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Theorem caratheodorylem1 41383
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 12559 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 17 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 id 22 . 2 (𝜑𝜑)
5 2fveq3 6382 . . . . 5 (𝑗 = 𝑀 → (𝑂‘(𝐺𝑗)) = (𝑂‘(𝐺𝑀)))
6 oveq2 6852 . . . . . . 7 (𝑗 = 𝑀 → (𝑀...𝑗) = (𝑀...𝑀))
76mpteq1d 4899 . . . . . 6 (𝑗 = 𝑀 → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛))))
87fveq2d 6381 . . . . 5 (𝑗 = 𝑀 → (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) = (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛)))))
95, 8eqeq12d 2780 . . . 4 (𝑗 = 𝑀 → ((𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) ↔ (𝑂‘(𝐺𝑀)) = (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛))))))
109imbi2d 331 . . 3 (𝑗 = 𝑀 → ((𝜑 → (𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))))) ↔ (𝜑 → (𝑂‘(𝐺𝑀)) = (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛)))))))
11 2fveq3 6382 . . . . 5 (𝑗 = 𝑖 → (𝑂‘(𝐺𝑗)) = (𝑂‘(𝐺𝑖)))
12 oveq2 6852 . . . . . . 7 (𝑗 = 𝑖 → (𝑀...𝑗) = (𝑀...𝑖))
1312mpteq1d 4899 . . . . . 6 (𝑗 = 𝑖 → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))
1413fveq2d 6381 . . . . 5 (𝑗 = 𝑖 → (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))))
1511, 14eqeq12d 2780 . . . 4 (𝑗 = 𝑖 → ((𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) ↔ (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))))
1615imbi2d 331 . . 3 (𝑗 = 𝑖 → ((𝜑 → (𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))))) ↔ (𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))))))
17 2fveq3 6382 . . . . 5 (𝑗 = (𝑖 + 1) → (𝑂‘(𝐺𝑗)) = (𝑂‘(𝐺‘(𝑖 + 1))))
18 oveq2 6852 . . . . . . 7 (𝑗 = (𝑖 + 1) → (𝑀...𝑗) = (𝑀...(𝑖 + 1)))
1918mpteq1d 4899 . . . . . 6 (𝑗 = (𝑖 + 1) → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛))))
2019fveq2d 6381 . . . . 5 (𝑗 = (𝑖 + 1) → (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) = (Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛)))))
2117, 20eqeq12d 2780 . . . 4 (𝑗 = (𝑖 + 1) → ((𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) ↔ (𝑂‘(𝐺‘(𝑖 + 1))) = (Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛))))))
2221imbi2d 331 . . 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)))))))
23 2fveq3 6382 . . . . 5 (𝑗 = 𝑁 → (𝑂‘(𝐺𝑗)) = (𝑂‘(𝐺𝑁)))
24 oveq2 6852 . . . . . . 7 (𝑗 = 𝑁 → (𝑀...𝑗) = (𝑀...𝑁))
2524mpteq1d 4899 . . . . . 6 (𝑗 = 𝑁 → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛))))
2625fveq2d 6381 . . . . 5 (𝑗 = 𝑁 → (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛)))))
2723, 26eqeq12d 2780 . . . 4 (𝑗 = 𝑁 → ((𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) ↔ (𝑂‘(𝐺𝑁)) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛))))))
2827imbi2d 331 . . 3 (𝑗 = 𝑁 → ((𝜑 → (𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))))) ↔ (𝜑 → (𝑂‘(𝐺𝑁)) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛)))))))
29 eluzel2 11894 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
301, 29syl 17 . . . . . . . 8 (𝜑𝑀 ∈ ℤ)
31 fzsn 12593 . . . . . . . 8 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
3230, 31syl 17 . . . . . . 7 (𝜑 → (𝑀...𝑀) = {𝑀})
3332mpteq1d 4899 . . . . . 6 (𝜑 → (𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛))))
3433fveq2d 6381 . . . . 5 (𝜑 → (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛)))) = (Σ^‘(𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛)))))
35 caratheodorylem1.o . . . . . . . . 9 (𝜑𝑂 ∈ OutMeas)
3635adantr 472 . . . . . . . 8 ((𝜑𝑛 ∈ {𝑀}) → 𝑂 ∈ OutMeas)
37 eqid 2765 . . . . . . . 8 dom 𝑂 = dom 𝑂
38 caratheodorylem1.s . . . . . . . . . . . 12 𝑆 = (CaraGen‘𝑂)
3938caragenss 41361 . . . . . . . . . . 11 (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
4036, 39syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ {𝑀}) → 𝑆 ⊆ dom 𝑂)
41 caratheodorylem1.e . . . . . . . . . . . 12 (𝜑𝐸:𝑍𝑆)
4241adantr 472 . . . . . . . . . . 11 ((𝜑𝑛 ∈ {𝑀}) → 𝐸:𝑍𝑆)
43 elsni 4353 . . . . . . . . . . . . 13 (𝑛 ∈ {𝑀} → 𝑛 = 𝑀)
4443adantl 473 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ {𝑀}) → 𝑛 = 𝑀)
45 uzid 11904 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
4630, 45syl 17 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ (ℤ𝑀))
47 caratheodorylem1.z . . . . . . . . . . . . . 14 𝑍 = (ℤ𝑀)
4846, 47syl6eleqr 2855 . . . . . . . . . . . . 13 (𝜑𝑀𝑍)
4948adantr 472 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ {𝑀}) → 𝑀𝑍)
5044, 49eqeltrd 2844 . . . . . . . . . . 11 ((𝜑𝑛 ∈ {𝑀}) → 𝑛𝑍)
5142, 50ffvelrnd 6552 . . . . . . . . . 10 ((𝜑𝑛 ∈ {𝑀}) → (𝐸𝑛) ∈ 𝑆)
5240, 51sseldd 3764 . . . . . . . . 9 ((𝜑𝑛 ∈ {𝑀}) → (𝐸𝑛) ∈ dom 𝑂)
53 elssuni 4627 . . . . . . . . 9 ((𝐸𝑛) ∈ dom 𝑂 → (𝐸𝑛) ⊆ dom 𝑂)
5452, 53syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ {𝑀}) → (𝐸𝑛) ⊆ dom 𝑂)
5536, 37, 54omecl 41360 . . . . . . 7 ((𝜑𝑛 ∈ {𝑀}) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
56 eqid 2765 . . . . . . 7 (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛)))
5755, 56fmptd 6576 . . . . . 6 (𝜑 → (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛))):{𝑀}⟶(0[,]+∞))
5830, 57sge0sn 41236 . . . . 5 (𝜑 → (Σ^‘(𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛)))) = ((𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛)))‘𝑀))
59 eqidd 2766 . . . . . 6 (𝜑 → (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛))))
6032iuneq1d 4703 . . . . . . . . . 10 (𝜑 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖) = 𝑖 ∈ {𝑀} (𝐸𝑖))
61 fveq2 6377 . . . . . . . . . . . 12 (𝑖 = 𝑀 → (𝐸𝑖) = (𝐸𝑀))
6261iunxsng 4760 . . . . . . . . . . 11 (𝑀𝑍 𝑖 ∈ {𝑀} (𝐸𝑖) = (𝐸𝑀))
6348, 62syl 17 . . . . . . . . . 10 (𝜑 𝑖 ∈ {𝑀} (𝐸𝑖) = (𝐸𝑀))
64 eqidd 2766 . . . . . . . . . 10 (𝜑 → (𝐸𝑀) = (𝐸𝑀))
6560, 63, 643eqtrrd 2804 . . . . . . . . 9 (𝜑 → (𝐸𝑀) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
6665adantr 472 . . . . . . . 8 ((𝜑𝑛 = 𝑀) → (𝐸𝑀) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
67 fveq2 6377 . . . . . . . . 9 (𝑛 = 𝑀 → (𝐸𝑛) = (𝐸𝑀))
6867adantl 473 . . . . . . . 8 ((𝜑𝑛 = 𝑀) → (𝐸𝑛) = (𝐸𝑀))
69 caratheodorylem1.g . . . . . . . . . . 11 𝐺 = (𝑛𝑍 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖))
7069a1i 11 . . . . . . . . . 10 (𝜑𝐺 = (𝑛𝑍 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖)))
71 oveq2 6852 . . . . . . . . . . . 12 (𝑛 = 𝑀 → (𝑀...𝑛) = (𝑀...𝑀))
7271iuneq1d 4703 . . . . . . . . . . 11 (𝑛 = 𝑀 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
7372adantl 473 . . . . . . . . . 10 ((𝜑𝑛 = 𝑀) → 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
74 ovex 6876 . . . . . . . . . . . 12 (𝑀...𝑀) ∈ V
75 fvex 6390 . . . . . . . . . . . 12 (𝐸𝑖) ∈ V
7674, 75iunex 7347 . . . . . . . . . . 11 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖) ∈ V
7776a1i 11 . . . . . . . . . 10 (𝜑 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖) ∈ V)
7870, 73, 48, 77fvmptd 6479 . . . . . . . . 9 (𝜑 → (𝐺𝑀) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
7978adantr 472 . . . . . . . 8 ((𝜑𝑛 = 𝑀) → (𝐺𝑀) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
8066, 68, 793eqtr4d 2809 . . . . . . 7 ((𝜑𝑛 = 𝑀) → (𝐸𝑛) = (𝐺𝑀))
8180fveq2d 6381 . . . . . 6 ((𝜑𝑛 = 𝑀) → (𝑂‘(𝐸𝑛)) = (𝑂‘(𝐺𝑀)))
82 snidg 4366 . . . . . . 7 (𝑀𝑍𝑀 ∈ {𝑀})
8348, 82syl 17 . . . . . 6 (𝜑𝑀 ∈ {𝑀})
84 fvexd 6392 . . . . . 6 (𝜑 → (𝑂‘(𝐺𝑀)) ∈ V)
8559, 81, 83, 84fvmptd 6479 . . . . 5 (𝜑 → ((𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛)))‘𝑀) = (𝑂‘(𝐺𝑀)))
8634, 58, 853eqtrrd 2804 . . . 4 (𝜑 → (𝑂‘(𝐺𝑀)) = (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛)))))
8786a1i 11 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (𝑂‘(𝐺𝑀)) = (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛))))))
88 simp3 1168 . . . . 5 ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) ∧ 𝜑) → 𝜑)
89 simp1 1166 . . . . 5 ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) ∧ 𝜑) → 𝑖 ∈ (𝑀..^𝑁))
90 id 22 . . . . . . 7 ((𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) → (𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))))
9190imp 395 . . . . . 6 (((𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) ∧ 𝜑) → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))))
92913adant1 1160 . . . . 5 ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) ∧ 𝜑) → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))))
93 elfzoel1 12679 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℤ)
94 elfzoelz 12681 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (𝑀..^𝑁) → 𝑖 ∈ ℤ)
9594peano2zd 11735 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ∈ ℤ)
9693, 95, 953jca 1158 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝑀..^𝑁) → (𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ))
9793zred 11732 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℝ)
9895zred 11732 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ∈ ℝ)
9994zred 11732 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (𝑀..^𝑁) → 𝑖 ∈ ℝ)
100 elfzole1 12689 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (𝑀..^𝑁) → 𝑀𝑖)
10199ltp1d 11210 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (𝑀..^𝑁) → 𝑖 < (𝑖 + 1))
10297, 99, 98, 100, 101lelttrd 10451 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (𝑀..^𝑁) → 𝑀 < (𝑖 + 1))
10397, 98, 102ltled 10441 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ≤ (𝑖 + 1))
104 leid 10389 . . . . . . . . . . . . . . . 16 ((𝑖 + 1) ∈ ℝ → (𝑖 + 1) ≤ (𝑖 + 1))
10598, 104syl 17 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ≤ (𝑖 + 1))
10696, 103, 105jca32 511 . . . . . . . . . . . . . 14 (𝑖 ∈ (𝑀..^𝑁) → ((𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ) ∧ (𝑀 ≤ (𝑖 + 1) ∧ (𝑖 + 1) ≤ (𝑖 + 1))))
107 elfz2 12543 . . . . . . . . . . . . . 14 ((𝑖 + 1) ∈ (𝑀...(𝑖 + 1)) ↔ ((𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ) ∧ (𝑀 ≤ (𝑖 + 1) ∧ (𝑖 + 1) ≤ (𝑖 + 1))))
108106, 107sylibr 225 . . . . . . . . . . . . 13 (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ∈ (𝑀...(𝑖 + 1)))
109108adantl 473 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ (𝑀...(𝑖 + 1)))
110 fveq2 6377 . . . . . . . . . . . . 13 (𝑗 = (𝑖 + 1) → (𝐸𝑗) = (𝐸‘(𝑖 + 1)))
111110ssiun2s 4722 . . . . . . . . . . . 12 ((𝑖 + 1) ∈ (𝑀...(𝑖 + 1)) → (𝐸‘(𝑖 + 1)) ⊆ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗))
112109, 111syl 17 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ⊆ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗))
113 fveq2 6377 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑗 → (𝐸𝑖) = (𝐸𝑗))
114113cbviunv 4717 . . . . . . . . . . . . . . . 16 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖) = 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗)
115114mpteq2i 4902 . . . . . . . . . . . . . . 15 (𝑛𝑍 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖)) = (𝑛𝑍 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗))
11669, 115eqtri 2787 . . . . . . . . . . . . . 14 𝐺 = (𝑛𝑍 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗))
117116a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝐺 = (𝑛𝑍 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗)))
118 oveq2 6852 . . . . . . . . . . . . . . 15 (𝑛 = (𝑖 + 1) → (𝑀...𝑛) = (𝑀...(𝑖 + 1)))
119118iuneq1d 4703 . . . . . . . . . . . . . 14 (𝑛 = (𝑖 + 1) → 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗) = 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗))
120119adantl 473 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 = (𝑖 + 1)) → 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗) = 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗))
12130adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℤ)
12294adantl 473 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ ℤ)
123122peano2zd 11735 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ ℤ)
124121zred 11732 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
125123zred 11732 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ ℝ)
126122zred 11732 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ ℝ)
127100adantl 473 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑀𝑖)
128126ltp1d 11210 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑖 < (𝑖 + 1))
129124, 126, 125, 127, 128lelttrd 10451 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑀 < (𝑖 + 1))
130124, 125, 129ltled 10441 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ≤ (𝑖 + 1))
131121, 123, 1303jca 1158 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑖 + 1)))
132 eluz2 11895 . . . . . . . . . . . . . . 15 ((𝑖 + 1) ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑖 + 1)))
133131, 132sylibr 225 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ (ℤ𝑀))
13447eqcomi 2774 . . . . . . . . . . . . . 14 (ℤ𝑀) = 𝑍
135133, 134syl6eleq 2854 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ 𝑍)
136 ovex 6876 . . . . . . . . . . . . . . 15 (𝑀...(𝑖 + 1)) ∈ V
137 fvex 6390 . . . . . . . . . . . . . . 15 (𝐸𝑗) ∈ V
138136, 137iunex 7347 . . . . . . . . . . . . . 14 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ∈ V
139138a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ∈ V)
140117, 120, 135, 139fvmptd 6479 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑖 + 1)) = 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗))
141140eqcomd 2771 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) = (𝐺‘(𝑖 + 1)))
142112, 141sseqtrd 3803 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ⊆ (𝐺‘(𝑖 + 1)))
143 sseqin2 3981 . . . . . . . . . . 11 ((𝐸‘(𝑖 + 1)) ⊆ (𝐺‘(𝑖 + 1)) ↔ ((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1))) = (𝐸‘(𝑖 + 1)))
144143biimpi 207 . . . . . . . . . 10 ((𝐸‘(𝑖 + 1)) ⊆ (𝐺‘(𝑖 + 1)) → ((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1))) = (𝐸‘(𝑖 + 1)))
145142, 144syl 17 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1))) = (𝐸‘(𝑖 + 1)))
146145fveq2d 6381 . . . . . . . 8 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) = (𝑂‘(𝐸‘(𝑖 + 1))))
147 nfcv 2907 . . . . . . . . . . . . 13 𝑗(𝐸‘(𝑖 + 1))
148 elfzouz 12685 . . . . . . . . . . . . . 14 (𝑖 ∈ (𝑀..^𝑁) → 𝑖 ∈ (ℤ𝑀))
149148adantl 473 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ (ℤ𝑀))
150147, 149, 110iunp1 39889 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) = ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ (𝐸‘(𝑖 + 1))))
151140, 150eqtrd 2799 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑖 + 1)) = ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ (𝐸‘(𝑖 + 1))))
152151difeq1d 3891 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = (( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ (𝐸‘(𝑖 + 1))) ∖ (𝐸‘(𝑖 + 1))))
153 caratheodorylem1.dj . . . . . . . . . . . . . . 15 (𝜑Disj 𝑛𝑍 (𝐸𝑛))
154 fveq2 6377 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → (𝐸𝑛) = (𝐸𝑗))
155154cbvdisjv 4790 . . . . . . . . . . . . . . 15 (Disj 𝑛𝑍 (𝐸𝑛) ↔ Disj 𝑗𝑍 (𝐸𝑗))
156153, 155sylib 209 . . . . . . . . . . . . . 14 (𝜑Disj 𝑗𝑍 (𝐸𝑗))
157156adantr 472 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → Disj 𝑗𝑍 (𝐸𝑗))
158 fzssuz 12592 . . . . . . . . . . . . . . 15 (𝑀...𝑖) ⊆ (ℤ𝑀)
159158, 134sseqtri 3799 . . . . . . . . . . . . . 14 (𝑀...𝑖) ⊆ 𝑍
160159a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑀...𝑖) ⊆ 𝑍)
161 fzp1nel 12634 . . . . . . . . . . . . . . . 16 ¬ (𝑖 + 1) ∈ (𝑀...𝑖)
162161a1i 11 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝑀..^𝑁) → ¬ (𝑖 + 1) ∈ (𝑀...𝑖))
163162adantl 473 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ¬ (𝑖 + 1) ∈ (𝑀...𝑖))
164135, 163eldifd 3745 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ (𝑍 ∖ (𝑀...𝑖)))
165157, 160, 164, 110disjiun2 39880 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∩ (𝐸‘(𝑖 + 1))) = ∅)
166 undif4 4197 . . . . . . . . . . . 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))))
167165, 166syl 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))))
168167eqcomd 2771 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ (𝐸‘(𝑖 + 1))) ∖ (𝐸‘(𝑖 + 1))) = ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))))
169 simpl 474 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝜑)
170149, 134syl6eleq 2854 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑖𝑍)
171116a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → 𝐺 = (𝑛𝑍 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗)))
172 simpr 477 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖𝑍) ∧ 𝑛 = 𝑖) → 𝑛 = 𝑖)
173172oveq2d 6860 . . . . . . . . . . . . . . . 16 (((𝜑𝑖𝑍) ∧ 𝑛 = 𝑖) → (𝑀...𝑛) = (𝑀...𝑖))
174173iuneq1d 4703 . . . . . . . . . . . . . . 15 (((𝜑𝑖𝑍) ∧ 𝑛 = 𝑖) → 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗) = 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗))
175 simpr 477 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → 𝑖𝑍)
176 ovex 6876 . . . . . . . . . . . . . . . . 17 (𝑀...𝑖) ∈ V
177176, 137iunex 7347 . . . . . . . . . . . . . . . 16 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∈ V
178177a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∈ V)
179171, 174, 175, 178fvmptd 6479 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍) → (𝐺𝑖) = 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗))
180169, 170, 179syl2anc 579 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐺𝑖) = 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗))
181180eqcomd 2771 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) = (𝐺𝑖))
182 difid 4115 . . . . . . . . . . . . 13 ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = ∅
183182a1i 11 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = ∅)
184181, 183uneq12d 3932 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = ((𝐺𝑖) ∪ ∅))
185 un0 4131 . . . . . . . . . . . 12 ((𝐺𝑖) ∪ ∅) = (𝐺𝑖)
186185a1i 11 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝐺𝑖) ∪ ∅) = (𝐺𝑖))
187184, 186eqtrd 2799 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = (𝐺𝑖))
188152, 168, 1873eqtrd 2803 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = (𝐺𝑖))
189188fveq2d 6381 . . . . . . . 8 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = (𝑂‘(𝐺𝑖)))
190146, 189oveq12d 6862 . . . . . . 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))))
1911903adant3 1162 . . . . . 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))))
19235adantr 472 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑂 ∈ OutMeas)
19341adantr 472 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝐸:𝑍𝑆)
194193, 135ffvelrnd 6552 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ∈ 𝑆)
195 simpll 783 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝜑)
19693adantr 472 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑀 ∈ ℤ)
197 elfzelz 12552 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (𝑀...(𝑖 + 1)) → 𝑗 ∈ ℤ)
198197adantl 473 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗 ∈ ℤ)
199 elfzle1 12554 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (𝑀...(𝑖 + 1)) → 𝑀𝑗)
200199adantl 473 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑀𝑗)
201196, 198, 2003jca 1158 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → (𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑀𝑗))
202 eluz2 11895 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑀𝑗))
203201, 202sylibr 225 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗 ∈ (ℤ𝑀))
204203, 134syl6eleq 2854 . . . . . . . . . . . . . 14 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗𝑍)
205204adantll 705 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗𝑍)
20635, 39syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑆 ⊆ dom 𝑂)
207206adantr 472 . . . . . . . . . . . . . . 15 ((𝜑𝑗𝑍) → 𝑆 ⊆ dom 𝑂)
20841ffvelrnda 6551 . . . . . . . . . . . . . . 15 ((𝜑𝑗𝑍) → (𝐸𝑗) ∈ 𝑆)
209207, 208sseldd 3764 . . . . . . . . . . . . . 14 ((𝜑𝑗𝑍) → (𝐸𝑗) ∈ dom 𝑂)
210 elssuni 4627 . . . . . . . . . . . . . 14 ((𝐸𝑗) ∈ dom 𝑂 → (𝐸𝑗) ⊆ dom 𝑂)
211209, 210syl 17 . . . . . . . . . . . . 13 ((𝜑𝑗𝑍) → (𝐸𝑗) ⊆ dom 𝑂)
212195, 205, 211syl2anc 579 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → (𝐸𝑗) ⊆ dom 𝑂)
213212ralrimiva 3113 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ∀𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ⊆ dom 𝑂)
214 iunss 4719 . . . . . . . . . . 11 ( 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ⊆ dom 𝑂 ↔ ∀𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ⊆ dom 𝑂)
215213, 214sylibr 225 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ⊆ dom 𝑂)
216140, 215eqsstrd 3801 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑖 + 1)) ⊆ dom 𝑂)
217192, 38, 37, 194, 216caragensplit 41357 . . . . . . . 8 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))))) = (𝑂‘(𝐺‘(𝑖 + 1))))
218217eqcomd 2771 . . . . . . 7 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘(𝐺‘(𝑖 + 1))) = ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))))))
2192183adant3 1162 . . . . . 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))))))
220192adantr 472 . . . . . . . . . 10 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → 𝑂 ∈ OutMeas)
221169adantr 472 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → 𝜑)
222 elfzuz 12548 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑀...(𝑖 + 1)) → 𝑛 ∈ (ℤ𝑀))
223222, 134syl6eleq 2854 . . . . . . . . . . . 12 (𝑛 ∈ (𝑀...(𝑖 + 1)) → 𝑛𝑍)
224223adantl 473 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → 𝑛𝑍)
22541, 206fssd 6239 . . . . . . . . . . . . 13 (𝜑𝐸:𝑍⟶dom 𝑂)
226225ffvelrnda 6551 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → (𝐸𝑛) ∈ dom 𝑂)
227226, 53syl 17 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (𝐸𝑛) ⊆ dom 𝑂)
228221, 224, 227syl2anc 579 . . . . . . . . . 10 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → (𝐸𝑛) ⊆ dom 𝑂)
229220, 37, 228omecl 41360 . . . . . . . . 9 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
230 2fveq3 6382 . . . . . . . . 9 (𝑛 = (𝑖 + 1) → (𝑂‘(𝐸𝑛)) = (𝑂‘(𝐸‘(𝑖 + 1))))
231149, 229, 230sge0p1 41271 . . . . . . . 8 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛)))) = ((Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))))
2322313adant3 1162 . . . . . . 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)))))
233 id 22 . . . . . . . . . 10 ((𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))) → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))))
234233eqcomd 2771 . . . . . . . . 9 ((𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))) → (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))) = (𝑂‘(𝐺𝑖)))
235234oveq1d 6859 . . . . . . . 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)))))
2362353ad2ant3 1165 . . . . . . 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)))))
237 simpl 474 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...𝑖)) → 𝜑)
238159sseli 3759 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (𝑀...𝑖) → 𝑗𝑍)
239238adantl 473 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...𝑖)) → 𝑗𝑍)
240237, 239, 211syl2anc 579 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (𝑀...𝑖)) → (𝐸𝑗) ⊆ dom 𝑂)
241240adantlr 706 . . . . . . . . . . . . . 14 (((𝜑𝑖𝑍) ∧ 𝑗 ∈ (𝑀...𝑖)) → (𝐸𝑗) ⊆ dom 𝑂)
242241ralrimiva 3113 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍) → ∀𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ⊆ dom 𝑂)
243 iunss 4719 . . . . . . . . . . . . 13 ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ⊆ dom 𝑂 ↔ ∀𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ⊆ dom 𝑂)
244242, 243sylibr 225 . . . . . . . . . . . 12 ((𝜑𝑖𝑍) → 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ⊆ dom 𝑂)
245179, 244eqsstrd 3801 . . . . . . . . . . 11 ((𝜑𝑖𝑍) → (𝐺𝑖) ⊆ dom 𝑂)
246169, 170, 245syl2anc 579 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐺𝑖) ⊆ dom 𝑂)
247192, 37, 246omexrcl 41364 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘(𝐺𝑖)) ∈ ℝ*)
248112, 215sstrd 3773 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ⊆ dom 𝑂)
249192, 37, 248omexrcl 41364 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘(𝐸‘(𝑖 + 1))) ∈ ℝ*)
250247, 249xaddcomd 40181 . . . . . . . 8 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝑂‘(𝐺𝑖)) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))) = ((𝑂‘(𝐸‘(𝑖 + 1))) +𝑒 (𝑂‘(𝐺𝑖))))
2512503adant3 1162 . . . . . . 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))))
252232, 236, 2513eqtrd 2803 . . . . . 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))))
253191, 219, 2523eqtr4d 2809 . . . . 5 ((𝜑𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) → (𝑂‘(𝐺‘(𝑖 + 1))) = (Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛)))))
25488, 89, 92, 253syl3anc 1490 . . . 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)))))
2552543exp 1148 . . 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)))))))
25610, 16, 22, 28, 87, 255fzind2 12797 . 2 (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (𝑂‘(𝐺𝑁)) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛))))))
2573, 4, 256sylc 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 1107   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  cdif 3731  cun 3732  cin 3733  wss 3734  c0 4081  {csn 4336   cuni 4596   ciun 4678  Disj wdisj 4779   class class class wbr 4811  cmpt 4890  dom cdm 5279  wf 6066  cfv 6070  (class class class)co 6844  cr 10190  0cc0 10191  1c1 10192   + caddc 10194  +∞cpnf 10327  cle 10331  cz 11626  cuz 11889   +𝑒 cxad 12147  [,]cicc 12383  ...cfz 12536  ..^cfzo 12676  Σ^csumge0 41219  OutMeascome 41346  CaraGenccaragen 41348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-inf2 8755  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268  ax-pre-sup 10269
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-disj 4780  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-om 7266  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-1o 7766  df-oadd 7770  df-er 7949  df-en 8163  df-dom 8164  df-sdom 8165  df-fin 8166  df-sup 8557  df-oi 8624  df-card 9018  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-div 10941  df-nn 11277  df-2 11337  df-3 11338  df-n0 11541  df-z 11627  df-uz 11890  df-rp 12032  df-xadd 12150  df-ico 12386  df-icc 12387  df-fz 12537  df-fzo 12677  df-seq 13012  df-exp 13071  df-hash 13325  df-cj 14127  df-re 14128  df-im 14129  df-sqrt 14263  df-abs 14264  df-clim 14507  df-sum 14705  df-sumge0 41220  df-ome 41347  df-caragen 41349
This theorem is referenced by:  caratheodorylem2  41384
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