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Theorem caratheodorylem1 46482
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 13569 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 17 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 id 22 . 2 (𝜑𝜑)
5 2fveq3 6912 . . . . 5 (𝑗 = 𝑀 → (𝑂‘(𝐺𝑗)) = (𝑂‘(𝐺𝑀)))
6 oveq2 7439 . . . . . . 7 (𝑗 = 𝑀 → (𝑀...𝑗) = (𝑀...𝑀))
76mpteq1d 5243 . . . . . 6 (𝑗 = 𝑀 → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛))))
87fveq2d 6911 . . . . 5 (𝑗 = 𝑀 → (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) = (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛)))))
95, 8eqeq12d 2751 . . . 4 (𝑗 = 𝑀 → ((𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) ↔ (𝑂‘(𝐺𝑀)) = (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛))))))
109imbi2d 340 . . 3 (𝑗 = 𝑀 → ((𝜑 → (𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))))) ↔ (𝜑 → (𝑂‘(𝐺𝑀)) = (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛)))))))
11 2fveq3 6912 . . . . 5 (𝑗 = 𝑖 → (𝑂‘(𝐺𝑗)) = (𝑂‘(𝐺𝑖)))
12 oveq2 7439 . . . . . . 7 (𝑗 = 𝑖 → (𝑀...𝑗) = (𝑀...𝑖))
1312mpteq1d 5243 . . . . . 6 (𝑗 = 𝑖 → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))
1413fveq2d 6911 . . . . 5 (𝑗 = 𝑖 → (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))))
1511, 14eqeq12d 2751 . . . 4 (𝑗 = 𝑖 → ((𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) ↔ (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))))
1615imbi2d 340 . . 3 (𝑗 = 𝑖 → ((𝜑 → (𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))))) ↔ (𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))))))
17 2fveq3 6912 . . . . 5 (𝑗 = (𝑖 + 1) → (𝑂‘(𝐺𝑗)) = (𝑂‘(𝐺‘(𝑖 + 1))))
18 oveq2 7439 . . . . . . 7 (𝑗 = (𝑖 + 1) → (𝑀...𝑗) = (𝑀...(𝑖 + 1)))
1918mpteq1d 5243 . . . . . 6 (𝑗 = (𝑖 + 1) → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛))))
2019fveq2d 6911 . . . . 5 (𝑗 = (𝑖 + 1) → (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) = (Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛)))))
2117, 20eqeq12d 2751 . . . 4 (𝑗 = (𝑖 + 1) → ((𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) ↔ (𝑂‘(𝐺‘(𝑖 + 1))) = (Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛))))))
2221imbi2d 340 . . 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 6912 . . . . 5 (𝑗 = 𝑁 → (𝑂‘(𝐺𝑗)) = (𝑂‘(𝐺𝑁)))
24 oveq2 7439 . . . . . . 7 (𝑗 = 𝑁 → (𝑀...𝑗) = (𝑀...𝑁))
2524mpteq1d 5243 . . . . . 6 (𝑗 = 𝑁 → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛))))
2625fveq2d 6911 . . . . 5 (𝑗 = 𝑁 → (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛)))))
2723, 26eqeq12d 2751 . . . 4 (𝑗 = 𝑁 → ((𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛)))) ↔ (𝑂‘(𝐺𝑁)) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛))))))
2827imbi2d 340 . . 3 (𝑗 = 𝑁 → ((𝜑 → (𝑂‘(𝐺𝑗)) = (Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸𝑛))))) ↔ (𝜑 → (𝑂‘(𝐺𝑁)) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛)))))))
29 eluzel2 12881 . . . . . . . . 9 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
301, 29syl 17 . . . . . . . 8 (𝜑𝑀 ∈ ℤ)
31 fzsn 13603 . . . . . . . 8 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
3230, 31syl 17 . . . . . . 7 (𝜑 → (𝑀...𝑀) = {𝑀})
3332mpteq1d 5243 . . . . . 6 (𝜑 → (𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛))))
3433fveq2d 6911 . . . . 5 (𝜑 → (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛)))) = (Σ^‘(𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛)))))
35 caratheodorylem1.o . . . . . . . . 9 (𝜑𝑂 ∈ OutMeas)
3635adantr 480 . . . . . . . 8 ((𝜑𝑛 ∈ {𝑀}) → 𝑂 ∈ OutMeas)
37 eqid 2735 . . . . . . . 8 dom 𝑂 = dom 𝑂
38 caratheodorylem1.s . . . . . . . . . . . 12 𝑆 = (CaraGen‘𝑂)
3938caragenss 46460 . . . . . . . . . . 11 (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
4036, 39syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ {𝑀}) → 𝑆 ⊆ dom 𝑂)
41 caratheodorylem1.e . . . . . . . . . . . 12 (𝜑𝐸:𝑍𝑆)
4241adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ {𝑀}) → 𝐸:𝑍𝑆)
43 elsni 4648 . . . . . . . . . . . . 13 (𝑛 ∈ {𝑀} → 𝑛 = 𝑀)
4443adantl 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ {𝑀}) → 𝑛 = 𝑀)
45 uzid 12891 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
4630, 45syl 17 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ (ℤ𝑀))
47 caratheodorylem1.z . . . . . . . . . . . . . 14 𝑍 = (ℤ𝑀)
4846, 47eleqtrrdi 2850 . . . . . . . . . . . . 13 (𝜑𝑀𝑍)
4948adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ {𝑀}) → 𝑀𝑍)
5044, 49eqeltrd 2839 . . . . . . . . . . 11 ((𝜑𝑛 ∈ {𝑀}) → 𝑛𝑍)
5142, 50ffvelcdmd 7105 . . . . . . . . . 10 ((𝜑𝑛 ∈ {𝑀}) → (𝐸𝑛) ∈ 𝑆)
5240, 51sseldd 3996 . . . . . . . . 9 ((𝜑𝑛 ∈ {𝑀}) → (𝐸𝑛) ∈ dom 𝑂)
53 elssuni 4942 . . . . . . . . 9 ((𝐸𝑛) ∈ dom 𝑂 → (𝐸𝑛) ⊆ dom 𝑂)
5452, 53syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ {𝑀}) → (𝐸𝑛) ⊆ dom 𝑂)
5536, 37, 54omecl 46459 . . . . . . 7 ((𝜑𝑛 ∈ {𝑀}) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
56 eqid 2735 . . . . . . 7 (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛)))
5755, 56fmptd 7134 . . . . . 6 (𝜑 → (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛))):{𝑀}⟶(0[,]+∞))
5830, 57sge0sn 46335 . . . . 5 (𝜑 → (Σ^‘(𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛)))) = ((𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛)))‘𝑀))
59 eqidd 2736 . . . . . 6 (𝜑 → (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛))) = (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛))))
6032iuneq1d 5024 . . . . . . . . . 10 (𝜑 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖) = 𝑖 ∈ {𝑀} (𝐸𝑖))
61 fveq2 6907 . . . . . . . . . . . 12 (𝑖 = 𝑀 → (𝐸𝑖) = (𝐸𝑀))
6261iunxsng 5095 . . . . . . . . . . 11 (𝑀𝑍 𝑖 ∈ {𝑀} (𝐸𝑖) = (𝐸𝑀))
6348, 62syl 17 . . . . . . . . . 10 (𝜑 𝑖 ∈ {𝑀} (𝐸𝑖) = (𝐸𝑀))
64 eqidd 2736 . . . . . . . . . 10 (𝜑 → (𝐸𝑀) = (𝐸𝑀))
6560, 63, 643eqtrrd 2780 . . . . . . . . 9 (𝜑 → (𝐸𝑀) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
6665adantr 480 . . . . . . . 8 ((𝜑𝑛 = 𝑀) → (𝐸𝑀) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
67 fveq2 6907 . . . . . . . . 9 (𝑛 = 𝑀 → (𝐸𝑛) = (𝐸𝑀))
6867adantl 481 . . . . . . . 8 ((𝜑𝑛 = 𝑀) → (𝐸𝑛) = (𝐸𝑀))
69 caratheodorylem1.g . . . . . . . . . 10 𝐺 = (𝑛𝑍 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖))
70 oveq2 7439 . . . . . . . . . . 11 (𝑛 = 𝑀 → (𝑀...𝑛) = (𝑀...𝑀))
7170iuneq1d 5024 . . . . . . . . . 10 (𝑛 = 𝑀 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
72 ovex 7464 . . . . . . . . . . . 12 (𝑀...𝑀) ∈ V
73 fvex 6920 . . . . . . . . . . . 12 (𝐸𝑖) ∈ V
7472, 73iunex 7992 . . . . . . . . . . 11 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖) ∈ V
7574a1i 11 . . . . . . . . . 10 (𝜑 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖) ∈ V)
7669, 71, 48, 75fvmptd3 7039 . . . . . . . . 9 (𝜑 → (𝐺𝑀) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
7776adantr 480 . . . . . . . 8 ((𝜑𝑛 = 𝑀) → (𝐺𝑀) = 𝑖 ∈ (𝑀...𝑀)(𝐸𝑖))
7866, 68, 773eqtr4d 2785 . . . . . . 7 ((𝜑𝑛 = 𝑀) → (𝐸𝑛) = (𝐺𝑀))
7978fveq2d 6911 . . . . . 6 ((𝜑𝑛 = 𝑀) → (𝑂‘(𝐸𝑛)) = (𝑂‘(𝐺𝑀)))
80 snidg 4665 . . . . . . 7 (𝑀𝑍𝑀 ∈ {𝑀})
8148, 80syl 17 . . . . . 6 (𝜑𝑀 ∈ {𝑀})
82 fvexd 6922 . . . . . 6 (𝜑 → (𝑂‘(𝐺𝑀)) ∈ V)
8359, 79, 81, 82fvmptd 7023 . . . . 5 (𝜑 → ((𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸𝑛)))‘𝑀) = (𝑂‘(𝐺𝑀)))
8434, 58, 833eqtrrd 2780 . . . 4 (𝜑 → (𝑂‘(𝐺𝑀)) = (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛)))))
8584a1i 11 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (𝑂‘(𝐺𝑀)) = (Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸𝑛))))))
86 simp3 1137 . . . . 5 ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) ∧ 𝜑) → 𝜑)
87 simp1 1135 . . . . 5 ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) ∧ 𝜑) → 𝑖 ∈ (𝑀..^𝑁))
88 id 22 . . . . . . 7 ((𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) → (𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))))
8988imp 406 . . . . . 6 (((𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) ∧ 𝜑) → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))))
90893adant1 1129 . . . . 5 ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) ∧ 𝜑) → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))))
91 elfzoel1 13694 . . . . . . . . . . . . . 14 (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℤ)
92 elfzoelz 13696 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝑀..^𝑁) → 𝑖 ∈ ℤ)
9392peano2zd 12723 . . . . . . . . . . . . . 14 (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ∈ ℤ)
9491zred 12720 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℝ)
9593zred 12720 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ∈ ℝ)
9692zred 12720 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (𝑀..^𝑁) → 𝑖 ∈ ℝ)
97 elfzole1 13704 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (𝑀..^𝑁) → 𝑀𝑖)
9896ltp1d 12196 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (𝑀..^𝑁) → 𝑖 < (𝑖 + 1))
9994, 96, 95, 97, 98lelttrd 11417 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝑀..^𝑁) → 𝑀 < (𝑖 + 1))
10094, 95, 99ltled 11407 . . . . . . . . . . . . . 14 (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ≤ (𝑖 + 1))
101 leid 11355 . . . . . . . . . . . . . . 15 ((𝑖 + 1) ∈ ℝ → (𝑖 + 1) ≤ (𝑖 + 1))
10295, 101syl 17 . . . . . . . . . . . . . 14 (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ≤ (𝑖 + 1))
10391, 93, 93, 100, 102elfzd 13552 . . . . . . . . . . . . 13 (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ∈ (𝑀...(𝑖 + 1)))
104103adantl 481 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ (𝑀...(𝑖 + 1)))
105 fveq2 6907 . . . . . . . . . . . . 13 (𝑗 = (𝑖 + 1) → (𝐸𝑗) = (𝐸‘(𝑖 + 1)))
106105ssiun2s 5053 . . . . . . . . . . . 12 ((𝑖 + 1) ∈ (𝑀...(𝑖 + 1)) → (𝐸‘(𝑖 + 1)) ⊆ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗))
107104, 106syl 17 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ⊆ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗))
108 fveq2 6907 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (𝐸𝑖) = (𝐸𝑗))
109108cbviunv 5045 . . . . . . . . . . . . . . 15 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖) = 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗)
110109mpteq2i 5253 . . . . . . . . . . . . . 14 (𝑛𝑍 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖)) = (𝑛𝑍 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗))
11169, 110eqtri 2763 . . . . . . . . . . . . 13 𝐺 = (𝑛𝑍 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗))
112 oveq2 7439 . . . . . . . . . . . . . 14 (𝑛 = (𝑖 + 1) → (𝑀...𝑛) = (𝑀...(𝑖 + 1)))
113112iuneq1d 5024 . . . . . . . . . . . . 13 (𝑛 = (𝑖 + 1) → 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗) = 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗))
11430adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℤ)
11592adantl 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ ℤ)
116115peano2zd 12723 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ ℤ)
117114zred 12720 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
118116zred 12720 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ ℝ)
119115zred 12720 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ ℝ)
12097adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑀𝑖)
121119ltp1d 12196 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑖 < (𝑖 + 1))
122117, 119, 118, 120, 121lelttrd 11417 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑀 < (𝑖 + 1))
123117, 118, 122ltled 11407 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ≤ (𝑖 + 1))
124114, 116, 1233jca 1127 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑖 + 1)))
125 eluz2 12882 . . . . . . . . . . . . . . 15 ((𝑖 + 1) ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑖 + 1)))
126124, 125sylibr 234 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ (ℤ𝑀))
12747eqcomi 2744 . . . . . . . . . . . . . 14 (ℤ𝑀) = 𝑍
128126, 127eleqtrdi 2849 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ 𝑍)
129 ovex 7464 . . . . . . . . . . . . . . 15 (𝑀...(𝑖 + 1)) ∈ V
130 fvex 6920 . . . . . . . . . . . . . . 15 (𝐸𝑗) ∈ V
131129, 130iunex 7992 . . . . . . . . . . . . . 14 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ∈ V
132131a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ∈ V)
133111, 113, 128, 132fvmptd3 7039 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑖 + 1)) = 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗))
134133eqcomd 2741 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) = (𝐺‘(𝑖 + 1)))
135107, 134sseqtrd 4036 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ⊆ (𝐺‘(𝑖 + 1)))
136 sseqin2 4231 . . . . . . . . . . 11 ((𝐸‘(𝑖 + 1)) ⊆ (𝐺‘(𝑖 + 1)) ↔ ((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1))) = (𝐸‘(𝑖 + 1)))
137136biimpi 216 . . . . . . . . . 10 ((𝐸‘(𝑖 + 1)) ⊆ (𝐺‘(𝑖 + 1)) → ((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1))) = (𝐸‘(𝑖 + 1)))
138135, 137syl 17 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1))) = (𝐸‘(𝑖 + 1)))
139138fveq2d 6911 . . . . . . . 8 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) = (𝑂‘(𝐸‘(𝑖 + 1))))
140 nfcv 2903 . . . . . . . . . . . . 13 𝑗(𝐸‘(𝑖 + 1))
141 elfzouz 13700 . . . . . . . . . . . . . 14 (𝑖 ∈ (𝑀..^𝑁) → 𝑖 ∈ (ℤ𝑀))
142141adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ (ℤ𝑀))
143140, 142, 105iunp1 45006 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) = ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ (𝐸‘(𝑖 + 1))))
144133, 143eqtrd 2775 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑖 + 1)) = ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ (𝐸‘(𝑖 + 1))))
145144difeq1d 4135 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = (( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ (𝐸‘(𝑖 + 1))) ∖ (𝐸‘(𝑖 + 1))))
146 caratheodorylem1.dj . . . . . . . . . . . . . . 15 (𝜑Disj 𝑛𝑍 (𝐸𝑛))
147 fveq2 6907 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → (𝐸𝑛) = (𝐸𝑗))
148147cbvdisjv 5126 . . . . . . . . . . . . . . 15 (Disj 𝑛𝑍 (𝐸𝑛) ↔ Disj 𝑗𝑍 (𝐸𝑗))
149146, 148sylib 218 . . . . . . . . . . . . . 14 (𝜑Disj 𝑗𝑍 (𝐸𝑗))
150149adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → Disj 𝑗𝑍 (𝐸𝑗))
151 fzssuz 13602 . . . . . . . . . . . . . . 15 (𝑀...𝑖) ⊆ (ℤ𝑀)
152151, 127sseqtri 4032 . . . . . . . . . . . . . 14 (𝑀...𝑖) ⊆ 𝑍
153152a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑀...𝑖) ⊆ 𝑍)
154 fzp1nel 13648 . . . . . . . . . . . . . . . 16 ¬ (𝑖 + 1) ∈ (𝑀...𝑖)
155154a1i 11 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝑀..^𝑁) → ¬ (𝑖 + 1) ∈ (𝑀...𝑖))
156155adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ¬ (𝑖 + 1) ∈ (𝑀...𝑖))
157128, 156eldifd 3974 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ (𝑍 ∖ (𝑀...𝑖)))
158150, 153, 157, 105disjiun2 44998 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∩ (𝐸‘(𝑖 + 1))) = ∅)
159 undif4 4473 . . . . . . . . . . . 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))))
160158, 159syl 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))))
161160eqcomd 2741 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ (𝐸‘(𝑖 + 1))) ∖ (𝐸‘(𝑖 + 1))) = ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))))
162 simpl 482 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝜑)
163142, 127eleqtrdi 2849 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑖𝑍)
164111a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → 𝐺 = (𝑛𝑍 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗)))
165 simpr 484 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖𝑍) ∧ 𝑛 = 𝑖) → 𝑛 = 𝑖)
166165oveq2d 7447 . . . . . . . . . . . . . . . 16 (((𝜑𝑖𝑍) ∧ 𝑛 = 𝑖) → (𝑀...𝑛) = (𝑀...𝑖))
167166iuneq1d 5024 . . . . . . . . . . . . . . 15 (((𝜑𝑖𝑍) ∧ 𝑛 = 𝑖) → 𝑗 ∈ (𝑀...𝑛)(𝐸𝑗) = 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗))
168 simpr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → 𝑖𝑍)
169 ovex 7464 . . . . . . . . . . . . . . . . 17 (𝑀...𝑖) ∈ V
170169, 130iunex 7992 . . . . . . . . . . . . . . . 16 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∈ V
171170a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝑍) → 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∈ V)
172164, 167, 168, 171fvmptd 7023 . . . . . . . . . . . . . 14 ((𝜑𝑖𝑍) → (𝐺𝑖) = 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗))
173162, 163, 172syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐺𝑖) = 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗))
174173eqcomd 2741 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) = (𝐺𝑖))
175 difid 4382 . . . . . . . . . . . . 13 ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = ∅
176175a1i 11 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = ∅)
177174, 176uneq12d 4179 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = ((𝐺𝑖) ∪ ∅))
178 un0 4400 . . . . . . . . . . . 12 ((𝐺𝑖) ∪ ∅) = (𝐺𝑖)
179178a1i 11 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝐺𝑖) ∪ ∅) = (𝐺𝑖))
180177, 179eqtrd 2775 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = (𝐺𝑖))
181145, 161, 1803eqtrd 2779 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = (𝐺𝑖))
182181fveq2d 6911 . . . . . . . 8 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = (𝑂‘(𝐺𝑖)))
183139, 182oveq12d 7449 . . . . . . 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))))
1841833adant3 1131 . . . . . 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))))
18535adantr 480 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑂 ∈ OutMeas)
18641adantr 480 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝐸:𝑍𝑆)
187186, 128ffvelcdmd 7105 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ∈ 𝑆)
188 simpll 767 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝜑)
18991adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑀 ∈ ℤ)
190 elfzelz 13561 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (𝑀...(𝑖 + 1)) → 𝑗 ∈ ℤ)
191190adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗 ∈ ℤ)
192 elfzle1 13564 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (𝑀...(𝑖 + 1)) → 𝑀𝑗)
193192adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑀𝑗)
194189, 191, 1933jca 1127 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → (𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑀𝑗))
195 eluz2 12882 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑀𝑗))
196194, 195sylibr 234 . . . . . . . . . . . . . . 15 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗 ∈ (ℤ𝑀))
197196, 127eleqtrdi 2849 . . . . . . . . . . . . . 14 ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗𝑍)
198197adantll 714 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗𝑍)
19935, 39syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑆 ⊆ dom 𝑂)
200199adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑗𝑍) → 𝑆 ⊆ dom 𝑂)
20141ffvelcdmda 7104 . . . . . . . . . . . . . . 15 ((𝜑𝑗𝑍) → (𝐸𝑗) ∈ 𝑆)
202200, 201sseldd 3996 . . . . . . . . . . . . . 14 ((𝜑𝑗𝑍) → (𝐸𝑗) ∈ dom 𝑂)
203 elssuni 4942 . . . . . . . . . . . . . 14 ((𝐸𝑗) ∈ dom 𝑂 → (𝐸𝑗) ⊆ dom 𝑂)
204202, 203syl 17 . . . . . . . . . . . . 13 ((𝜑𝑗𝑍) → (𝐸𝑗) ⊆ dom 𝑂)
205188, 198, 204syl2anc 584 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → (𝐸𝑗) ⊆ dom 𝑂)
206205ralrimiva 3144 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ∀𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ⊆ dom 𝑂)
207 iunss 5050 . . . . . . . . . . 11 ( 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ⊆ dom 𝑂 ↔ ∀𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ⊆ dom 𝑂)
208206, 207sylibr 234 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸𝑗) ⊆ dom 𝑂)
209133, 208eqsstrd 4034 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑖 + 1)) ⊆ dom 𝑂)
210185, 38, 37, 187, 209caragensplit 46456 . . . . . . . 8 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))))) = (𝑂‘(𝐺‘(𝑖 + 1))))
211210eqcomd 2741 . . . . . . 7 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘(𝐺‘(𝑖 + 1))) = ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))))))
2122113adant3 1131 . . . . . 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))))))
213185adantr 480 . . . . . . . . . 10 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → 𝑂 ∈ OutMeas)
214162adantr 480 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → 𝜑)
215 elfzuz 13557 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑀...(𝑖 + 1)) → 𝑛 ∈ (ℤ𝑀))
216215, 127eleqtrdi 2849 . . . . . . . . . . . 12 (𝑛 ∈ (𝑀...(𝑖 + 1)) → 𝑛𝑍)
217216adantl 481 . . . . . . . . . . 11 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → 𝑛𝑍)
21841, 199fssd 6754 . . . . . . . . . . . . 13 (𝜑𝐸:𝑍⟶dom 𝑂)
219218ffvelcdmda 7104 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → (𝐸𝑛) ∈ dom 𝑂)
220219, 53syl 17 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (𝐸𝑛) ⊆ dom 𝑂)
221214, 217, 220syl2anc 584 . . . . . . . . . 10 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → (𝐸𝑛) ⊆ dom 𝑂)
222213, 37, 221omecl 46459 . . . . . . . . 9 (((𝜑𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → (𝑂‘(𝐸𝑛)) ∈ (0[,]+∞))
223 2fveq3 6912 . . . . . . . . 9 (𝑛 = (𝑖 + 1) → (𝑂‘(𝐸𝑛)) = (𝑂‘(𝐸‘(𝑖 + 1))))
224142, 222, 223sge0p1 46370 . . . . . . . 8 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛)))) = ((Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))))
2252243adant3 1131 . . . . . . 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)))))
226 id 22 . . . . . . . . . 10 ((𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))) → (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))))
227226eqcomd 2741 . . . . . . . . 9 ((𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))) → (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛)))) = (𝑂‘(𝐺𝑖)))
228227oveq1d 7446 . . . . . . . 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)))))
2292283ad2ant3 1134 . . . . . . 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)))))
230 simpl 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...𝑖)) → 𝜑)
231152sseli 3991 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (𝑀...𝑖) → 𝑗𝑍)
232231adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀...𝑖)) → 𝑗𝑍)
233230, 232, 204syl2anc 584 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (𝑀...𝑖)) → (𝐸𝑗) ⊆ dom 𝑂)
234233adantlr 715 . . . . . . . . . . . . . 14 (((𝜑𝑖𝑍) ∧ 𝑗 ∈ (𝑀...𝑖)) → (𝐸𝑗) ⊆ dom 𝑂)
235234ralrimiva 3144 . . . . . . . . . . . . 13 ((𝜑𝑖𝑍) → ∀𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ⊆ dom 𝑂)
236 iunss 5050 . . . . . . . . . . . . 13 ( 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ⊆ dom 𝑂 ↔ ∀𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ⊆ dom 𝑂)
237235, 236sylibr 234 . . . . . . . . . . . 12 ((𝜑𝑖𝑍) → 𝑗 ∈ (𝑀...𝑖)(𝐸𝑗) ⊆ dom 𝑂)
238172, 237eqsstrd 4034 . . . . . . . . . . 11 ((𝜑𝑖𝑍) → (𝐺𝑖) ⊆ dom 𝑂)
239162, 163, 238syl2anc 584 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐺𝑖) ⊆ dom 𝑂)
240185, 37, 239omexrcl 46463 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘(𝐺𝑖)) ∈ ℝ*)
241107, 208sstrd 4006 . . . . . . . . . 10 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ⊆ dom 𝑂)
242185, 37, 241omexrcl 46463 . . . . . . . . 9 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘(𝐸‘(𝑖 + 1))) ∈ ℝ*)
243240, 242xaddcomd 45274 . . . . . . . 8 ((𝜑𝑖 ∈ (𝑀..^𝑁)) → ((𝑂‘(𝐺𝑖)) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))) = ((𝑂‘(𝐸‘(𝑖 + 1))) +𝑒 (𝑂‘(𝐺𝑖))))
2442433adant3 1131 . . . . . . 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))))
245225, 229, 2443eqtrd 2779 . . . . . 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))))
246184, 212, 2453eqtr4d 2785 . . . . 5 ((𝜑𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺𝑖)) = (Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸𝑛))))) → (𝑂‘(𝐺‘(𝑖 + 1))) = (Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸𝑛)))))
24786, 87, 90, 246syl3anc 1370 . . . 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)))))
2482473exp 1118 . . 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)))))))
24910, 16, 22, 28, 85, 248fzind2 13821 . 2 (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (𝑂‘(𝐺𝑁)) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛))))))
2503, 4, 249sylc 65 1 (𝜑 → (𝑂‘(𝐺𝑁)) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cdif 3960  cun 3961  cin 3962  wss 3963  c0 4339  {csn 4631   cuni 4912   ciun 4996  Disj wdisj 5115   class class class wbr 5148  cmpt 5231  dom cdm 5689  wf 6559  cfv 6563  (class class class)co 7431  cr 11152  0cc0 11153  1c1 11154   + caddc 11156  +∞cpnf 11290  cle 11294  cz 12611  cuz 12876   +𝑒 cxad 13150  [,]cicc 13387  ...cfz 13544  ..^cfzo 13691  Σ^csumge0 46318  OutMeascome 46445  CaraGenccaragen 46447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-disj 5116  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-xadd 13153  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-sum 15720  df-sumge0 46319  df-ome 46446  df-caragen 46448
This theorem is referenced by:  caratheodorylem2  46483
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