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Theorem carsgclctunlem2 31570
Description: Lemma for carsgclctun 31572. (Contributed by Thierry Arnoux, 25-May-2020.)
Hypotheses
Ref Expression
carsgval.1 (𝜑𝑂𝑉)
carsgval.2 (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))
carsgsiga.1 (𝜑 → (𝑀‘∅) = 0)
carsgsiga.2 ((𝜑𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))
carsgsiga.3 ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))
carsgclctunlem2.1 (𝜑Disj 𝑘 ∈ ℕ 𝐴)
carsgclctunlem2.2 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ (toCaraSiga‘𝑀))
carsgclctunlem2.3 (𝜑𝐸 ∈ 𝒫 𝑂)
carsgclctunlem2.4 (𝜑 → (𝑀𝐸) ≠ +∞)
Assertion
Ref Expression
carsgclctunlem2 (𝜑 → ((𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) +𝑒 (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) ≤ (𝑀𝐸))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐸,𝑦   𝑥,𝑀,𝑦   𝑥,𝑂,𝑦   𝜑,𝑥,𝑦,𝑘   𝑘,𝐸   𝑘,𝑀   𝑘,𝑂   𝜑,𝑘
Allowed substitution hints:   𝐴(𝑘)   𝑉(𝑥,𝑦,𝑘)

Proof of Theorem carsgclctunlem2
Dummy variables 𝑒 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iunin2 4984 . . . . 5 𝑘 ∈ ℕ (𝐸𝐴) = (𝐸 𝑘 ∈ ℕ 𝐴)
21fveq2i 6666 . . . 4 (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) = (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))
3 iccssxr 12811 . . . . 5 (0[,]+∞) ⊆ ℝ*
4 carsgval.2 . . . . . 6 (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))
5 nnex 11636 . . . . . . . 8 ℕ ∈ V
65a1i 11 . . . . . . 7 (𝜑 → ℕ ∈ V)
7 carsgclctunlem2.3 . . . . . . . . 9 (𝜑𝐸 ∈ 𝒫 𝑂)
87adantr 483 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐸 ∈ 𝒫 𝑂)
98elpwincl1 30278 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝐸𝐴) ∈ 𝒫 𝑂)
106, 9elpwiuncl 30280 . . . . . 6 (𝜑 𝑘 ∈ ℕ (𝐸𝐴) ∈ 𝒫 𝑂)
114, 10ffvelrnd 6845 . . . . 5 (𝜑 → (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) ∈ (0[,]+∞))
123, 11sseldi 3963 . . . 4 (𝜑 → (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) ∈ ℝ*)
132, 12eqeltrrid 2916 . . 3 (𝜑 → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
144, 7ffvelrnd 6845 . . . . 5 (𝜑 → (𝑀𝐸) ∈ (0[,]+∞))
153, 14sseldi 3963 . . . 4 (𝜑 → (𝑀𝐸) ∈ ℝ*)
167elpwdifcl 30279 . . . . . . 7 (𝜑 → (𝐸 𝑘 ∈ ℕ 𝐴) ∈ 𝒫 𝑂)
174, 16ffvelrnd 6845 . . . . . 6 (𝜑 → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ (0[,]+∞))
183, 17sseldi 3963 . . . . 5 (𝜑 → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
1918xnegcld 12685 . . . 4 (𝜑 → -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
2015, 19xaddcld 12686 . . 3 (𝜑 → ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) ∈ ℝ*)
214adantr 483 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
2221, 9ffvelrnd 6845 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑀‘(𝐸𝐴)) ∈ (0[,]+∞))
2322ralrimiva 3180 . . . . . . 7 (𝜑 → ∀𝑘 ∈ ℕ (𝑀‘(𝐸𝐴)) ∈ (0[,]+∞))
24 nfcv 2975 . . . . . . . 8 𝑘
2524esumcl 31282 . . . . . . 7 ((ℕ ∈ V ∧ ∀𝑘 ∈ ℕ (𝑀‘(𝐸𝐴)) ∈ (0[,]+∞)) → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸𝐴)) ∈ (0[,]+∞))
266, 23, 25syl2anc 586 . . . . . 6 (𝜑 → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸𝐴)) ∈ (0[,]+∞))
273, 26sseldi 3963 . . . . 5 (𝜑 → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸𝐴)) ∈ ℝ*)
289ralrimiva 3180 . . . . . . . . 9 (𝜑 → ∀𝑘 ∈ ℕ (𝐸𝐴) ∈ 𝒫 𝑂)
29 dfiun3g 5828 . . . . . . . . 9 (∀𝑘 ∈ ℕ (𝐸𝐴) ∈ 𝒫 𝑂 𝑘 ∈ ℕ (𝐸𝐴) = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)))
3028, 29syl 17 . . . . . . . 8 (𝜑 𝑘 ∈ ℕ (𝐸𝐴) = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)))
3130fveq2d 6667 . . . . . . 7 (𝜑 → (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) = (𝑀 ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))))
32 nnct 13341 . . . . . . . . . 10 ℕ ≼ ω
33 mptct 9952 . . . . . . . . . 10 (ℕ ≼ ω → (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ≼ ω)
34 rnct 9939 . . . . . . . . . 10 ((𝑘 ∈ ℕ ↦ (𝐸𝐴)) ≼ ω → ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ≼ ω)
3532, 33, 34mp2b 10 . . . . . . . . 9 ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ≼ ω
3635a1i 11 . . . . . . . 8 (𝜑 → ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ≼ ω)
37 eqid 2819 . . . . . . . . . 10 (𝑘 ∈ ℕ ↦ (𝐸𝐴)) = (𝑘 ∈ ℕ ↦ (𝐸𝐴))
3837rnmptss 6879 . . . . . . . . 9 (∀𝑘 ∈ ℕ (𝐸𝐴) ∈ 𝒫 𝑂 → ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ⊆ 𝒫 𝑂)
3928, 38syl 17 . . . . . . . 8 (𝜑 → ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ⊆ 𝒫 𝑂)
40 mptexg 6976 . . . . . . . . . 10 (ℕ ∈ V → (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ∈ V)
41 rnexg 7606 . . . . . . . . . 10 ((𝑘 ∈ ℕ ↦ (𝐸𝐴)) ∈ V → ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ∈ V)
425, 40, 41mp2b 10 . . . . . . . . 9 ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ∈ V
43 breq1 5060 . . . . . . . . . . . 12 (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) → (𝑥 ≼ ω ↔ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ≼ ω))
44 sseq1 3990 . . . . . . . . . . . 12 (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) → (𝑥 ⊆ 𝒫 𝑂 ↔ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ⊆ 𝒫 𝑂))
4543, 443anbi23d 1433 . . . . . . . . . . 11 (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) → ((𝜑𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) ↔ (𝜑 ∧ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ≼ ω ∧ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ⊆ 𝒫 𝑂)))
46 unieq 4838 . . . . . . . . . . . . 13 (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) → 𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)))
4746fveq2d 6667 . . . . . . . . . . . 12 (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) → (𝑀 𝑥) = (𝑀 ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))))
48 esumeq1 31286 . . . . . . . . . . . 12 (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) → Σ*𝑦𝑥(𝑀𝑦) = Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))(𝑀𝑦))
4947, 48breq12d 5070 . . . . . . . . . . 11 (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) → ((𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦) ↔ (𝑀 ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))) ≤ Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))(𝑀𝑦)))
5045, 49imbi12d 347 . . . . . . . . . 10 (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) → (((𝜑𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦)) ↔ ((𝜑 ∧ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ≼ ω ∧ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ⊆ 𝒫 𝑂) → (𝑀 ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))) ≤ Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))(𝑀𝑦))))
51 carsgsiga.2 . . . . . . . . . 10 ((𝜑𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))
5250, 51vtoclg 3566 . . . . . . . . 9 (ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ∈ V → ((𝜑 ∧ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ≼ ω ∧ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ⊆ 𝒫 𝑂) → (𝑀 ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))) ≤ Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))(𝑀𝑦)))
5342, 52ax-mp 5 . . . . . . . 8 ((𝜑 ∧ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ≼ ω ∧ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ⊆ 𝒫 𝑂) → (𝑀 ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))) ≤ Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))(𝑀𝑦))
5436, 39, 53mpd3an23 1457 . . . . . . 7 (𝜑 → (𝑀 ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))) ≤ Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))(𝑀𝑦))
5531, 54eqbrtrd 5079 . . . . . 6 (𝜑 → (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) ≤ Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))(𝑀𝑦))
56 fveq2 6663 . . . . . . 7 (𝑦 = (𝐸𝐴) → (𝑀𝑦) = (𝑀‘(𝐸𝐴)))
57 simpr 487 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ (𝐸𝐴) = ∅) → (𝐸𝐴) = ∅)
5857fveq2d 6667 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ (𝐸𝐴) = ∅) → (𝑀‘(𝐸𝐴)) = (𝑀‘∅))
59 carsgsiga.1 . . . . . . . . 9 (𝜑 → (𝑀‘∅) = 0)
6059ad2antrr 724 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ (𝐸𝐴) = ∅) → (𝑀‘∅) = 0)
6158, 60eqtrd 2854 . . . . . . 7 (((𝜑𝑘 ∈ ℕ) ∧ (𝐸𝐴) = ∅) → (𝑀‘(𝐸𝐴)) = 0)
62 carsgclctunlem2.1 . . . . . . . . 9 (𝜑Disj 𝑘 ∈ ℕ 𝐴)
63 disjin 30328 . . . . . . . . 9 (Disj 𝑘 ∈ ℕ 𝐴Disj 𝑘 ∈ ℕ (𝐴𝐸))
6462, 63syl 17 . . . . . . . 8 (𝜑Disj 𝑘 ∈ ℕ (𝐴𝐸))
65 incom 4176 . . . . . . . . . 10 (𝐴𝐸) = (𝐸𝐴)
6665rgenw 3148 . . . . . . . . 9 𝑘 ∈ ℕ (𝐴𝐸) = (𝐸𝐴)
67 disjeq2 5026 . . . . . . . . 9 (∀𝑘 ∈ ℕ (𝐴𝐸) = (𝐸𝐴) → (Disj 𝑘 ∈ ℕ (𝐴𝐸) ↔ Disj 𝑘 ∈ ℕ (𝐸𝐴)))
6866, 67ax-mp 5 . . . . . . . 8 (Disj 𝑘 ∈ ℕ (𝐴𝐸) ↔ Disj 𝑘 ∈ ℕ (𝐸𝐴))
6964, 68sylib 220 . . . . . . 7 (𝜑Disj 𝑘 ∈ ℕ (𝐸𝐴))
7056, 6, 22, 9, 61, 69esumrnmpt2 31320 . . . . . 6 (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))(𝑀𝑦) = Σ*𝑘 ∈ ℕ(𝑀‘(𝐸𝐴)))
7155, 70breqtrd 5083 . . . . 5 (𝜑 → (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) ≤ Σ*𝑘 ∈ ℕ(𝑀‘(𝐸𝐴)))
72 carsgval.1 . . . . . . . 8 (𝜑𝑂𝑉)
73 difssd 4107 . . . . . . . 8 (𝜑 → (𝐸 𝑘 ∈ ℕ 𝐴) ⊆ 𝐸)
74 carsgsiga.3 . . . . . . . 8 ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))
7572, 4, 73, 7, 74carsgmon 31565 . . . . . . 7 (𝜑 → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀𝐸))
7614, 17, 75xrge0subcld 30479 . . . . . 6 (𝜑 → ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) ∈ (0[,]+∞))
774adantr 483 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
787adantr 483 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → 𝐸 ∈ 𝒫 𝑂)
7978elpwincl1 30278 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐸 𝑘 ∈ (1...𝑛)𝐴) ∈ 𝒫 𝑂)
8077, 79ffvelrnd 6845 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ (0[,]+∞))
813, 80sseldi 3963 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
82 xrge0neqmnf 12832 . . . . . . . . . . 11 ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ (0[,]+∞) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞)
8380, 82syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞)
8478elpwdifcl 30279 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐸 𝑘 ∈ (1...𝑛)𝐴) ∈ 𝒫 𝑂)
8577, 84ffvelrnd 6845 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ (0[,]+∞))
863, 85sseldi 3963 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
87 xrge0neqmnf 12832 . . . . . . . . . . 11 ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ (0[,]+∞) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞)
8885, 87syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞)
8986xnegcld 12685 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
90 xnegneg 12599 . . . . . . . . . . . . . . . . 17 ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ* → -𝑒-𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)))
9186, 90syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → -𝑒-𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)))
9291adantr 483 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → -𝑒-𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)))
93 xnegeq 12592 . . . . . . . . . . . . . . . . 17 (-𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞ → -𝑒-𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -𝑒-∞)
9493adantl 484 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → -𝑒-𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -𝑒-∞)
95 xnegmnf 12595 . . . . . . . . . . . . . . . 16 -𝑒-∞ = +∞
9694, 95syl6eq 2870 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → -𝑒-𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = +∞)
9792, 96eqtr3d 2856 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = +∞)
9897oveq2d 7164 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))) = ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 +∞))
99 simpll 765 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
100 fz1ssnn 12930 . . . . . . . . . . . . . . . . . . . . . . 23 (1...𝑛) ⊆ ℕ
101100a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
102101sselda 3965 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
103 carsgclctunlem2.2 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ (toCaraSiga‘𝑀))
10499, 102, 103syl2anc 586 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (toCaraSiga‘𝑀))
105104ralrimiva 3180 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀))
106 dfiun3g 5828 . . . . . . . . . . . . . . . . . . 19 (∀𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀) → 𝑘 ∈ (1...𝑛)𝐴 = ran (𝑘 ∈ (1...𝑛) ↦ 𝐴))
107105, 106syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → 𝑘 ∈ (1...𝑛)𝐴 = ran (𝑘 ∈ (1...𝑛) ↦ 𝐴))
10872adantr 483 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → 𝑂𝑉)
10959adantr 483 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → (𝑀‘∅) = 0)
110513adant1r 1172 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))
111 fzfi 13332 . . . . . . . . . . . . . . . . . . . . 21 (1...𝑛) ∈ Fin
112 mptfi 8815 . . . . . . . . . . . . . . . . . . . . 21 ((1...𝑛) ∈ Fin → (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ Fin)
113 rnfi 8799 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ Fin → ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ Fin)
114111, 112, 113mp2b 10 . . . . . . . . . . . . . . . . . . . 20 ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ Fin
115114a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ Fin)
116 eqid 2819 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝑛) ↦ 𝐴) = (𝑘 ∈ (1...𝑛) ↦ 𝐴)
117116rnmptss 6879 . . . . . . . . . . . . . . . . . . . 20 (∀𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀) → ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ⊆ (toCaraSiga‘𝑀))
118105, 117syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ⊆ (toCaraSiga‘𝑀))
119108, 77, 109, 110, 115, 118fiunelcarsg 31567 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ (toCaraSiga‘𝑀))
120107, 119eqeltrd 2911 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → 𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀))
121108, 77elcarsg 31556 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → ( 𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀) ↔ ( 𝑘 ∈ (1...𝑛)𝐴𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝑒 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀𝑒))))
122120, 121mpbid 234 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ ℕ) → ( 𝑘 ∈ (1...𝑛)𝐴𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝑒 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀𝑒)))
123122simprd 498 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝑒 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀𝑒))
124 ineq1 4179 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 𝐸 → (𝑒 𝑘 ∈ (1...𝑛)𝐴) = (𝐸 𝑘 ∈ (1...𝑛)𝐴))
125124fveq2d 6667 . . . . . . . . . . . . . . . . . 18 (𝑒 = 𝐸 → (𝑀‘(𝑒 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)))
126 difeq1 4090 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 𝐸 → (𝑒 𝑘 ∈ (1...𝑛)𝐴) = (𝐸 𝑘 ∈ (1...𝑛)𝐴))
127126fveq2d 6667 . . . . . . . . . . . . . . . . . 18 (𝑒 = 𝐸 → (𝑀‘(𝑒 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)))
128125, 127oveq12d 7166 . . . . . . . . . . . . . . . . 17 (𝑒 = 𝐸 → ((𝑀‘(𝑒 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝑒 𝑘 ∈ (1...𝑛)𝐴))) = ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))))
129 fveq2 6663 . . . . . . . . . . . . . . . . 17 (𝑒 = 𝐸 → (𝑀𝑒) = (𝑀𝐸))
130128, 129eqeq12d 2835 . . . . . . . . . . . . . . . 16 (𝑒 = 𝐸 → (((𝑀‘(𝑒 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝑒 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀𝑒) ↔ ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀𝐸)))
131130rspcv 3616 . . . . . . . . . . . . . . 15 (𝐸 ∈ 𝒫 𝑂 → (∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝑒 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀𝑒) → ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀𝐸)))
13278, 123, 131sylc 65 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀𝐸))
133132adantr 483 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀𝐸))
134 xaddpnf1 12611 . . . . . . . . . . . . . . 15 (((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ* ∧ (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞) → ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 +∞) = +∞)
13581, 83, 134syl2anc 586 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 +∞) = +∞)
136135adantr 483 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 +∞) = +∞)
13798, 133, 1363eqtr3d 2862 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → (𝑀𝐸) = +∞)
138 carsgclctunlem2.4 . . . . . . . . . . . . . 14 (𝜑 → (𝑀𝐸) ≠ +∞)
139138ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → (𝑀𝐸) ≠ +∞)
140139neneqd 3019 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → ¬ (𝑀𝐸) = +∞)
141137, 140pm2.65da 815 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ¬ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞)
142141neqned 3021 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞)
143 xaddass 12634 . . . . . . . . . 10 ((((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ* ∧ (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞) ∧ ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ* ∧ (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞) ∧ (-𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ* ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞)) → (((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))) = ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)))))
14481, 83, 86, 88, 89, 142, 143syl222anc 1381 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))) = ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)))))
145 xnegid 12623 . . . . . . . . . . 11 ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ* → ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))) = 0)
14686, 145syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))) = 0)
147146oveq2d 7164 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)))) = ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 0))
148 xaddid1 12626 . . . . . . . . . 10 ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ* → ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 0) = (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)))
14981, 148syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 0) = (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)))
150144, 147, 1493eqtrd 2858 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)))
151132oveq1d 7163 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))) = ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))))
152107ineq2d 4187 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐸 𝑘 ∈ (1...𝑛)𝐴) = (𝐸 ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)))
153152fveq2d 6667 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 ran (𝑘 ∈ (1...𝑛) ↦ 𝐴))))
154 mptss 5903 . . . . . . . . . . . . 13 ((1...𝑛) ⊆ ℕ → (𝑘 ∈ (1...𝑛) ↦ 𝐴) ⊆ (𝑘 ∈ ℕ ↦ 𝐴))
155 rnss 5802 . . . . . . . . . . . . 13 ((𝑘 ∈ (1...𝑛) ↦ 𝐴) ⊆ (𝑘 ∈ ℕ ↦ 𝐴) → ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ⊆ ran (𝑘 ∈ ℕ ↦ 𝐴))
156100, 154, 155mp2b 10 . . . . . . . . . . . 12 ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ⊆ ran (𝑘 ∈ ℕ ↦ 𝐴)
157156a1i 11 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ⊆ ran (𝑘 ∈ ℕ ↦ 𝐴))
158 disjrnmpt 30327 . . . . . . . . . . . . 13 (Disj 𝑘 ∈ ℕ 𝐴Disj 𝑦 ∈ ran (𝑘 ∈ ℕ ↦ 𝐴)𝑦)
15962, 158syl 17 . . . . . . . . . . . 12 (𝜑Disj 𝑦 ∈ ran (𝑘 ∈ ℕ ↦ 𝐴)𝑦)
160159adantr 483 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → Disj 𝑦 ∈ ran (𝑘 ∈ ℕ ↦ 𝐴)𝑦)
161 disjss1 5028 . . . . . . . . . . 11 (ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ⊆ ran (𝑘 ∈ ℕ ↦ 𝐴) → (Disj 𝑦 ∈ ran (𝑘 ∈ ℕ ↦ 𝐴)𝑦Disj 𝑦 ∈ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)𝑦))
162157, 160, 161sylc 65 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → Disj 𝑦 ∈ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)𝑦)
163108, 77, 109, 110, 115, 118, 162, 78carsgclctunlem1 31568 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 ran (𝑘 ∈ (1...𝑛) ↦ 𝐴))) = Σ*𝑦 ∈ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)(𝑀‘(𝐸𝑦)))
164 ineq2 4181 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝐸𝑦) = (𝐸𝐴))
165164fveq2d 6667 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑀‘(𝐸𝑦)) = (𝑀‘(𝐸𝐴)))
166111elexi 3512 . . . . . . . . . . 11 (1...𝑛) ∈ V
167166a1i 11 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1...𝑛) ∈ V)
16899, 102, 22syl2anc 586 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑀‘(𝐸𝐴)) ∈ (0[,]+∞))
169 inss2 4204 . . . . . . . . . . . . . . 15 (𝐸𝐴) ⊆ 𝐴
170 sseq2 3991 . . . . . . . . . . . . . . 15 (𝐴 = ∅ → ((𝐸𝐴) ⊆ 𝐴 ↔ (𝐸𝐴) ⊆ ∅))
171169, 170mpbii 235 . . . . . . . . . . . . . 14 (𝐴 = ∅ → (𝐸𝐴) ⊆ ∅)
172 ss0 4350 . . . . . . . . . . . . . 14 ((𝐸𝐴) ⊆ ∅ → (𝐸𝐴) = ∅)
173171, 172syl 17 . . . . . . . . . . . . 13 (𝐴 = ∅ → (𝐸𝐴) = ∅)
174173adantl 484 . . . . . . . . . . . 12 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝐸𝐴) = ∅)
175174fveq2d 6667 . . . . . . . . . . 11 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝑀‘(𝐸𝐴)) = (𝑀‘∅))
176109ad2antrr 724 . . . . . . . . . . 11 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝑀‘∅) = 0)
177175, 176eqtrd 2854 . . . . . . . . . 10 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝑀‘(𝐸𝐴)) = 0)
17862adantr 483 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → Disj 𝑘 ∈ ℕ 𝐴)
179 disjss1 5028 . . . . . . . . . . 11 ((1...𝑛) ⊆ ℕ → (Disj 𝑘 ∈ ℕ 𝐴Disj 𝑘 ∈ (1...𝑛)𝐴))
180101, 178, 179sylc 65 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → Disj 𝑘 ∈ (1...𝑛)𝐴)
181165, 167, 168, 104, 177, 180esumrnmpt2 31320 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → Σ*𝑦 ∈ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)(𝑀‘(𝐸𝑦)) = Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝐸𝐴)))
182153, 163, 1813eqtrd 2858 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝐸𝐴)))
183150, 151, 1823eqtr3rd 2863 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝐸𝐴)) = ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))))
18417adantr 483 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ (0[,]+∞))
1853, 184sseldi 3963 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
186185xnegcld 12685 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
18715adantr 483 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝑀𝐸) ∈ ℝ*)
188 iunss1 4924 . . . . . . . . . . . 12 ((1...𝑛) ⊆ ℕ → 𝑘 ∈ (1...𝑛)𝐴 𝑘 ∈ ℕ 𝐴)
189100, 188mp1i 13 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝑘 ∈ (1...𝑛)𝐴 𝑘 ∈ ℕ 𝐴)
190189sscond 4116 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐸 𝑘 ∈ ℕ 𝐴) ⊆ (𝐸 𝑘 ∈ (1...𝑛)𝐴))
191743adant1r 1172 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))
192108, 77, 190, 84, 191carsgmon 31565 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)))
193 xleneg 12603 . . . . . . . . . 10 (((𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ* ∧ (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*) → ((𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ↔ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ≤ -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))))
194193biimpa 479 . . . . . . . . 9 ((((𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ* ∧ (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*) ∧ (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))) → -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ≤ -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)))
195185, 86, 192, 194syl21anc 835 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ≤ -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)))
196 xleadd2a 12639 . . . . . . . 8 (((-𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ* ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ* ∧ (𝑀𝐸) ∈ ℝ*) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ≤ -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) → ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))) ≤ ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))))
19789, 186, 187, 195, 196syl31anc 1368 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))) ≤ ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))))
198183, 197eqbrtrd 5079 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝐸𝐴)) ≤ ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))))
19976, 22, 198esumgect 31342 . . . . 5 (𝜑 → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸𝐴)) ≤ ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))))
20012, 27, 20, 71, 199xrletrd 12547 . . . 4 (𝜑 → (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) ≤ ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))))
2012, 200eqbrtrrid 5093 . . 3 (𝜑 → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ≤ ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))))
202 xleadd1a 12638 . . 3 ((((𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ* ∧ ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) ∈ ℝ* ∧ (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*) ∧ (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ≤ ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)))) → ((𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) +𝑒 (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) ≤ (((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) +𝑒 (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))))
20313, 20, 18, 201, 202syl31anc 1368 . 2 (𝜑 → ((𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) +𝑒 (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) ≤ (((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) +𝑒 (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))))
204 xrge0npcan 30674 . . 3 (((𝑀𝐸) ∈ (0[,]+∞) ∧ (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ (0[,]+∞) ∧ (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀𝐸)) → (((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) +𝑒 (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) = (𝑀𝐸))
20514, 17, 75, 204syl3anc 1366 . 2 (𝜑 → (((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) +𝑒 (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) = (𝑀𝐸))
206203, 205breqtrd 5083 1 (𝜑 → ((𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) +𝑒 (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) ≤ (𝑀𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1082   = wceq 1531  wcel 2108  wne 3014  wral 3136  Vcvv 3493  cdif 3931  cin 3933  wss 3934  c0 4289  𝒫 cpw 4537   cuni 4830   ciun 4910  Disj wdisj 5022   class class class wbr 5057  cmpt 5137  ran crn 5549  wf 6344  cfv 6348  (class class class)co 7148  ωcom 7572  cdom 8499  Fincfn 8501  0cc0 10529  1c1 10530  +∞cpnf 10664  -∞cmnf 10665  *cxr 10666  cle 10668  cn 11630  -𝑒cxne 12496   +𝑒 cxad 12497  [,]cicc 12733  ...cfz 12884  Σ*cesum 31279  toCaraSigaccarsg 31552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096  ax-ac2 9877  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607  ax-addf 10608  ax-mulf 10609
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-fal 1544  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-disj 5023  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401  df-om 7573  df-1st 7681  df-2nd 7682  df-supp 7823  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-pm 8401  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-fi 8867  df-sup 8898  df-inf 8899  df-oi 8966  df-dju 9322  df-card 9360  df-acn 9363  df-ac 9534  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-q 12341  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-ioo 12734  df-ioc 12735  df-ico 12736  df-icc 12737  df-fz 12885  df-fzo 13026  df-fl 13154  df-mod 13230  df-seq 13362  df-exp 13422  df-fac 13626  df-bc 13655  df-hash 13683  df-shft 14418  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-limsup 14820  df-clim 14837  df-rlim 14838  df-sum 15035  df-ef 15413  df-sin 15415  df-cos 15416  df-pi 15418  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-starv 16572  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-unif 16580  df-hom 16581  df-cco 16582  df-rest 16688  df-topn 16689  df-0g 16707  df-gsum 16708  df-topgen 16709  df-pt 16710  df-prds 16713  df-ordt 16766  df-xrs 16767  df-qtop 16772  df-imas 16773  df-xps 16775  df-mre 16849  df-mrc 16850  df-acs 16852  df-ps 17802  df-tsr 17803  df-plusf 17843  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948  df-submnd 17949  df-grp 18098  df-minusg 18099  df-sbg 18100  df-mulg 18217  df-subg 18268  df-cntz 18439  df-cmn 18900  df-abl 18901  df-mgp 19232  df-ur 19244  df-ring 19291  df-cring 19292  df-subrg 19525  df-abv 19580  df-lmod 19628  df-scaf 19629  df-sra 19936  df-rgmod 19937  df-psmet 20529  df-xmet 20530  df-met 20531  df-bl 20532  df-mopn 20533  df-fbas 20534  df-fg 20535  df-cnfld 20538  df-top 21494  df-topon 21511  df-topsp 21533  df-bases 21546  df-cld 21619  df-ntr 21620  df-cls 21621  df-nei 21698  df-lp 21736  df-perf 21737  df-cn 21827  df-cnp 21828  df-haus 21915  df-tx 22162  df-hmeo 22355  df-fil 22446  df-fm 22538  df-flim 22539  df-flf 22540  df-tmd 22672  df-tgp 22673  df-tsms 22727  df-trg 22760  df-xms 22922  df-ms 22923  df-tms 22924  df-nm 23184  df-ngp 23185  df-nrg 23187  df-nlm 23188  df-ii 23477  df-cncf 23478  df-limc 24456  df-dv 24457  df-log 25132  df-esum 31280  df-carsg 31553
This theorem is referenced by:  carsgclctunlem3  31571
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