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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  ℝcr 9920  0cc0 9921  1c1 9922   + caddc 9924  +∞cpnf 10056   ≤ cle 10060  ℤcz 11362  ℤ≥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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