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) = Σ*𝑧 ∈ ran (𝑘 ∈ ℕ ↦ (𝑓𝑘))(𝑀𝑧))
12 fvex 6239 . . . . . . . . . 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 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝐴 ⊆ (toCaraSiga‘𝑀))
16 carsgval.1 . . . . . . . . . . . . 13 (𝜑𝑂𝑉)
1716adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝑂𝑉)
18 carsgval.2 . . . . . . . . . . . . 13 (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))
1918adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
20 carsgsiga.1 . . . . . . . . . . . . 13 (𝜑 → (𝑀‘∅) = 0)
2120adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑀‘∅) = 0)
2217, 19, 210elcarsg 30497 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ∅ ∈ (toCaraSiga‘𝑀))
2322snssd 4372 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → {∅} ⊆ (toCaraSiga‘𝑀))
2415, 23unssd 3822 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝐴 ∪ {∅}) ⊆ (toCaraSiga‘𝑀))
2513, 24ssexd 4838 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝐴 ∪ {∅}) ∈ V)
2619adantr 480 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
2716, 18carsgcl 30494 . . . . . . . . . . . . 13 (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂)
2814, 27sstrd 3646 . . . . . . . . . . . 12 (𝜑𝐴 ⊆ 𝒫 𝑂)
2928adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝐴 ⊆ 𝒫 𝑂)
30 0elpw 4864 . . . . . . . . . . . . 13 ∅ ∈ 𝒫 𝑂
3130a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ∅ ∈ 𝒫 𝑂)
3231snssd 4372 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → {∅} ⊆ 𝒫 𝑂)
3329, 32unssd 3822 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
3433sselda 3636 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → 𝑧 ∈ 𝒫 𝑂)
3526, 34ffvelrnd 6400 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → (𝑀𝑧) ∈ (0[,]+∞))
36 frn 6091 . . . . . . . . 9 (𝑓:ℕ⟶(𝐴 ∪ {∅}) → ran 𝑓 ⊆ (𝐴 ∪ {∅}))
378, 36syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran 𝑓 ⊆ (𝐴 ∪ {∅}))
387, 25, 35, 37esummono 30244 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) ≤ Σ*𝑧 ∈ (𝐴 ∪ {∅})(𝑀𝑧))
39 ctex 8012 . . . . . . . . . 10 (𝐴 ≼ ω → 𝐴 ∈ V)
401, 39syl 17 . . . . . . . . 9 (𝜑𝐴 ∈ V)
4140adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝐴 ∈ V)
4213, 23ssexd 4838 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → {∅} ∈ V)
4319adantr 480 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧𝐴) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
4429sselda 3636 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧𝐴) → 𝑧 ∈ 𝒫 𝑂)
4543, 44ffvelrnd 6400 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧𝐴) → (𝑀𝑧) ∈ (0[,]+∞))
46 elsni 4227 . . . . . . . . . . 11 (𝑧 ∈ {∅} → 𝑧 = ∅)
4746adantl 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ {∅}) → 𝑧 = ∅)
4847fveq2d 6233 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ {∅}) → (𝑀𝑧) = (𝑀‘∅))
4921adantr 480 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ {∅}) → (𝑀‘∅) = 0)
5048, 49eqtrd 2685 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ {∅}) → (𝑀𝑧) = 0)
5141, 42, 45, 50esumpad 30245 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ (𝐴 ∪ {∅})(𝑀𝑧) = Σ*𝑧𝐴(𝑀𝑧))
5238, 51breqtrd 4711 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) ≤ Σ*𝑧𝐴(𝑀𝑧))
5337, 24sstrd 3646 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran 𝑓 ⊆ (toCaraSiga‘𝑀))
54 ssexg 4837 . . . . . . . 8 ((ran 𝑓 ⊆ (toCaraSiga‘𝑀) ∧ (toCaraSiga‘𝑀) ∈ V) → ran 𝑓 ∈ V)
5553, 12, 54sylancl 695 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran 𝑓 ∈ V)
5619adantr 480 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ ran 𝑓) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
5737, 33sstrd 3646 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran 𝑓 ⊆ 𝒫 𝑂)
5857sselda 3636 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ ran 𝑓) → 𝑧 ∈ 𝒫 𝑂)
5956, 58ffvelrnd 6400 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑧 ∈ ran 𝑓) → (𝑀𝑧) ∈ (0[,]+∞))
60 simpr2 1088 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝐴 ⊆ ran 𝑓)
617, 55, 59, 60esummono 30244 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧𝐴(𝑀𝑧) ≤ Σ*𝑧 ∈ ran 𝑓(𝑀𝑧))
6252, 61jca 553 . . . . 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)))
63 iccssxr 12294 . . . . . . 7 (0[,]+∞) ⊆ ℝ*
6459ralrimiva 2995 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ∀𝑧 ∈ ran 𝑓(𝑀𝑧) ∈ (0[,]+∞))
65 nfcv 2793 . . . . . . . . 9 𝑧ran 𝑓
6665esumcl 30220 . . . . . . . 8 ((ran 𝑓 ∈ V ∧ ∀𝑧 ∈ ran 𝑓(𝑀𝑧) ∈ (0[,]+∞)) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) ∈ (0[,]+∞))
6755, 64, 66syl2anc 694 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) ∈ (0[,]+∞))
6863, 67sseldi 3634 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) ∈ ℝ*)
6945ralrimiva 2995 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ∀𝑧𝐴 (𝑀𝑧) ∈ (0[,]+∞))
70 nfcv 2793 . . . . . . . . 9 𝑧𝐴
7170esumcl 30220 . . . . . . . 8 ((𝐴 ∈ V ∧ ∀𝑧𝐴 (𝑀𝑧) ∈ (0[,]+∞)) → Σ*𝑧𝐴(𝑀𝑧) ∈ (0[,]+∞))
7241, 69, 71syl2anc 694 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧𝐴(𝑀𝑧) ∈ (0[,]+∞))
7363, 72sseldi 3634 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧𝐴(𝑀𝑧) ∈ ℝ*)
74 xrletri3 12023 . . . . . 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))))
7568, 73, 74syl2anc 694 . . . . 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))))
7662, 75mpbird 247 . . . 4 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀𝑧) = Σ*𝑧𝐴(𝑀𝑧))
77 fveq2 6229 . . . . 5 (𝑧 = (𝑓𝑘) → (𝑀𝑧) = (𝑀‘(𝑓𝑘)))
78 nnex 11064 . . . . . 6 ℕ ∈ V
7978a1i 11 . . . . 5 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ℕ ∈ V)
8019adantr 480 . . . . . 6 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
8133adantr 480 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
828adantr 480 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → 𝑓:ℕ⟶(𝐴 ∪ {∅}))
83 simpr 476 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
8482, 83ffvelrnd 6400 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ (𝐴 ∪ {∅}))
8581, 84sseldd 3637 . . . . . 6 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ 𝒫 𝑂)
8680, 85ffvelrnd 6400 . . . . 5 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑀‘(𝑓𝑘)) ∈ (0[,]+∞))
87 simpr 476 . . . . . . 7 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓𝑘) = ∅) → (𝑓𝑘) = ∅)
8887fveq2d 6233 . . . . . 6 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓𝑘) = ∅) → (𝑀‘(𝑓𝑘)) = (𝑀‘∅))
8921ad2antrr 762 . . . . . 6 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓𝑘) = ∅) → (𝑀‘∅) = 0)
9088, 89eqtrd 2685 . . . . 5 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓𝑘) = ∅) → (𝑀‘(𝑓𝑘)) = 0)
91 cnvimass 5520 . . . . . . . 8 (𝑓𝐴) ⊆ dom 𝑓
9291a1i 11 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓𝐴) ⊆ dom 𝑓)
93 fdm 6089 . . . . . . . 8 (𝑓:ℕ⟶(𝐴 ∪ {∅}) → dom 𝑓 = ℕ)
948, 93syl 17 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → dom 𝑓 = ℕ)
9592, 94sseqtrd 3674 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓𝐴) ⊆ ℕ)
96 ffun 6086 . . . . . . . . . . 11 (𝑓:ℕ⟶(𝐴 ∪ {∅}) → Fun 𝑓)
978, 96syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Fun 𝑓)
9897adantr 480 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (𝑓𝐴))) → Fun 𝑓)
99 difpreima 6383 . . . . . . . . . . . . 13 (Fun 𝑓 → (𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) = ((𝑓 “ (𝐴 ∪ {∅})) ∖ (𝑓𝐴)))
1008, 96, 993syl 18 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) = ((𝑓 “ (𝐴 ∪ {∅})) ∖ (𝑓𝐴)))
101 fimacnv 6387 . . . . . . . . . . . . . 14 (𝑓:ℕ⟶(𝐴 ∪ {∅}) → (𝑓 “ (𝐴 ∪ {∅})) = ℕ)
1028, 101syl 17 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓 “ (𝐴 ∪ {∅})) = ℕ)
103102difeq1d 3760 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ((𝑓 “ (𝐴 ∪ {∅})) ∖ (𝑓𝐴)) = (ℕ ∖ (𝑓𝐴)))
104100, 103eqtrd 2685 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) = (ℕ ∖ (𝑓𝐴)))
105 uncom 3790 . . . . . . . . . . . . . . . 16 ({∅} ∪ 𝐴) = (𝐴 ∪ {∅})
106105difeq1i 3757 . . . . . . . . . . . . . . 15 (({∅} ∪ 𝐴) ∖ 𝐴) = ((𝐴 ∪ {∅}) ∖ 𝐴)
107 difun2 4081 . . . . . . . . . . . . . . 15 (({∅} ∪ 𝐴) ∖ 𝐴) = ({∅} ∖ 𝐴)
108106, 107eqtr3i 2675 . . . . . . . . . . . . . 14 ((𝐴 ∪ {∅}) ∖ 𝐴) = ({∅} ∖ 𝐴)
109 difss 3770 . . . . . . . . . . . . . 14 ({∅} ∖ 𝐴) ⊆ {∅}
110108, 109eqsstri 3668 . . . . . . . . . . . . 13 ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅}
111110a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅})
112 sspreima 29575 . . . . . . . . . . . 12 ((Fun 𝑓 ∧ ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅}) → (𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) ⊆ (𝑓 “ {∅}))
11397, 111, 112syl2anc 694 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) ⊆ (𝑓 “ {∅}))
114104, 113eqsstr3d 3673 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (ℕ ∖ (𝑓𝐴)) ⊆ (𝑓 “ {∅}))
115114sselda 3636 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (𝑓𝐴))) → 𝑘 ∈ (𝑓 “ {∅}))
116 fvimacnvi 6371 . . . . . . . . 9 ((Fun 𝑓𝑘 ∈ (𝑓 “ {∅})) → (𝑓𝑘) ∈ {∅})
11798, 115, 116syl2anc 694 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (𝑓𝐴))) → (𝑓𝑘) ∈ {∅})
118 elsni 4227 . . . . . . . 8 ((𝑓𝑘) ∈ {∅} → (𝑓𝑘) = ∅)
119117, 118syl 17 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (𝑓𝐴))) → (𝑓𝑘) = ∅)
120119ralrimiva 2995 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ∀𝑘 ∈ (ℕ ∖ (𝑓𝐴))(𝑓𝑘) = ∅)
121 carsggect.3 . . . . . . . 8 (𝜑Disj 𝑦𝐴 𝑦)
122121adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Disj 𝑦𝐴 𝑦)
123 simpr3 1089 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Fun (𝑓𝐴))
124 fresf1o 29561 . . . . . . . . . 10 ((Fun 𝑓𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto𝐴)
12597, 60, 123, 124syl3anc 1366 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓 ↾ (𝑓𝐴)):(𝑓𝐴)–1-1-onto𝐴)
126 simpr 476 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑦 = ((𝑓 ↾ (𝑓𝐴))‘𝑘)) → 𝑦 = ((𝑓 ↾ (𝑓𝐴))‘𝑘))
127125, 126disjrdx 29530 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (Disj 𝑘 ∈ (𝑓𝐴)((𝑓 ↾ (𝑓𝐴))‘𝑘) ↔ Disj 𝑦𝐴 𝑦))
128 fvres 6245 . . . . . . . . . 10 (𝑘 ∈ (𝑓𝐴) → ((𝑓 ↾ (𝑓𝐴))‘𝑘) = (𝑓𝑘))
129128adantl 481 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑘 ∈ (𝑓𝐴)) → ((𝑓 ↾ (𝑓𝐴))‘𝑘) = (𝑓𝑘))
130129disjeq2dv 4657 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (Disj 𝑘 ∈ (𝑓𝐴)((𝑓 ↾ (𝑓𝐴))‘𝑘) ↔ Disj 𝑘 ∈ (𝑓𝐴)(𝑓𝑘)))
131127, 130bitr3d 270 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (Disj 𝑦𝐴 𝑦Disj 𝑘 ∈ (𝑓𝐴)(𝑓𝑘)))
132122, 131mpbid 222 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Disj 𝑘 ∈ (𝑓𝐴)(𝑓𝑘))
133 disjss3 4684 . . . . . . 7 (((𝑓𝐴) ⊆ ℕ ∧ ∀𝑘 ∈ (ℕ ∖ (𝑓𝐴))(𝑓𝑘) = ∅) → (Disj 𝑘 ∈ (𝑓𝐴)(𝑓𝑘) ↔ Disj 𝑘 ∈ ℕ (𝑓𝑘)))
134133biimpa 500 . . . . . 6 ((((𝑓𝐴) ⊆ ℕ ∧ ∀𝑘 ∈ (ℕ ∖ (𝑓𝐴))(𝑓𝑘) = ∅) ∧ Disj 𝑘 ∈ (𝑓𝐴)(𝑓𝑘)) → Disj 𝑘 ∈ ℕ (𝑓𝑘))
13595, 120, 132, 134syl21anc 1365 . . . . 5 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Disj 𝑘 ∈ ℕ (𝑓𝑘))
13677, 79, 86, 85, 90, 135esumrnmpt2 30258 . . . 4 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧 ∈ ran (𝑘 ∈ ℕ ↦ (𝑓𝑘))(𝑀𝑧) = Σ*𝑘 ∈ ℕ(𝑀‘(𝑓𝑘)))
13711, 76, 1363eqtr3rd 2694 . . 3 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑘 ∈ ℕ(𝑀‘(𝑓𝑘)) = Σ*𝑧𝐴(𝑀𝑧))
138 uniiun 4605 . . . . . . 7 𝐴 = 𝑥𝐴 𝑥
13928sselda 3636 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝑥 ∈ 𝒫 𝑂)
14040, 139elpwiuncl 29485 . . . . . . 7 (𝜑 𝑥𝐴 𝑥 ∈ 𝒫 𝑂)
141138, 140syl5eqel 2734 . . . . . 6 (𝜑 𝐴 ∈ 𝒫 𝑂)
142141adantr 480 . . . . 5 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝐴 ∈ 𝒫 𝑂)
14319, 142ffvelrnd 6400 . . . 4 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑀 𝐴) ∈ (0[,]+∞))
144 carsgsiga.2 . . . . . . . . . 10 ((𝜑𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))
1451443adant1r 1359 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))
146 fveq2 6229 . . . . . . . . . 10 (𝑦 = 𝑧 → (𝑀𝑦) = (𝑀𝑧))
147 nfcv 2793 . . . . . . . . . 10 𝑧𝑥
148 nfcv 2793 . . . . . . . . . 10 𝑦𝑥
149 nfcv 2793 . . . . . . . . . 10 𝑧(𝑀𝑦)
150 nfcv 2793 . . . . . . . . . 10 𝑦(𝑀𝑧)
151146, 147, 148, 149, 150cbvesum 30232 . . . . . . . . 9 Σ*𝑦𝑥(𝑀𝑦) = Σ*𝑧𝑥(𝑀𝑧)
152145, 151syl6breq 4726 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑧𝑥(𝑀𝑧))
153 ffn 6083 . . . . . . . . . 10 (𝑓:ℕ⟶(𝐴 ∪ {∅}) → 𝑓 Fn ℕ)
154 fz1ssnn 12410 . . . . . . . . . . 11 (1...𝑛) ⊆ ℕ
155 fnssres 6042 . . . . . . . . . . 11 ((𝑓 Fn ℕ ∧ (1...𝑛) ⊆ ℕ) → (𝑓 ↾ (1...𝑛)) Fn (1...𝑛))
156154, 155mpan2 707 . . . . . . . . . 10 (𝑓 Fn ℕ → (𝑓 ↾ (1...𝑛)) Fn (1...𝑛))
1578, 153, 1563syl 18 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓 ↾ (1...𝑛)) Fn (1...𝑛))
158 fzfi 12811 . . . . . . . . . 10 (1...𝑛) ∈ Fin
159 fnfi 8279 . . . . . . . . . 10 (((𝑓 ↾ (1...𝑛)) Fn (1...𝑛) ∧ (1...𝑛) ∈ Fin) → (𝑓 ↾ (1...𝑛)) ∈ Fin)
160158, 159mpan2 707 . . . . . . . . 9 ((𝑓 ↾ (1...𝑛)) Fn (1...𝑛) → (𝑓 ↾ (1...𝑛)) ∈ Fin)
161 rnfi 8290 . . . . . . . . 9 ((𝑓 ↾ (1...𝑛)) ∈ Fin → ran (𝑓 ↾ (1...𝑛)) ∈ Fin)
162157, 160, 1613syl 18 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran (𝑓 ↾ (1...𝑛)) ∈ Fin)
163 resss 5457 . . . . . . . . . . 11 (𝑓 ↾ (1...𝑛)) ⊆ 𝑓
164 rnss 5386 . . . . . . . . . . 11 ((𝑓 ↾ (1...𝑛)) ⊆ 𝑓 → ran (𝑓 ↾ (1...𝑛)) ⊆ ran 𝑓)
165163, 164ax-mp 5 . . . . . . . . . 10 ran (𝑓 ↾ (1...𝑛)) ⊆ ran 𝑓
166165a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ ran 𝑓)
167166, 53sstrd 3646 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ (toCaraSiga‘𝑀))
168166, 37sstrd 3646 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ (𝐴 ∪ {∅}))
169 nfcv 2793 . . . . . . . . . . . . 13 𝑧𝑦
170 nfcv 2793 . . . . . . . . . . . . 13 𝑦𝑧
171 id 22 . . . . . . . . . . . . 13 (𝑦 = 𝑧𝑦 = 𝑧)
172169, 170, 171cbvdisj 4662 . . . . . . . . . . . 12 (Disj 𝑦𝐴 𝑦Disj 𝑧𝐴 𝑧)
173 disjun0 29534 . . . . . . . . . . . 12 (Disj 𝑧𝐴 𝑧Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
174172, 173sylbi 207 . . . . . . . . . . 11 (Disj 𝑦𝐴 𝑦Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
175121, 174syl 17 . . . . . . . . . 10 (𝜑Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
176175adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
177 disjss1 4658 . . . . . . . . 9 (ran (𝑓 ↾ (1...𝑛)) ⊆ (𝐴 ∪ {∅}) → (Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧Disj 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))𝑧))
178168, 176, 177sylc 65 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Disj 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))𝑧)
179 pwidg 4206 . . . . . . . . 9 (𝑂𝑉𝑂 ∈ 𝒫 𝑂)
18017, 179syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → 𝑂 ∈ 𝒫 𝑂)
18117, 19, 21, 152, 162, 167, 178, 180carsgclctunlem1 30507 . . . . . . 7 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑀‘(𝑂 ran (𝑓 ↾ (1...𝑛)))) = Σ*𝑧 ∈ ran (𝑓 ↾ (1...𝑛))(𝑀‘(𝑂𝑧)))
182181adantr 480 . . . . . 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)))
183168unissd 4494 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ (𝐴 ∪ {∅}))
184 uniun 4488 . . . . . . . . . . . 12 (𝐴 ∪ {∅}) = ( 𝐴 {∅})
1852unisn 4483 . . . . . . . . . . . . 13 {∅} = ∅
186185uneq2i 3797 . . . . . . . . . . . 12 ( 𝐴 {∅}) = ( 𝐴 ∪ ∅)
187 un0 4000 . . . . . . . . . . . 12 ( 𝐴 ∪ ∅) = 𝐴
188184, 186, 1873eqtri 2677 . . . . . . . . . . 11 (𝐴 ∪ {∅}) = 𝐴
189183, 188syl6sseq 3684 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ 𝐴)
190189adantr 480 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑓 ↾ (1...𝑛)) ⊆ 𝐴)
191 uniss 4490 . . . . . . . . . . . 12 (𝐴 ⊆ 𝒫 𝑂 𝐴 𝒫 𝑂)
192 unipw 4948 . . . . . . . . . . . 12 𝒫 𝑂 = 𝑂
193191, 192syl6sseq 3684 . . . . . . . . . . 11 (𝐴 ⊆ 𝒫 𝑂 𝐴𝑂)
19428, 193syl 17 . . . . . . . . . 10 (𝜑 𝐴𝑂)
195194ad2antrr 762 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → 𝐴𝑂)
196190, 195sstrd 3646 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑓 ↾ (1...𝑛)) ⊆ 𝑂)
197 sseqin2 3850 . . . . . . . 8 ( ran (𝑓 ↾ (1...𝑛)) ⊆ 𝑂 ↔ (𝑂 ran (𝑓 ↾ (1...𝑛))) = ran (𝑓 ↾ (1...𝑛)))
198196, 197sylib 208 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑂 ran (𝑓 ↾ (1...𝑛))) = ran (𝑓 ↾ (1...𝑛)))
199198fveq2d 6233 . . . . . 6 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝑂 ran (𝑓 ↾ (1...𝑛)))) = (𝑀 ran (𝑓 ↾ (1...𝑛))))
200 nfv 1883 . . . . . . . 8 𝑧((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ)
201168adantr 480 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑓 ↾ (1...𝑛)) ⊆ (𝐴 ∪ {∅}))
20228ad2antrr 762 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ 𝒫 𝑂)
20330a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ∅ ∈ 𝒫 𝑂)
204203snssd 4372 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → {∅} ⊆ 𝒫 𝑂)
205202, 204unssd 3822 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
206201, 205sstrd 3646 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑓 ↾ (1...𝑛)) ⊆ 𝒫 𝑂)
207206sselda 3636 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → 𝑧 ∈ 𝒫 𝑂)
208207elpwid 4203 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → 𝑧𝑂)
209 sseqin2 3850 . . . . . . . . . . 11 (𝑧𝑂 ↔ (𝑂𝑧) = 𝑧)
210208, 209sylib 208 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → (𝑂𝑧) = 𝑧)
211210fveq2d 6233 . . . . . . . . 9 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → (𝑀‘(𝑂𝑧)) = (𝑀𝑧))
212211ralrimiva 2995 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ∀𝑧 ∈ ran (𝑓 ↾ (1...𝑛))(𝑀‘(𝑂𝑧)) = (𝑀𝑧))
213200, 212esumeq2d 30227 . . . . . . 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))
2149reseq1d 5427 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑓 ↾ (1...𝑛)) = ((𝑘 ∈ ℕ ↦ (𝑓𝑘)) ↾ (1...𝑛)))
215214adantr 480 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑓 ↾ (1...𝑛)) = ((𝑘 ∈ ℕ ↦ (𝑓𝑘)) ↾ (1...𝑛)))
216 resmpt 5484 . . . . . . . . . . . 12 ((1...𝑛) ⊆ ℕ → ((𝑘 ∈ ℕ ↦ (𝑓𝑘)) ↾ (1...𝑛)) = (𝑘 ∈ (1...𝑛) ↦ (𝑓𝑘)))
217154, 216ax-mp 5 . . . . . . . . . . 11 ((𝑘 ∈ ℕ ↦ (𝑓𝑘)) ↾ (1...𝑛)) = (𝑘 ∈ (1...𝑛) ↦ (𝑓𝑘))
218215, 217syl6eq 2701 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑓 ↾ (1...𝑛)) = (𝑘 ∈ (1...𝑛) ↦ (𝑓𝑘)))
219218eqcomd 2657 . . . . . . . . 9 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ (1...𝑛) ↦ (𝑓𝑘)) = (𝑓 ↾ (1...𝑛)))
220219rneqd 5385 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑘 ∈ (1...𝑛) ↦ (𝑓𝑘)) = ran (𝑓 ↾ (1...𝑛)))
221200, 220esumeq1d 30225 . . . . . . 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))
222158a1i 11 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
22319ad2antrr 762 . . . . . . . . 9 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
224154a1i 11 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
225224sselda 3636 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
22685adantlr 751 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ 𝒫 𝑂)
227225, 226syldan 486 . . . . . . . . 9 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑓𝑘) ∈ 𝒫 𝑂)
228223, 227ffvelrnd 6400 . . . . . . . 8 ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑀‘(𝑓𝑘)) ∈ (0[,]+∞))
229 simpr 476 . . . . . . . . . 10 (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓𝑘) = ∅) → (𝑓𝑘) = ∅)
230229fveq2d 6233 . . . . . . . . 9 (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓𝑘) = ∅) → (𝑀‘(𝑓𝑘)) = (𝑀‘∅))
23121ad3antrrr 766 . . . . . . . . 9 (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓𝑘) = ∅) → (𝑀‘∅) = 0)
232230, 231eqtrd 2685 . . . . . . . 8 (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓𝑘) = ∅) → (𝑀‘(𝑓𝑘)) = 0)
233 disjss1 4658 . . . . . . . . . . 11 ((1...𝑛) ⊆ ℕ → (Disj 𝑘 ∈ ℕ (𝑓𝑘) → Disj 𝑘 ∈ (1...𝑛)(𝑓𝑘)))
234154, 233ax-mp 5 . . . . . . . . . 10 (Disj 𝑘 ∈ ℕ (𝑓𝑘) → Disj 𝑘 ∈ (1...𝑛)(𝑓𝑘))
235135, 234syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Disj 𝑘 ∈ (1...𝑛)(𝑓𝑘))
236235adantr 480 . . . . . . . 8 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → Disj 𝑘 ∈ (1...𝑛)(𝑓𝑘))
23777, 222, 228, 227, 232, 236esumrnmpt2 30258 . . . . . . 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)))
238213, 221, 2373eqtr2d 2691 . . . . . 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)))
239182, 199, 2383eqtr3d 2693 . . . . 5 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑀 ran (𝑓 ↾ (1...𝑛))) = Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝑓𝑘)))
240 carsggect.4 . . . . . . . 8 ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))
2412403adant1r 1359 . . . . . . 7 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))
24217, 19, 189, 142, 241carsgmon 30504 . . . . . 6 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → (𝑀 ran (𝑓 ↾ (1...𝑛))) ≤ (𝑀 𝐴))
243242adantr 480 . . . . 5 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑀 ran (𝑓 ↾ (1...𝑛))) ≤ (𝑀 𝐴))
244239, 243eqbrtrrd 4709 . . . 4 (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) ∧ 𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝑓𝑘)) ≤ (𝑀 𝐴))
245143, 86, 244esumgect 30280 . . 3 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑘 ∈ ℕ(𝑀‘(𝑓𝑘)) ≤ (𝑀 𝐴))
246137, 245eqbrtrrd 4709 . 2 ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴))) → Σ*𝑧𝐴(𝑀𝑧) ≤ (𝑀 𝐴))
2476, 246exlimddv 1903 1 (𝜑 → Σ*𝑧𝐴(𝑀𝑧) ≤ (𝑀 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wral 2941  Vcvv 3231  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210   cuni 4468   ciun 4552  Disj wdisj 4652   class class class wbr 4685  cmpt 4762  ccnv 5142  dom cdm 5143  ran crn 5144  cres 5145  cima 5146  Fun wfun 5920   Fn wfn 5921  wf 5922  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  ωcom 7107  cdom 7995  Fincfn 7997  0cc0 9974  1c1 9975  +∞cpnf 10109  *cxr 10111  cle 10113  cn 11058  [,]cicc 12216  ...cfz 12364  Σ*cesum 30217  toCaraSigaccarsg 30491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-shft 13851  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-sum 14461  df-ef 14842  df-sin 14844  df-cos 14845  df-pi 14847  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-ordt 16208  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-ps 17247  df-tsr 17248  df-plusf 17288  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-subg 17638  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-cring 18596  df-subrg 18826  df-abv 18865  df-lmod 18913  df-scaf 18914  df-sra 19220  df-rgmod 19221  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lp 20988  df-perf 20989  df-cn 21079  df-cnp 21080  df-haus 21167  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-tmd 21923  df-tgp 21924  df-tsms 21977  df-trg 22010  df-xms 22172  df-ms 22173  df-tms 22174  df-nm 22434  df-ngp 22435  df-nrg 22437  df-nlm 22438  df-ii 22727  df-cncf 22728  df-limc 23675  df-dv 23676  df-log 24348  df-esum 30218  df-carsg 30492
This theorem is referenced by:  omsmeas  30513
  Copyright terms: Public domain W3C validator