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Theorem caratheodorylem1 40503
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 12334 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 17 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 id 22 . 2 (𝜑𝜑)
5 fveq2 6178 . . . . . 6 (𝑗 = 𝑀 → (𝐺𝑗) = (𝐺𝑀))
65fveq2d 6182 . . . . 5 (𝑗 = 𝑀 → (𝑂‘(𝐺𝑗)) = (𝑂‘(𝐺𝑀)))
7 oveq2 6643 . . . . . . 7 (𝑗 = 𝑀 → (𝑀...𝑗) = (𝑀...𝑀))
87mpteq1d 4729 . . . . . 6 (𝑗 = 𝑀 → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛))))
98fveq2d 6182 . . . . 5 (𝑗 = 𝑀 → (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) = (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛)))))
106, 9eqeq12d 2635 . . . 4 (𝑗 = 𝑀 → ((𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) ↔ (𝑂‘(𝐺𝑀)) = (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛))))))
1110imbi2d 330 . . 3 (𝑗 = 𝑀 → ((𝜑 → (𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))))) ↔ (𝜑 → (𝑂‘(𝐺𝑀)) = (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛)))))))
12 fveq2 6178 . . . . . 6 (𝑗 = 𝑖 → (𝐺𝑗) = (𝐺𝑖))
1312fveq2d 6182 . . . . 5 (𝑗 = 𝑖 → (𝑂‘(𝐺𝑗)) = (𝑂‘(𝐺𝑖)))
14 oveq2 6643 . . . . . . 7 (𝑗 = 𝑖 → (𝑀...𝑗) = (𝑀...𝑖))
1514mpteq1d 4729 . . . . . 6 (𝑗 = 𝑖 → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))
1615fveq2d 6182 . . . . 5 (𝑗 = 𝑖 → (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))))
1713, 16eqeq12d 2635 . . . 4 (𝑗 = 𝑖 → ((𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) ↔ (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))))
1817imbi2d 330 . . 3 (𝑗 = 𝑖 → ((𝜑 → (𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))))) ↔ (𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))))))
19 fveq2 6178 . . . . . 6 (𝑗 = (𝑖 + 1) → (𝐺𝑗) = (𝐺‘(𝑖 + 1)))
2019fveq2d 6182 . . . . 5 (𝑗 = (𝑖 + 1) → (𝑂‘(𝐺𝑗)) = (𝑂‘(𝐺‘(𝑖 + 1))))
21 oveq2 6643 . . . . . . 7 (𝑗 = (𝑖 + 1) → (𝑀...𝑗) = (𝑀...(𝑖 + 1)))
2221mpteq1d 4729 . . . . . 6 (𝑗 = (𝑖 + 1) → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛))))
2322fveq2d 6182 . . . . 5 (𝑗 = (𝑖 + 1) → (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) = (Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛)))))
2420, 23eqeq12d 2635 . . . 4 (𝑗 = (𝑖 + 1) → ((𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) ↔ (𝑂‘(𝐺‘(𝑖 + 1))) = (Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛))))))
2524imbi2d 330 . . 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 6178 . . . . . 6 (𝑗 = 𝑁 → (𝐺𝑗) = (𝐺𝑁))
2726fveq2d 6182 . . . . 5 (𝑗 = 𝑁 → (𝑂‘(𝐺𝑗)) = (𝑂‘(𝐺𝑁)))
28 oveq2 6643 . . . . . . 7 (𝑗 = 𝑁 → (𝑀...𝑗) = (𝑀...𝑁))
2928mpteq1d 4729 . . . . . 6 (𝑗 = 𝑁 → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛))))
3029fveq2d 6182 . . . . 5 (𝑗 = 𝑁 → (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛)))))
3127, 30eqeq12d 2635 . . . 4 (𝑗 = 𝑁 → ((𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) ↔ (𝑂‘(𝐺𝑁)) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛))))))
3231imbi2d 330 . . 3 (𝑗 = 𝑁 → ((𝜑 → (𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))))) ↔ (𝜑 → (𝑂‘(𝐺𝑁)) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛)))))))
33 eluzel2 11677 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
341, 33syl 17 . . . . . . . 8 (𝜑𝑀 ∈ ℤ)
35 fzsn 12368 . . . . . . . 8 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
3634, 35syl 17 . . . . . . 7 (𝜑 → (𝑀...𝑀) = {𝑀})
3736mpteq1d 4729 . . . . . 6 (𝜑 → (𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛))))
3837fveq2d 6182 . . . . 5 (𝜑 → (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛)))) = (Σ^‘(𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛)))))
39 caratheodorylem1.o . . . . . . . . 9 (𝜑𝑂 ∈ OutMeas)
4039adantr 481 . . . . . . . 8 ((𝜑𝑛 ∈ {𝑀}) → 𝑂 ∈ OutMeas)
41 eqid 2620 . . . . . . . 8 dom 𝑂 = dom 𝑂
42 caratheodorylem1.s . . . . . . . . . . . 12 𝑆 = (CaraGen‘𝑂)
4342caragenss 40481 . . . . . . . . . . 11 (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
4440, 43syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ {𝑀}) → 𝑆 ⊆ dom 𝑂)
45 caratheodorylem1.e . . . . . . . . . . . 12 (𝜑𝐸:𝑍𝑆)
4645adantr 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ {𝑀}) → 𝐸:𝑍𝑆)
47 elsni 4185 . . . . . . . . . . . . 13 (𝑛 ∈ {𝑀} → 𝑛 = 𝑀)
4847adantl 482 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ {𝑀}) → 𝑛 = 𝑀)
49 uzid 11687 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
5034, 49syl 17 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ (ℤ𝑀))
51 caratheodorylem1.z . . . . . . . . . . . . . 14 𝑍 = (ℤ𝑀)
5250, 51syl6eleqr 2710 . . . . . . . . . . . . 13 (𝜑𝑀𝑍)
5352adantr 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ {𝑀}) → 𝑀𝑍)
5448, 53eqeltrd 2699 . . . . . . . . . . 11 ((𝜑𝑛 ∈ {𝑀}) → 𝑛𝑍)
5546, 54ffvelrnd 6346 . . . . . . . . . 10 ((𝜑𝑛 ∈ {𝑀}) → (𝐸𝑛) ∈ 𝑆)
5644, 55sseldd 3596 . . . . . . . . 9 ((𝜑𝑛 ∈ {𝑀}) → (𝐸𝑛) ∈ dom 𝑂)
57 elssuni 4458 . . . . . . . . 9 ((𝐸𝑛) ∈ dom 𝑂 → (𝐸𝑛) ⊆ dom 𝑂)
5856, 57syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ {𝑀}) → (𝐸𝑛) ⊆ dom 𝑂)
5940, 41, 58omecl 40480 . . . . . . 7 ((𝜑𝑛 ∈ {𝑀}) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
60 eqid 2620 . . . . . . 7 (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛)))
6159, 60fmptd 6371 . . . . . 6 (𝜑 → (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛))):{𝑀}⟶(0[,]+∞))
6234, 61sge0sn 40359 . . . . 5 (𝜑 → (Σ^‘(𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛)))) = ((𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛)))‘𝑀))
63 eqidd 2621 . . . . . 6 (𝜑 → (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛))))
6436iuneq1d 4536 . . . . . . . . . 10 (𝜑 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖) = 𝑖 ∈ {𝑀} (𝐸𝑖))
65 fveq2 6178 . . . . . . . . . . . 12 (𝑖 = 𝑀 → (𝐸𝑖) = (𝐸𝑀))
6665iunxsng 4593 . . . . . . . . . . 11 (𝑀𝑍 𝑖 ∈ {𝑀} (𝐸𝑖) = (𝐸𝑀))
6752, 66syl 17 . . . . . . . . . 10 (𝜑 𝑖 ∈ {𝑀} (𝐸𝑖) = (𝐸𝑀))
68 eqidd 2621 . . . . . . . . . 10 (𝜑 → (𝐸𝑀) = (𝐸𝑀))
6964, 67, 683eqtrrd 2659 . . . . . . . . 9 (𝜑 → (𝐸𝑀) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
7069adantr 481 . . . . . . . 8 ((𝜑𝑛 = 𝑀) → (𝐸𝑀) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
71 fveq2 6178 . . . . . . . . 9 (𝑛 = 𝑀 → (𝐸𝑛) = (𝐸𝑀))
7271adantl 482 . . . . . . . 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 6643 . . . . . . . . . . . 12 (𝑛 = 𝑀 → (𝑀...𝑛) = (𝑀...𝑀))
7675iuneq1d 4536 . . . . . . . . . . 11 (𝑛 = 𝑀 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
7776adantl 482 . . . . . . . . . 10 ((𝜑𝑛 = 𝑀) → 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
78 ovex 6663 . . . . . . . . . . . 12 (𝑀...𝑀) ∈ V
79 fvex 6188 . . . . . . . . . . . 12 (𝐸𝑖) ∈ V
8078, 79iunex 7132 . . . . . . . . . . 11 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖) ∈ V
8180a1i 11 . . . . . . . . . 10 (𝜑 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖) ∈ V)
8274, 77, 52, 81fvmptd 6275 . . . . . . . . 9 (𝜑 → (𝐺𝑀) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
8382adantr 481 . . . . . . . 8 ((𝜑𝑛 = 𝑀) → (𝐺𝑀) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
8470, 72, 833eqtr4d 2664 . . . . . . 7 ((𝜑𝑛 = 𝑀) → (𝐸𝑛) = (𝐺𝑀))
8584fveq2d 6182 . . . . . 6 ((𝜑𝑛 = 𝑀) → (𝑂‘(𝐸𝑛)) = (𝑂‘(𝐺𝑀)))
86 snidg 4197 . . . . . . 7 (𝑀𝑍𝑀 ∈ {𝑀})
8752, 86syl 17 . . . . . 6 (𝜑𝑀 ∈ {𝑀})
88 fvexd 6190 . . . . . 6 (𝜑 → (𝑂‘(𝐺𝑀)) ∈ V)
8963, 85, 87, 88fvmptd 6275 . . . . 5 (𝜑 → ((𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛)))‘𝑀) = (𝑂‘(𝐺𝑀)))
9038, 62, 893eqtrrd 2659 . . . 4 (𝜑 → (𝑂‘(𝐺𝑀)) = (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛)))))
9190a1i 11 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (𝑂‘(𝐺𝑀)) = (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛))))))
92 simp3 1061 . . . . 5 ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) ∧ 𝜑) → 𝜑)
93 simp1 1059 . . . . 5 ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) ∧ 𝜑) → 𝑖 ∈ (𝑀..^𝑁))
94 id 22 . . . . . . 7 ((𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) → (𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))))
9594imp 445 . . . . . 6 (((𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) ∧ 𝜑) → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))))
96953adant1 1077 . . . . 5 ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) ∧ 𝜑) → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))))
97 elfzoel1 12452 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℤ)
98 elfzoelz 12454 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (𝑀..^𝑁) → 𝑖 ∈ ℤ)
9998peano2zd 11470 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ∈ ℤ)
10097, 99, 993jca 1240 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝑀..^𝑁) → (𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ))
10197zred 11467 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℝ)
10299zred 11467 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ∈ ℝ)
10398zred 11467 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (𝑀..^𝑁) → 𝑖 ∈ ℝ)
104 elfzole1 12462 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (𝑀..^𝑁) → 𝑀𝑖)
105103ltp1d 10939 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (𝑀..^𝑁) → 𝑖 < (𝑖 + 1))
106101, 103, 102, 104, 105lelttrd 10180 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (𝑀..^𝑁) → 𝑀 < (𝑖 + 1))
107101, 102, 106ltled 10170 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ≤ (𝑖 + 1))
108 leid 10118 . . . . . . . . . . . . . . . 16 ((𝑖 + 1) ∈ ℝ → (𝑖 + 1) ≤ (𝑖 + 1))
109102, 108syl 17 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ≤ (𝑖 + 1))
110100, 107, 109jca32 557 . . . . . . . . . . . . . 14 (𝑖 ∈ (𝑀..^𝑁) → ((𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ) ∧ (𝑀 ≤ (𝑖 + 1) ∧ (𝑖 + 1) ≤ (𝑖 + 1))))
111 elfz2 12318 . . . . . . . . . . . . . 14 ((𝑖 + 1) ∈ (𝑀...(𝑖 + 1)) ↔ ((𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ) ∧ (𝑀 ≤ (𝑖 + 1) ∧ (𝑖 + 1) ≤ (𝑖 + 1))))
112110, 111sylibr 224 . . . . . . . . . . . . 13 (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ∈ (𝑀...(𝑖 + 1)))
113112adantl 482 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ (𝑀...(𝑖 + 1)))
114 fveq2 6178 . . . . . . . . . . . . 13 (𝑗 = (𝑖 + 1) → (𝐸𝑗) = (𝐸‘(𝑖 + 1)))
115114ssiun2s 4555 . . . . . . . . . . . 12 ((𝑖 + 1) ∈ (𝑀...(𝑖 + 1)) → (𝐸‘(𝑖 + 1)) ⊆ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗))
116113, 115syl 17 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ⊆ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗))
117 fveq2 6178 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑗 → (𝐸𝑖) = (𝐸𝑗))
118117cbviunv 4550 . . . . . . . . . . . . . . . 16 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖) = 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗)
119118mpteq2i 4732 . . . . . . . . . . . . . . 15 (𝑛𝑍 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖)) = (𝑛𝑍 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗))
12073, 119eqtri 2642 . . . . . . . . . . . . . 14 𝐺 = (𝑛𝑍 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗))
121120a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝐺 = (𝑛𝑍 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗)))
122 oveq2 6643 . . . . . . . . . . . . . . 15 (𝑛 = (𝑖 + 1) → (𝑀...𝑛) = (𝑀...(𝑖 + 1)))
123122iuneq1d 4536 . . . . . . . . . . . . . 14 (𝑛 = (𝑖 + 1) → 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗) = 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗))
124123adantl 482 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 = (𝑖 + 1)) → 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗) = 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗))
12534adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℤ)
12698adantl 482 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ ℤ)
127126peano2zd 11470 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ ℤ)
128125zred 11467 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
129127zred 11467 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ ℝ)
130126zred 11467 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ ℝ)
131104adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑀𝑖)
132130ltp1d 10939 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑖 < (𝑖 + 1))
133128, 130, 129, 131, 132lelttrd 10180 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑀 < (𝑖 + 1))
134128, 129, 133ltled 10170 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ≤ (𝑖 + 1))
135125, 127, 1343jca 1240 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑖 + 1)))
136 eluz2 11678 . . . . . . . . . . . . . . 15 ((𝑖 + 1) ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑖 + 1)))
137135, 136sylibr 224 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ (ℤ𝑀))
13851eqcomi 2629 . . . . . . . . . . . . . 14 (ℤ𝑀) = 𝑍
139137, 138syl6eleq 2709 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ 𝑍)
140 ovex 6663 . . . . . . . . . . . . . . 15 (𝑀...(𝑖 + 1)) ∈ V
141 fvex 6188 . . . . . . . . . . . . . . 15 (𝐸𝑗) ∈ V
142140, 141iunex 7132 . . . . . . . . . . . . . 14 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ∈ V
143142a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ∈ V)
144121, 124, 139, 143fvmptd 6275 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑖 + 1)) = 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗))
145144eqcomd 2626 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) = (𝐺‘(𝑖 + 1)))
146116, 145sseqtrd 3633 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ⊆ (𝐺‘(𝑖 + 1)))
147 sseqin2 3809 . . . . . . . . . . 11 ((𝐸‘(𝑖 + 1)) ⊆ (𝐺‘(𝑖 + 1)) ↔ ((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1))) = (𝐸‘(𝑖 + 1)))
148147biimpi 206 . . . . . . . . . 10 ((𝐸‘(𝑖 + 1)) ⊆ (𝐺‘(𝑖 + 1)) → ((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1))) = (𝐸‘(𝑖 + 1)))
149146, 148syl 17 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1))) = (𝐸‘(𝑖 + 1)))
150149fveq2d 6182 . . . . . . . 8 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) = (𝑂‘(𝐸‘(𝑖 + 1))))
151 nfcv 2762 . . . . . . . . . . . . 13 𝑗(𝐸‘(𝑖 + 1))
152 elfzouz 12458 . . . . . . . . . . . . . 14 (𝑖 ∈ (𝑀..^𝑁) → 𝑖 ∈ (ℤ𝑀))
153152adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ (ℤ𝑀))
154151, 153, 114iunp1 39055 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) = ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ (𝐸‘(𝑖 + 1))))
155144, 154eqtrd 2654 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑖 + 1)) = ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ (𝐸‘(𝑖 + 1))))
156155difeq1d 3719 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = (( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ (𝐸‘(𝑖 + 1))) ∖ (𝐸‘(𝑖 + 1))))
157 caratheodorylem1.dj . . . . . . . . . . . . . . 15 (𝜑Disj 𝑛𝑍 (𝐸𝑛))
158 fveq2 6178 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → (𝐸𝑛) = (𝐸𝑗))
159158cbvdisjv 4622 . . . . . . . . . . . . . . 15 (Disj 𝑛𝑍 (𝐸𝑛) ↔ Disj 𝑗𝑍 (𝐸𝑗))
160157, 159sylib 208 . . . . . . . . . . . . . 14 (𝜑Disj 𝑗𝑍 (𝐸𝑗))
161160adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → Disj 𝑗𝑍 (𝐸𝑗))
162 fzssuz 12367 . . . . . . . . . . . . . . 15 (𝑀...𝑖) ⊆ (ℤ𝑀)
163162, 138sseqtri 3629 . . . . . . . . . . . . . 14 (𝑀...𝑖) ⊆ 𝑍
164163a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑀...𝑖) ⊆ 𝑍)
165 fzp1nel 12408 . . . . . . . . . . . . . . . 16 ¬ (𝑖 + 1) ∈ (𝑀...𝑖)
166165a1i 11 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝑀..^𝑁) → ¬ (𝑖 + 1) ∈ (𝑀...𝑖))
167166adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ¬ (𝑖 + 1) ∈ (𝑀...𝑖))
168139, 167eldifd 3578 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ (𝑍 ∖ (𝑀...𝑖)))
169161, 164, 168, 114disjiun2 39046 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∩ (𝐸‘(𝑖 + 1))) = ∅)
170 undif4 4026 . . . . . . . . . . . 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))))
171169, 170syl 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))))
172171eqcomd 2626 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ (𝐸‘(𝑖 + 1))) ∖ (𝐸‘(𝑖 + 1))) = ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))))
173 simpl 473 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝜑)
174153, 138syl6eleq 2709 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑖𝑍)
175120a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → 𝐺 = (𝑛𝑍 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗)))
176 simpr 477 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖𝑍) ∧ 𝑛 = 𝑖) → 𝑛 = 𝑖)
177176oveq2d 6651 . . . . . . . . . . . . . . . 16 (((𝜑𝑖𝑍) ∧ 𝑛 = 𝑖) → (𝑀...𝑛) = (𝑀...𝑖))
178177iuneq1d 4536 . . . . . . . . . . . . . . 15 (((𝜑𝑖𝑍) ∧ 𝑛 = 𝑖) → 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗) = 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗))
179 simpr 477 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → 𝑖𝑍)
180 ovex 6663 . . . . . . . . . . . . . . . . 17 (𝑀...𝑖) ∈ V
181180, 141iunex 7132 . . . . . . . . . . . . . . . 16 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∈ V
182181a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∈ V)
183175, 178, 179, 182fvmptd 6275 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍) → (𝐺𝑖) = 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗))
184173, 174, 183syl2anc 692 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐺𝑖) = 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗))
185184eqcomd 2626 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) = (𝐺𝑖))
186 difid 3939 . . . . . . . . . . . . 13 ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = ∅
187186a1i 11 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = ∅)
188185, 187uneq12d 3760 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = ((𝐺𝑖) ∪ ∅))
189 un0 3958 . . . . . . . . . . . 12 ((𝐺𝑖) ∪ ∅) = (𝐺𝑖)
190189a1i 11 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝐺𝑖) ∪ ∅) = (𝐺𝑖))
191188, 190eqtrd 2654 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = (𝐺𝑖))
192156, 172, 1913eqtrd 2658 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = (𝐺𝑖))
193192fveq2d 6182 . . . . . . . 8 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = (𝑂‘(𝐺𝑖)))
194150, 193oveq12d 6653 . . . . . . 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))))
1951943adant3 1079 . . . . . 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))))
19639adantr 481 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑂 ∈ OutMeas)
19745adantr 481 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝐸:𝑍𝑆)
198197, 139ffvelrnd 6346 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ∈ 𝑆)
199 simpll 789 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝜑)
20097adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑀 ∈ ℤ)
201 elfzelz 12327 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (𝑀...(𝑖 + 1)) → 𝑗 ∈ ℤ)
202201adantl 482 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗 ∈ ℤ)
203 elfzle1 12329 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (𝑀...(𝑖 + 1)) → 𝑀𝑗)
204203adantl 482 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑀𝑗)
205200, 202, 2043jca 1240 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → (𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑀𝑗))
206 eluz2 11678 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑀𝑗))
207205, 206sylibr 224 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗 ∈ (ℤ𝑀))
208207, 138syl6eleq 2709 . . . . . . . . . . . . . 14 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗𝑍)
209208adantll 749 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗𝑍)
21039, 43syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑆 ⊆ dom 𝑂)
211210adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑗𝑍) → 𝑆 ⊆ dom 𝑂)
21245ffvelrnda 6345 . . . . . . . . . . . . . . 15 ((𝜑𝑗𝑍) → (𝐸𝑗) ∈ 𝑆)
213211, 212sseldd 3596 . . . . . . . . . . . . . 14 ((𝜑𝑗𝑍) → (𝐸𝑗) ∈ dom 𝑂)
214 elssuni 4458 . . . . . . . . . . . . . 14 ((𝐸𝑗) ∈ dom 𝑂 → (𝐸𝑗) ⊆ dom 𝑂)
215213, 214syl 17 . . . . . . . . . . . . 13 ((𝜑𝑗𝑍) → (𝐸𝑗) ⊆ dom 𝑂)
216199, 209, 215syl2anc 692 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → (𝐸𝑗) ⊆ dom 𝑂)
217216ralrimiva 2963 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ∀𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ⊆ dom 𝑂)
218 iunss 4552 . . . . . . . . . . 11 ( 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ⊆ dom 𝑂 ↔ ∀𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ⊆ dom 𝑂)
219217, 218sylibr 224 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ⊆ dom 𝑂)
220144, 219eqsstrd 3631 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑖 + 1)) ⊆ dom 𝑂)
221196, 42, 41, 198, 220caragensplit 40477 . . . . . . . 8 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))))) = (𝑂‘(𝐺‘(𝑖 + 1))))
222221eqcomd 2626 . . . . . . 7 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘(𝐺‘(𝑖 + 1))) = ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))))))
2232223adant3 1079 . . . . . 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))))))
224196adantr 481 . . . . . . . . . 10 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → 𝑂 ∈ OutMeas)
225173adantr 481 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → 𝜑)
226 elfzuz 12323 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑀...(𝑖 + 1)) → 𝑛 ∈ (ℤ𝑀))
227226, 138syl6eleq 2709 . . . . . . . . . . . 12 (𝑛 ∈ (𝑀...(𝑖 + 1)) → 𝑛𝑍)
228227adantl 482 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → 𝑛𝑍)
22945, 210fssd 6044 . . . . . . . . . . . . 13 (𝜑𝐸:𝑍⟶dom 𝑂)
230229ffvelrnda 6345 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → (𝐸𝑛) ∈ dom 𝑂)
231230, 57syl 17 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (𝐸𝑛) ⊆ dom 𝑂)
232225, 228, 231syl2anc 692 . . . . . . . . . 10 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → (𝐸𝑛) ⊆ dom 𝑂)
233224, 41, 232omecl 40480 . . . . . . . . 9 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
234 fveq2 6178 . . . . . . . . . 10 (𝑛 = (𝑖 + 1) → (𝐸𝑛) = (𝐸‘(𝑖 + 1)))
235234fveq2d 6182 . . . . . . . . 9 (𝑛 = (𝑖 + 1) → (𝑂‘(𝐸𝑛)) = (𝑂‘(𝐸‘(𝑖 + 1))))
236153, 233, 235sge0p1 40394 . . . . . . . 8 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛)))) = ((Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))))
2372363adant3 1079 . . . . . . 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)))))
238 id 22 . . . . . . . . . 10 ((𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))) → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))))
239238eqcomd 2626 . . . . . . . . 9 ((𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))) → (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))) = (𝑂‘(𝐺𝑖)))
240239oveq1d 6650 . . . . . . . 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)))))
2412403ad2ant3 1082 . . . . . . 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)))))
242 simpl 473 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...𝑖)) → 𝜑)
243163sseli 3591 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (𝑀...𝑖) → 𝑗𝑍)
244243adantl 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...𝑖)) → 𝑗𝑍)
245242, 244, 215syl2anc 692 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (𝑀...𝑖)) → (𝐸𝑗) ⊆ dom 𝑂)
246245adantlr 750 . . . . . . . . . . . . . 14 (((𝜑𝑖𝑍) ∧ 𝑗 ∈ (𝑀...𝑖)) → (𝐸𝑗) ⊆ dom 𝑂)
247246ralrimiva 2963 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍) → ∀𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ⊆ dom 𝑂)
248 iunss 4552 . . . . . . . . . . . . 13 ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ⊆ dom 𝑂 ↔ ∀𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ⊆ dom 𝑂)
249247, 248sylibr 224 . . . . . . . . . . . 12 ((𝜑𝑖𝑍) → 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ⊆ dom 𝑂)
250183, 249eqsstrd 3631 . . . . . . . . . . 11 ((𝜑𝑖𝑍) → (𝐺𝑖) ⊆ dom 𝑂)
251173, 174, 250syl2anc 692 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐺𝑖) ⊆ dom 𝑂)
252196, 41, 251omexrcl 40484 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘(𝐺𝑖)) ∈ ℝ*)
253116, 219sstrd 3605 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ⊆ dom 𝑂)
254196, 41, 253omexrcl 40484 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘(𝐸‘(𝑖 + 1))) ∈ ℝ*)
255252, 254xaddcomd 39353 . . . . . . . 8 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝑂‘(𝐺𝑖)) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))) = ((𝑂‘(𝐸‘(𝑖 + 1))) +𝑒 (𝑂‘(𝐺𝑖))))
2562553adant3 1079 . . . . . . 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))))
257237, 241, 2563eqtrd 2658 . . . . . 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))))
258195, 223, 2573eqtr4d 2664 . . . . 5 ((𝜑𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) → (𝑂‘(𝐺‘(𝑖 + 1))) = (Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛)))))
25992, 93, 96, 258syl3anc 1324 . . . 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)))))
2602593exp 1262 . . 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)))))))
26111, 18, 25, 32, 91, 260fzind2 12569 . 2 (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (𝑂‘(𝐺𝑁)) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛))))))
2623, 4, 261sylc 65 1 (𝜑 → (𝑂‘(𝐺𝑁)) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  Vcvv 3195  cdif 3564  cun 3565  cin 3566  wss 3567  c0 3907  {csn 4168   cuni 4427   ciun 4511  Disj wdisj 4611   class class class wbr 4644  cmpt 4720  dom cdm 5104  wf 5872  cfv 5876  (class class class)co 6635  cr 9920  0cc0 9921  1c1 9922   + caddc 9924  +∞cpnf 10056  cle 10060  cz 11362  cuz 11672   +𝑒 cxad 11929  [,]cicc 12163  ...cfz 12311  ..^cfzo 12449  Σ^csumge0 40342  OutMeascome 40466  CaraGenccaragen 40468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-disj 4612  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-sup 8333  df-oi 8400  df-card 8750  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-n0 11278  df-z 11363  df-uz 11673  df-rp 11818  df-xadd 11932  df-ico 12166  df-icc 12167  df-fz 12312  df-fzo 12450  df-seq 12785  df-exp 12844  df-hash 13101  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-clim 14200  df-sum 14398  df-sumge0 40343  df-ome 40467  df-caragen 40469
This theorem is referenced by:  caratheodorylem2  40504
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