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Theorem caratheodorylem1 39215
Description: Lemma used to prove that Caratheodory's construction is sigma-additive. This is the proof of the statement in the middle of Step (e) in the proof of Theorem 113C of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
caratheodorylem1.o (𝜑𝑂 ∈ OutMeas)
caratheodorylem1.s 𝑆 = (CaraGen‘𝑂)
caratheodorylem1.z 𝑍 = (ℤ𝑀)
caratheodorylem1.e (𝜑𝐸:𝑍𝑆)
caratheodorylem1.dj (𝜑Disj 𝑛𝑍 (𝐸𝑛))
caratheodorylem1.g 𝐺 = (𝑛𝑍 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖))
caratheodorylem1.n (𝜑𝑁 ∈ (ℤ𝑀))
Assertion
Ref Expression
caratheodorylem1 (𝜑 → (𝑂‘(𝐺𝑁)) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛)))))
Distinct variable groups:   𝑖,𝐸,𝑛   𝑖,𝐺,𝑛   𝑖,𝑀,𝑛   𝑖,𝑁,𝑛   𝑖,𝑂,𝑛   𝑛,𝑍   𝜑,𝑖,𝑛
Allowed substitution hints:   𝑆(𝑖,𝑛)   𝑍(𝑖)

Proof of Theorem caratheodorylem1
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 caratheodorylem1.n . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
2 eluzfz2 12170 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 17 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 id 22 . 2 (𝜑𝜑)
5 fveq2 6083 . . . . . 6 (𝑗 = 𝑀 → (𝐺𝑗) = (𝐺𝑀))
65fveq2d 6087 . . . . 5 (𝑗 = 𝑀 → (𝑂‘(𝐺𝑗)) = (𝑂‘(𝐺𝑀)))
7 oveq2 6530 . . . . . . 7 (𝑗 = 𝑀 → (𝑀...𝑗) = (𝑀...𝑀))
87mpteq1d 4655 . . . . . 6 (𝑗 = 𝑀 → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛))))
98fveq2d 6087 . . . . 5 (𝑗 = 𝑀 → (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) = (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛)))))
106, 9eqeq12d 2619 . . . 4 (𝑗 = 𝑀 → ((𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) ↔ (𝑂‘(𝐺𝑀)) = (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛))))))
1110imbi2d 328 . . 3 (𝑗 = 𝑀 → ((𝜑 → (𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))))) ↔ (𝜑 → (𝑂‘(𝐺𝑀)) = (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛)))))))
12 fveq2 6083 . . . . . 6 (𝑗 = 𝑖 → (𝐺𝑗) = (𝐺𝑖))
1312fveq2d 6087 . . . . 5 (𝑗 = 𝑖 → (𝑂‘(𝐺𝑗)) = (𝑂‘(𝐺𝑖)))
14 oveq2 6530 . . . . . . 7 (𝑗 = 𝑖 → (𝑀...𝑗) = (𝑀...𝑖))
1514mpteq1d 4655 . . . . . 6 (𝑗 = 𝑖 → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))
1615fveq2d 6087 . . . . 5 (𝑗 = 𝑖 → (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))))
1713, 16eqeq12d 2619 . . . 4 (𝑗 = 𝑖 → ((𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) ↔ (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))))
1817imbi2d 328 . . 3 (𝑗 = 𝑖 → ((𝜑 → (𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))))) ↔ (𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))))))
19 fveq2 6083 . . . . . 6 (𝑗 = (𝑖 + 1) → (𝐺𝑗) = (𝐺‘(𝑖 + 1)))
2019fveq2d 6087 . . . . 5 (𝑗 = (𝑖 + 1) → (𝑂‘(𝐺𝑗)) = (𝑂‘(𝐺‘(𝑖 + 1))))
21 oveq2 6530 . . . . . . 7 (𝑗 = (𝑖 + 1) → (𝑀...𝑗) = (𝑀...(𝑖 + 1)))
2221mpteq1d 4655 . . . . . 6 (𝑗 = (𝑖 + 1) → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛))))
2322fveq2d 6087 . . . . 5 (𝑗 = (𝑖 + 1) → (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) = (Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛)))))
2420, 23eqeq12d 2619 . . . 4 (𝑗 = (𝑖 + 1) → ((𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) ↔ (𝑂‘(𝐺‘(𝑖 + 1))) = (Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛))))))
2524imbi2d 328 . . 3 (𝑗 = (𝑖 + 1) → ((𝜑 → (𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))))) ↔ (𝜑 → (𝑂‘(𝐺‘(𝑖 + 1))) = (Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛)))))))
26 fveq2 6083 . . . . . 6 (𝑗 = 𝑁 → (𝐺𝑗) = (𝐺𝑁))
2726fveq2d 6087 . . . . 5 (𝑗 = 𝑁 → (𝑂‘(𝐺𝑗)) = (𝑂‘(𝐺𝑁)))
28 oveq2 6530 . . . . . . 7 (𝑗 = 𝑁 → (𝑀...𝑗) = (𝑀...𝑁))
2928mpteq1d 4655 . . . . . 6 (𝑗 = 𝑁 → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛))))
3029fveq2d 6087 . . . . 5 (𝑗 = 𝑁 → (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛)))))
3127, 30eqeq12d 2619 . . . 4 (𝑗 = 𝑁 → ((𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) ↔ (𝑂‘(𝐺𝑁)) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛))))))
3231imbi2d 328 . . 3 (𝑗 = 𝑁 → ((𝜑 → (𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))))) ↔ (𝜑 → (𝑂‘(𝐺𝑁)) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛)))))))
33 eluzel2 11519 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
341, 33syl 17 . . . . . . . 8 (𝜑𝑀 ∈ ℤ)
35 fzsn 12204 . . . . . . . 8 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
3634, 35syl 17 . . . . . . 7 (𝜑 → (𝑀...𝑀) = {𝑀})
3736mpteq1d 4655 . . . . . 6 (𝜑 → (𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛))))
3837fveq2d 6087 . . . . 5 (𝜑 → (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛)))) = (Σ^‘(𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛)))))
39 caratheodorylem1.o . . . . . . . . 9 (𝜑𝑂 ∈ OutMeas)
4039adantr 479 . . . . . . . 8 ((𝜑𝑛 ∈ {𝑀}) → 𝑂 ∈ OutMeas)
41 eqid 2604 . . . . . . . 8 dom 𝑂 = dom 𝑂
42 caratheodorylem1.s . . . . . . . . . . . 12 𝑆 = (CaraGen‘𝑂)
4342caragenss 39193 . . . . . . . . . . 11 (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
4440, 43syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ {𝑀}) → 𝑆 ⊆ dom 𝑂)
45 caratheodorylem1.e . . . . . . . . . . . 12 (𝜑𝐸:𝑍𝑆)
4645adantr 479 . . . . . . . . . . 11 ((𝜑𝑛 ∈ {𝑀}) → 𝐸:𝑍𝑆)
47 elsni 4136 . . . . . . . . . . . . 13 (𝑛 ∈ {𝑀} → 𝑛 = 𝑀)
4847adantl 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ {𝑀}) → 𝑛 = 𝑀)
49 uzid 11529 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
5034, 49syl 17 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ (ℤ𝑀))
51 caratheodorylem1.z . . . . . . . . . . . . . 14 𝑍 = (ℤ𝑀)
5250, 51syl6eleqr 2693 . . . . . . . . . . . . 13 (𝜑𝑀𝑍)
5352adantr 479 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ {𝑀}) → 𝑀𝑍)
5448, 53eqeltrd 2682 . . . . . . . . . . 11 ((𝜑𝑛 ∈ {𝑀}) → 𝑛𝑍)
5546, 54ffvelrnd 6248 . . . . . . . . . 10 ((𝜑𝑛 ∈ {𝑀}) → (𝐸𝑛) ∈ 𝑆)
5644, 55sseldd 3563 . . . . . . . . 9 ((𝜑𝑛 ∈ {𝑀}) → (𝐸𝑛) ∈ dom 𝑂)
57 elssuni 4392 . . . . . . . . 9 ((𝐸𝑛) ∈ dom 𝑂 → (𝐸𝑛) ⊆ dom 𝑂)
5856, 57syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ {𝑀}) → (𝐸𝑛) ⊆ dom 𝑂)
5940, 41, 58omecl 39192 . . . . . . 7 ((𝜑𝑛 ∈ {𝑀}) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
60 eqid 2604 . . . . . . 7 (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛)))
6159, 60fmptd 6272 . . . . . 6 (𝜑 → (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛))):{𝑀}⟶(0[,]+∞))
6234, 61sge0sn 39071 . . . . 5 (𝜑 → (Σ^‘(𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛)))) = ((𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛)))‘𝑀))
63 eqidd 2605 . . . . . 6 (𝜑 → (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛))))
6436iuneq1d 4470 . . . . . . . . . 10 (𝜑 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖) = 𝑖 ∈ {𝑀} (𝐸𝑖))
65 fveq2 6083 . . . . . . . . . . . 12 (𝑖 = 𝑀 → (𝐸𝑖) = (𝐸𝑀))
6665iunxsng 4527 . . . . . . . . . . 11 (𝑀𝑍 𝑖 ∈ {𝑀} (𝐸𝑖) = (𝐸𝑀))
6752, 66syl 17 . . . . . . . . . 10 (𝜑 𝑖 ∈ {𝑀} (𝐸𝑖) = (𝐸𝑀))
68 eqidd 2605 . . . . . . . . . 10 (𝜑 → (𝐸𝑀) = (𝐸𝑀))
6964, 67, 683eqtrrd 2643 . . . . . . . . 9 (𝜑 → (𝐸𝑀) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
7069adantr 479 . . . . . . . 8 ((𝜑𝑛 = 𝑀) → (𝐸𝑀) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
71 fveq2 6083 . . . . . . . . 9 (𝑛 = 𝑀 → (𝐸𝑛) = (𝐸𝑀))
7271adantl 480 . . . . . . . 8 ((𝜑𝑛 = 𝑀) → (𝐸𝑛) = (𝐸𝑀))
73 caratheodorylem1.g . . . . . . . . . . 11 𝐺 = (𝑛𝑍 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖))
7473a1i 11 . . . . . . . . . 10 (𝜑𝐺 = (𝑛𝑍 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖)))
75 oveq2 6530 . . . . . . . . . . . 12 (𝑛 = 𝑀 → (𝑀...𝑛) = (𝑀...𝑀))
7675iuneq1d 4470 . . . . . . . . . . 11 (𝑛 = 𝑀 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
7776adantl 480 . . . . . . . . . 10 ((𝜑𝑛 = 𝑀) → 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
78 ovex 6550 . . . . . . . . . . . 12 (𝑀...𝑀) ∈ V
79 fvex 6093 . . . . . . . . . . . 12 (𝐸𝑖) ∈ V
8078, 79iunex 7011 . . . . . . . . . . 11 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖) ∈ V
8180a1i 11 . . . . . . . . . 10 (𝜑 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖) ∈ V)
8274, 77, 52, 81fvmptd 6177 . . . . . . . . 9 (𝜑 → (𝐺𝑀) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
8382adantr 479 . . . . . . . 8 ((𝜑𝑛 = 𝑀) → (𝐺𝑀) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
8470, 72, 833eqtr4d 2648 . . . . . . 7 ((𝜑𝑛 = 𝑀) → (𝐸𝑛) = (𝐺𝑀))
8584fveq2d 6087 . . . . . 6 ((𝜑𝑛 = 𝑀) → (𝑂‘(𝐸𝑛)) = (𝑂‘(𝐺𝑀)))
86 snidg 4147 . . . . . . 7 (𝑀𝑍𝑀 ∈ {𝑀})
8752, 86syl 17 . . . . . 6 (𝜑𝑀 ∈ {𝑀})
88 fvex 6093 . . . . . . 7 (𝑂‘(𝐺𝑀)) ∈ V
8988a1i 11 . . . . . 6 (𝜑 → (𝑂‘(𝐺𝑀)) ∈ V)
9063, 85, 87, 89fvmptd 6177 . . . . 5 (𝜑 → ((𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛)))‘𝑀) = (𝑂‘(𝐺𝑀)))
9138, 62, 903eqtrrd 2643 . . . 4 (𝜑 → (𝑂‘(𝐺𝑀)) = (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛)))))
9291a1i 11 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (𝑂‘(𝐺𝑀)) = (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛))))))
93 simp3 1055 . . . . 5 ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) ∧ 𝜑) → 𝜑)
94 simp1 1053 . . . . 5 ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) ∧ 𝜑) → 𝑖 ∈ (𝑀..^𝑁))
95 id 22 . . . . . . 7 ((𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) → (𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))))
9695imp 443 . . . . . 6 (((𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) ∧ 𝜑) → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))))
97963adant1 1071 . . . . 5 ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) ∧ 𝜑) → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))))
98 elfzoel1 12287 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℤ)
99 elfzoelz 12289 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (𝑀..^𝑁) → 𝑖 ∈ ℤ)
10099peano2zd 11312 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ∈ ℤ)
10198, 100, 1003jca 1234 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝑀..^𝑁) → (𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ))
10298zred 11309 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℝ)
103100zred 11309 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ∈ ℝ)
10499zred 11309 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (𝑀..^𝑁) → 𝑖 ∈ ℝ)
105 elfzole1 12297 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (𝑀..^𝑁) → 𝑀𝑖)
106104ltp1d 10798 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (𝑀..^𝑁) → 𝑖 < (𝑖 + 1))
107102, 104, 103, 105, 106lelttrd 10041 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (𝑀..^𝑁) → 𝑀 < (𝑖 + 1))
108102, 103, 107ltled 10031 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ≤ (𝑖 + 1))
109 leid 9979 . . . . . . . . . . . . . . . 16 ((𝑖 + 1) ∈ ℝ → (𝑖 + 1) ≤ (𝑖 + 1))
110103, 109syl 17 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ≤ (𝑖 + 1))
111101, 108, 110jca32 555 . . . . . . . . . . . . . 14 (𝑖 ∈ (𝑀..^𝑁) → ((𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ) ∧ (𝑀 ≤ (𝑖 + 1) ∧ (𝑖 + 1) ≤ (𝑖 + 1))))
112 elfz2 12154 . . . . . . . . . . . . . 14 ((𝑖 + 1) ∈ (𝑀...(𝑖 + 1)) ↔ ((𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ) ∧ (𝑀 ≤ (𝑖 + 1) ∧ (𝑖 + 1) ≤ (𝑖 + 1))))
113111, 112sylibr 222 . . . . . . . . . . . . 13 (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ∈ (𝑀...(𝑖 + 1)))
114113adantl 480 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ (𝑀...(𝑖 + 1)))
115 fveq2 6083 . . . . . . . . . . . . 13 (𝑗 = (𝑖 + 1) → (𝐸𝑗) = (𝐸‘(𝑖 + 1)))
116115ssiun2s 4489 . . . . . . . . . . . 12 ((𝑖 + 1) ∈ (𝑀...(𝑖 + 1)) → (𝐸‘(𝑖 + 1)) ⊆ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗))
117114, 116syl 17 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ⊆ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗))
118 fveq2 6083 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑗 → (𝐸𝑖) = (𝐸𝑗))
119118cbviunv 4484 . . . . . . . . . . . . . . . 16 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖) = 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗)
120119mpteq2i 4658 . . . . . . . . . . . . . . 15 (𝑛𝑍 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖)) = (𝑛𝑍 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗))
12173, 120eqtri 2626 . . . . . . . . . . . . . 14 𝐺 = (𝑛𝑍 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗))
122121a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝐺 = (𝑛𝑍 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗)))
123 oveq2 6530 . . . . . . . . . . . . . . 15 (𝑛 = (𝑖 + 1) → (𝑀...𝑛) = (𝑀...(𝑖 + 1)))
124123iuneq1d 4470 . . . . . . . . . . . . . 14 (𝑛 = (𝑖 + 1) → 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗) = 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗))
125124adantl 480 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 = (𝑖 + 1)) → 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗) = 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗))
12634adantr 479 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℤ)
12799adantl 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ ℤ)
128127peano2zd 11312 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ ℤ)
129126zred 11309 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
130128zred 11309 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ ℝ)
131127zred 11309 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ ℝ)
132105adantl 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑀𝑖)
133131ltp1d 10798 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑖 < (𝑖 + 1))
134129, 131, 130, 132, 133lelttrd 10041 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑀 < (𝑖 + 1))
135129, 130, 134ltled 10031 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ≤ (𝑖 + 1))
136126, 128, 1353jca 1234 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑖 + 1)))
137 eluz2 11520 . . . . . . . . . . . . . . 15 ((𝑖 + 1) ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑖 + 1)))
138136, 137sylibr 222 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ (ℤ𝑀))
13951eqcomi 2613 . . . . . . . . . . . . . 14 (ℤ𝑀) = 𝑍
140138, 139syl6eleq 2692 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ 𝑍)
141 ovex 6550 . . . . . . . . . . . . . . 15 (𝑀...(𝑖 + 1)) ∈ V
142 fvex 6093 . . . . . . . . . . . . . . 15 (𝐸𝑗) ∈ V
143141, 142iunex 7011 . . . . . . . . . . . . . 14 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ∈ V
144143a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ∈ V)
145122, 125, 140, 144fvmptd 6177 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑖 + 1)) = 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗))
146145eqcomd 2610 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) = (𝐺‘(𝑖 + 1)))
147117, 146sseqtrd 3598 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ⊆ (𝐺‘(𝑖 + 1)))
148 sseqin2 3773 . . . . . . . . . . 11 ((𝐸‘(𝑖 + 1)) ⊆ (𝐺‘(𝑖 + 1)) ↔ ((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1))) = (𝐸‘(𝑖 + 1)))
149148biimpi 204 . . . . . . . . . 10 ((𝐸‘(𝑖 + 1)) ⊆ (𝐺‘(𝑖 + 1)) → ((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1))) = (𝐸‘(𝑖 + 1)))
150147, 149syl 17 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1))) = (𝐸‘(𝑖 + 1)))
151150fveq2d 6087 . . . . . . . 8 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) = (𝑂‘(𝐸‘(𝑖 + 1))))
152 nfcv 2745 . . . . . . . . . . . . 13 𝑗(𝐸‘(𝑖 + 1))
153 elfzouz 12293 . . . . . . . . . . . . . 14 (𝑖 ∈ (𝑀..^𝑁) → 𝑖 ∈ (ℤ𝑀))
154153adantl 480 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ (ℤ𝑀))
155152, 154, 115iunp1 38058 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) = ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ (𝐸‘(𝑖 + 1))))
156145, 155eqtrd 2638 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑖 + 1)) = ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ (𝐸‘(𝑖 + 1))))
157156difeq1d 3683 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = (( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ (𝐸‘(𝑖 + 1))) ∖ (𝐸‘(𝑖 + 1))))
158 caratheodorylem1.dj . . . . . . . . . . . . . . 15 (𝜑Disj 𝑛𝑍 (𝐸𝑛))
159 fveq2 6083 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → (𝐸𝑛) = (𝐸𝑗))
160159cbvdisjv 4553 . . . . . . . . . . . . . . 15 (Disj 𝑛𝑍 (𝐸𝑛) ↔ Disj 𝑗𝑍 (𝐸𝑗))
161158, 160sylib 206 . . . . . . . . . . . . . 14 (𝜑Disj 𝑗𝑍 (𝐸𝑗))
162161adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → Disj 𝑗𝑍 (𝐸𝑗))
163 fzssuz 12203 . . . . . . . . . . . . . . 15 (𝑀...𝑖) ⊆ (ℤ𝑀)
164163, 139sseqtri 3594 . . . . . . . . . . . . . 14 (𝑀...𝑖) ⊆ 𝑍
165164a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑀...𝑖) ⊆ 𝑍)
166 fzp1nel 12243 . . . . . . . . . . . . . . . 16 ¬ (𝑖 + 1) ∈ (𝑀...𝑖)
167166a1i 11 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝑀..^𝑁) → ¬ (𝑖 + 1) ∈ (𝑀...𝑖))
168167adantl 480 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ¬ (𝑖 + 1) ∈ (𝑀...𝑖))
169140, 168eldifd 3545 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ (𝑍 ∖ (𝑀...𝑖)))
170162, 165, 169, 115disjiun2 38049 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∩ (𝐸‘(𝑖 + 1))) = ∅)
171 undif4 3981 . . . . . . . . . . . 12 (( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∩ (𝐸‘(𝑖 + 1))) = ∅ → ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = (( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ (𝐸‘(𝑖 + 1))) ∖ (𝐸‘(𝑖 + 1))))
172170, 171syl 17 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = (( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ (𝐸‘(𝑖 + 1))) ∖ (𝐸‘(𝑖 + 1))))
173172eqcomd 2610 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ (𝐸‘(𝑖 + 1))) ∖ (𝐸‘(𝑖 + 1))) = ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))))
174 simpl 471 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝜑)
175154, 139syl6eleq 2692 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑖𝑍)
176121a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → 𝐺 = (𝑛𝑍 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗)))
177 simpr 475 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖𝑍) ∧ 𝑛 = 𝑖) → 𝑛 = 𝑖)
178177oveq2d 6538 . . . . . . . . . . . . . . . 16 (((𝜑𝑖𝑍) ∧ 𝑛 = 𝑖) → (𝑀...𝑛) = (𝑀...𝑖))
179178iuneq1d 4470 . . . . . . . . . . . . . . 15 (((𝜑𝑖𝑍) ∧ 𝑛 = 𝑖) → 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗) = 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗))
180 simpr 475 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → 𝑖𝑍)
181 ovex 6550 . . . . . . . . . . . . . . . . 17 (𝑀...𝑖) ∈ V
182181, 142iunex 7011 . . . . . . . . . . . . . . . 16 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∈ V
183182a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∈ V)
184176, 179, 180, 183fvmptd 6177 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍) → (𝐺𝑖) = 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗))
185174, 175, 184syl2anc 690 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐺𝑖) = 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗))
186185eqcomd 2610 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) = (𝐺𝑖))
187 difid 3896 . . . . . . . . . . . . 13 ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = ∅
188187a1i 11 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = ∅)
189186, 188uneq12d 3724 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = ((𝐺𝑖) ∪ ∅))
190 un0 3913 . . . . . . . . . . . 12 ((𝐺𝑖) ∪ ∅) = (𝐺𝑖)
191190a1i 11 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝐺𝑖) ∪ ∅) = (𝐺𝑖))
192189, 191eqtrd 2638 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = (𝐺𝑖))
193157, 173, 1923eqtrd 2642 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = (𝐺𝑖))
194193fveq2d 6087 . . . . . . . 8 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = (𝑂‘(𝐺𝑖)))
195151, 194oveq12d 6540 . . . . . . 7 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))))) = ((𝑂‘(𝐸‘(𝑖 + 1))) +𝑒 (𝑂‘(𝐺𝑖))))
1961953adant3 1073 . . . . . 6 ((𝜑𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) → ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))))) = ((𝑂‘(𝐸‘(𝑖 + 1))) +𝑒 (𝑂‘(𝐺𝑖))))
19739adantr 479 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑂 ∈ OutMeas)
19845adantr 479 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝐸:𝑍𝑆)
199198, 140ffvelrnd 6248 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ∈ 𝑆)
200 simpll 785 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝜑)
20198adantr 479 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑀 ∈ ℤ)
202 elfzelz 12163 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (𝑀...(𝑖 + 1)) → 𝑗 ∈ ℤ)
203202adantl 480 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗 ∈ ℤ)
204 elfzle1 12165 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (𝑀...(𝑖 + 1)) → 𝑀𝑗)
205204adantl 480 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑀𝑗)
206201, 203, 2053jca 1234 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → (𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑀𝑗))
207 eluz2 11520 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑀𝑗))
208206, 207sylibr 222 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗 ∈ (ℤ𝑀))
209208, 139syl6eleq 2692 . . . . . . . . . . . . . 14 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗𝑍)
210209adantll 745 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗𝑍)
21139, 43syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑆 ⊆ dom 𝑂)
212211adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑗𝑍) → 𝑆 ⊆ dom 𝑂)
21345ffvelrnda 6247 . . . . . . . . . . . . . . 15 ((𝜑𝑗𝑍) → (𝐸𝑗) ∈ 𝑆)
214212, 213sseldd 3563 . . . . . . . . . . . . . 14 ((𝜑𝑗𝑍) → (𝐸𝑗) ∈ dom 𝑂)
215 elssuni 4392 . . . . . . . . . . . . . 14 ((𝐸𝑗) ∈ dom 𝑂 → (𝐸𝑗) ⊆ dom 𝑂)
216214, 215syl 17 . . . . . . . . . . . . 13 ((𝜑𝑗𝑍) → (𝐸𝑗) ⊆ dom 𝑂)
217200, 210, 216syl2anc 690 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → (𝐸𝑗) ⊆ dom 𝑂)
218217ralrimiva 2943 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ∀𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ⊆ dom 𝑂)
219 iunss 4486 . . . . . . . . . . 11 ( 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ⊆ dom 𝑂 ↔ ∀𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ⊆ dom 𝑂)
220218, 219sylibr 222 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ⊆ dom 𝑂)
221145, 220eqsstrd 3596 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑖 + 1)) ⊆ dom 𝑂)
222197, 42, 41, 199, 221caragensplit 39189 . . . . . . . 8 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))))) = (𝑂‘(𝐺‘(𝑖 + 1))))
223222eqcomd 2610 . . . . . . 7 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘(𝐺‘(𝑖 + 1))) = ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))))))
2242233adant3 1073 . . . . . 6 ((𝜑𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) → (𝑂‘(𝐺‘(𝑖 + 1))) = ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))))))
225197adantr 479 . . . . . . . . . 10 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → 𝑂 ∈ OutMeas)
226174adantr 479 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → 𝜑)
227 elfzuz 12159 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑀...(𝑖 + 1)) → 𝑛 ∈ (ℤ𝑀))
228227, 139syl6eleq 2692 . . . . . . . . . . . 12 (𝑛 ∈ (𝑀...(𝑖 + 1)) → 𝑛𝑍)
229228adantl 480 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → 𝑛𝑍)
23045, 211fssd 5951 . . . . . . . . . . . . 13 (𝜑𝐸:𝑍⟶dom 𝑂)
231230ffvelrnda 6247 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → (𝐸𝑛) ∈ dom 𝑂)
232231, 57syl 17 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (𝐸𝑛) ⊆ dom 𝑂)
233226, 229, 232syl2anc 690 . . . . . . . . . 10 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → (𝐸𝑛) ⊆ dom 𝑂)
234225, 41, 233omecl 39192 . . . . . . . . 9 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
235 fveq2 6083 . . . . . . . . . 10 (𝑛 = (𝑖 + 1) → (𝐸𝑛) = (𝐸‘(𝑖 + 1)))
236235fveq2d 6087 . . . . . . . . 9 (𝑛 = (𝑖 + 1) → (𝑂‘(𝐸𝑛)) = (𝑂‘(𝐸‘(𝑖 + 1))))
237154, 234, 236sge0p1 39106 . . . . . . . 8 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛)))) = ((Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))))
2382373adant3 1073 . . . . . . 7 ((𝜑𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) → (Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛)))) = ((Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))))
239 id 22 . . . . . . . . . 10 ((𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))) → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))))
240239eqcomd 2610 . . . . . . . . 9 ((𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))) → (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))) = (𝑂‘(𝐺𝑖)))
241240oveq1d 6537 . . . . . . . 8 ((𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))) → ((Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))) = ((𝑂‘(𝐺𝑖)) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))))
2422413ad2ant3 1076 . . . . . . 7 ((𝜑𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) → ((Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))) = ((𝑂‘(𝐺𝑖)) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))))
243 simpl 471 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...𝑖)) → 𝜑)
244164sseli 3558 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (𝑀...𝑖) → 𝑗𝑍)
245244adantl 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...𝑖)) → 𝑗𝑍)
246243, 245, 216syl2anc 690 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (𝑀...𝑖)) → (𝐸𝑗) ⊆ dom 𝑂)
247246adantlr 746 . . . . . . . . . . . . . 14 (((𝜑𝑖𝑍) ∧ 𝑗 ∈ (𝑀...𝑖)) → (𝐸𝑗) ⊆ dom 𝑂)
248247ralrimiva 2943 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍) → ∀𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ⊆ dom 𝑂)
249 iunss 4486 . . . . . . . . . . . . 13 ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ⊆ dom 𝑂 ↔ ∀𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ⊆ dom 𝑂)
250248, 249sylibr 222 . . . . . . . . . . . 12 ((𝜑𝑖𝑍) → 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ⊆ dom 𝑂)
251184, 250eqsstrd 3596 . . . . . . . . . . 11 ((𝜑𝑖𝑍) → (𝐺𝑖) ⊆ dom 𝑂)
252174, 175, 251syl2anc 690 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐺𝑖) ⊆ dom 𝑂)
253197, 41, 252omexrcl 39196 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘(𝐺𝑖)) ∈ ℝ*)
254117, 220sstrd 3572 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ⊆ dom 𝑂)
255197, 41, 254omexrcl 39196 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘(𝐸‘(𝑖 + 1))) ∈ ℝ*)
256253, 255xaddcomd 38280 . . . . . . . 8 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝑂‘(𝐺𝑖)) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))) = ((𝑂‘(𝐸‘(𝑖 + 1))) +𝑒 (𝑂‘(𝐺𝑖))))
2572563adant3 1073 . . . . . . 7 ((𝜑𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) → ((𝑂‘(𝐺𝑖)) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))) = ((𝑂‘(𝐸‘(𝑖 + 1))) +𝑒 (𝑂‘(𝐺𝑖))))
258238, 242, 2573eqtrd 2642 . . . . . 6 ((𝜑𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) → (Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛)))) = ((𝑂‘(𝐸‘(𝑖 + 1))) +𝑒 (𝑂‘(𝐺𝑖))))
259196, 224, 2583eqtr4d 2648 . . . . 5 ((𝜑𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) → (𝑂‘(𝐺‘(𝑖 + 1))) = (Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛)))))
26093, 94, 97, 259syl3anc 1317 . . . 4 ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) ∧ 𝜑) → (𝑂‘(𝐺‘(𝑖 + 1))) = (Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛)))))
2612603exp 1255 . . 3 (𝑖 ∈ (𝑀..^𝑁) → ((𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) → (𝜑 → (𝑂‘(𝐺‘(𝑖 + 1))) = (Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛)))))))
26211, 18, 25, 32, 92, 261fzind2 12398 . 2 (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (𝑂‘(𝐺𝑁)) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛))))))
2633, 4, 262sylc 62 1 (𝜑 → (𝑂‘(𝐺𝑁)) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1975  wral 2890  Vcvv 3167  cdif 3531  cun 3532  cin 3533  wss 3534  c0 3868  {csn 4119   cuni 4361   ciun 4444  Disj wdisj 4542   class class class wbr 4572  cmpt 4632  dom cdm 5023  wf 5781  cfv 5785  (class class class)co 6522  cr 9786  0cc0 9787  1c1 9788   + caddc 9790  +∞cpnf 9922  cle 9926  cz 11205  cuz 11514   +𝑒 cxad 11771  [,]cicc 12000  ...cfz 12147  ..^cfzo 12284  Σ^csumge0 39054  OutMeascome 39178  CaraGenccaragen 39180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819  ax-inf2 8393  ax-cnex 9843  ax-resscn 9844  ax-1cn 9845  ax-icn 9846  ax-addcl 9847  ax-addrcl 9848  ax-mulcl 9849  ax-mulrcl 9850  ax-mulcom 9851  ax-addass 9852  ax-mulass 9853  ax-distr 9854  ax-i2m1 9855  ax-1ne0 9856  ax-1rid 9857  ax-rnegex 9858  ax-rrecex 9859  ax-cnre 9860  ax-pre-lttri 9861  ax-pre-lttrn 9862  ax-pre-ltadd 9863  ax-pre-mulgt0 9864  ax-pre-sup 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-nel 2777  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-int 4400  df-iun 4446  df-disj 4543  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-se 4983  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-pred 5578  df-ord 5624  df-on 5625  df-lim 5626  df-suc 5627  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-isom 5794  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-om 6930  df-1st 7031  df-2nd 7032  df-wrecs 7266  df-recs 7327  df-rdg 7365  df-1o 7419  df-oadd 7423  df-er 7601  df-en 7814  df-dom 7815  df-sdom 7816  df-fin 7817  df-sup 8203  df-oi 8270  df-card 8620  df-pnf 9927  df-mnf 9928  df-xr 9929  df-ltxr 9930  df-le 9931  df-sub 10114  df-neg 10115  df-div 10529  df-nn 10863  df-2 10921  df-3 10922  df-n0 11135  df-z 11206  df-uz 11515  df-rp 11660  df-xadd 11774  df-ico 12003  df-icc 12004  df-fz 12148  df-fzo 12285  df-seq 12614  df-exp 12673  df-hash 12930  df-cj 13628  df-re 13629  df-im 13630  df-sqrt 13764  df-abs 13765  df-clim 14008  df-sum 14206  df-sumge0 39055  df-ome 39179  df-caragen 39181
This theorem is referenced by:  caratheodorylem2  39216
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