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Theorem carsgclctunlem2 32919
Description: Lemma for carsgclctun 32921. (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 5031 . . . . 5 𝑘 ∈ ℕ (𝐸𝐴) = (𝐸 𝑘 ∈ ℕ 𝐴)
21fveq2i 6845 . . . 4 (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) = (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))
3 iccssxr 13347 . . . . 5 (0[,]+∞) ⊆ ℝ*
4 carsgval.2 . . . . . 6 (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))
5 nnex 12159 . . . . . . . 8 ℕ ∈ V
65a1i 11 . . . . . . 7 (𝜑 → ℕ ∈ V)
7 carsgclctunlem2.3 . . . . . . . . 9 (𝜑𝐸 ∈ 𝒫 𝑂)
87adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐸 ∈ 𝒫 𝑂)
98elpwincl1 31453 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝐸𝐴) ∈ 𝒫 𝑂)
106, 9elpwiuncl 31455 . . . . . 6 (𝜑 𝑘 ∈ ℕ (𝐸𝐴) ∈ 𝒫 𝑂)
114, 10ffvelcdmd 7036 . . . . 5 (𝜑 → (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) ∈ (0[,]+∞))
123, 11sselid 3942 . . . 4 (𝜑 → (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) ∈ ℝ*)
132, 12eqeltrrid 2843 . . 3 (𝜑 → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
144, 7ffvelcdmd 7036 . . . . 5 (𝜑 → (𝑀𝐸) ∈ (0[,]+∞))
153, 14sselid 3942 . . . 4 (𝜑 → (𝑀𝐸) ∈ ℝ*)
167elpwdifcl 31454 . . . . . . 7 (𝜑 → (𝐸 𝑘 ∈ ℕ 𝐴) ∈ 𝒫 𝑂)
174, 16ffvelcdmd 7036 . . . . . 6 (𝜑 → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ (0[,]+∞))
183, 17sselid 3942 . . . . 5 (𝜑 → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
1918xnegcld 13219 . . . 4 (𝜑 → -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
2015, 19xaddcld 13220 . . 3 (𝜑 → ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) ∈ ℝ*)
214adantr 481 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
2221, 9ffvelcdmd 7036 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑀‘(𝐸𝐴)) ∈ (0[,]+∞))
2322ralrimiva 3143 . . . . . . 7 (𝜑 → ∀𝑘 ∈ ℕ (𝑀‘(𝐸𝐴)) ∈ (0[,]+∞))
24 nfcv 2907 . . . . . . . 8 𝑘
2524esumcl 32629 . . . . . . 7 ((ℕ ∈ V ∧ ∀𝑘 ∈ ℕ (𝑀‘(𝐸𝐴)) ∈ (0[,]+∞)) → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸𝐴)) ∈ (0[,]+∞))
266, 23, 25syl2anc 584 . . . . . 6 (𝜑 → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸𝐴)) ∈ (0[,]+∞))
273, 26sselid 3942 . . . . 5 (𝜑 → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸𝐴)) ∈ ℝ*)
289ralrimiva 3143 . . . . . . . . 9 (𝜑 → ∀𝑘 ∈ ℕ (𝐸𝐴) ∈ 𝒫 𝑂)
29 dfiun3g 5919 . . . . . . . . 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 6846 . . . . . . 7 (𝜑 → (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) = (𝑀 ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))))
32 nnct 13886 . . . . . . . . . 10 ℕ ≼ ω
33 mptct 10474 . . . . . . . . . 10 (ℕ ≼ ω → (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ≼ ω)
34 rnct 10461 . . . . . . . . . 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 2736 . . . . . . . . . 10 (𝑘 ∈ ℕ ↦ (𝐸𝐴)) = (𝑘 ∈ ℕ ↦ (𝐸𝐴))
3837rnmptss 7070 . . . . . . . . 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 7171 . . . . . . . . . 10 (ℕ ∈ V → (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ∈ V)
41 rnexg 7841 . . . . . . . . . 10 ((𝑘 ∈ ℕ ↦ (𝐸𝐴)) ∈ V → ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ∈ V)
425, 40, 41mp2b 10 . . . . . . . . 9 ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ∈ V
43 breq1 5108 . . . . . . . . . . . 12 (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) → (𝑥 ≼ ω ↔ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ≼ ω))
44 sseq1 3969 . . . . . . . . . . . 12 (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) → (𝑥 ⊆ 𝒫 𝑂 ↔ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ⊆ 𝒫 𝑂))
4543, 443anbi23d 1439 . . . . . . . . . . 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 4876 . . . . . . . . . . . . 13 (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) → 𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)))
4746fveq2d 6846 . . . . . . . . . . . 12 (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) → (𝑀 𝑥) = (𝑀 ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))))
48 esumeq1 32633 . . . . . . . . . . . 12 (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) → Σ*𝑦𝑥(𝑀𝑦) = Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))(𝑀𝑦))
4947, 48breq12d 5118 . . . . . . . . . . 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 344 . . . . . . . . . 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 3525 . . . . . . . . 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 1463 . . . . . . 7 (𝜑 → (𝑀 ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))) ≤ Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))(𝑀𝑦))
5531, 54eqbrtrd 5127 . . . . . 6 (𝜑 → (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) ≤ Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))(𝑀𝑦))
56 fveq2 6842 . . . . . . 7 (𝑦 = (𝐸𝐴) → (𝑀𝑦) = (𝑀‘(𝐸𝐴)))
57 simpr 485 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ (𝐸𝐴) = ∅) → (𝐸𝐴) = ∅)
5857fveq2d 6846 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ (𝐸𝐴) = ∅) → (𝑀‘(𝐸𝐴)) = (𝑀‘∅))
59 carsgsiga.1 . . . . . . . . 9 (𝜑 → (𝑀‘∅) = 0)
6059ad2antrr 724 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ (𝐸𝐴) = ∅) → (𝑀‘∅) = 0)
6158, 60eqtrd 2776 . . . . . . 7 (((𝜑𝑘 ∈ ℕ) ∧ (𝐸𝐴) = ∅) → (𝑀‘(𝐸𝐴)) = 0)
62 carsgclctunlem2.1 . . . . . . . . 9 (𝜑Disj 𝑘 ∈ ℕ 𝐴)
63 disjin 31504 . . . . . . . . 9 (Disj 𝑘 ∈ ℕ 𝐴Disj 𝑘 ∈ ℕ (𝐴𝐸))
6462, 63syl 17 . . . . . . . 8 (𝜑Disj 𝑘 ∈ ℕ (𝐴𝐸))
65 incom 4161 . . . . . . . . . 10 (𝐴𝐸) = (𝐸𝐴)
6665rgenw 3068 . . . . . . . . 9 𝑘 ∈ ℕ (𝐴𝐸) = (𝐸𝐴)
67 disjeq2 5074 . . . . . . . . 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 217 . . . . . . 7 (𝜑Disj 𝑘 ∈ ℕ (𝐸𝐴))
7056, 6, 22, 9, 61, 69esumrnmpt2 32667 . . . . . 6 (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))(𝑀𝑦) = Σ*𝑘 ∈ ℕ(𝑀‘(𝐸𝐴)))
7155, 70breqtrd 5131 . . . . 5 (𝜑 → (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) ≤ Σ*𝑘 ∈ ℕ(𝑀‘(𝐸𝐴)))
72 carsgval.1 . . . . . . . 8 (𝜑𝑂𝑉)
73 difssd 4092 . . . . . . . 8 (𝜑 → (𝐸 𝑘 ∈ ℕ 𝐴) ⊆ 𝐸)
74 carsgsiga.3 . . . . . . . 8 ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))
7572, 4, 73, 7, 74carsgmon 32914 . . . . . . 7 (𝜑 → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀𝐸))
7614, 17, 75xrge0subcld 31668 . . . . . 6 (𝜑 → ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) ∈ (0[,]+∞))
774adantr 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
787adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → 𝐸 ∈ 𝒫 𝑂)
7978elpwincl1 31453 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐸 𝑘 ∈ (1...𝑛)𝐴) ∈ 𝒫 𝑂)
8077, 79ffvelcdmd 7036 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ (0[,]+∞))
813, 80sselid 3942 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
82 xrge0neqmnf 13369 . . . . . . . . . . 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 31454 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐸 𝑘 ∈ (1...𝑛)𝐴) ∈ 𝒫 𝑂)
8577, 84ffvelcdmd 7036 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ (0[,]+∞))
863, 85sselid 3942 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
87 xrge0neqmnf 13369 . . . . . . . . . . 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 13219 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
90 xnegneg 13133 . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 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 13126 . . . . . . . . . . . . . . . . 17 (-𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞ → -𝑒-𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -𝑒-∞)
9493adantl 482 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → -𝑒-𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -𝑒-∞)
95 xnegmnf 13129 . . . . . . . . . . . . . . . 16 -𝑒-∞ = +∞
9694, 95eqtrdi 2792 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → -𝑒-𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = +∞)
9792, 96eqtr3d 2778 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = +∞)
9897oveq2d 7373 . . . . . . . . . . . . 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 13472 . . . . . . . . . . . . . . . . . . . . . . 23 (1...𝑛) ⊆ ℕ
101100a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
102101sselda 3944 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
103 carsgclctunlem2.2 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ (toCaraSiga‘𝑀))
10499, 102, 103syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (toCaraSiga‘𝑀))
105104ralrimiva 3143 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀))
106 dfiun3g 5919 . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → 𝑂𝑉)
10959adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → (𝑀‘∅) = 0)
110513adant1r 1177 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))
111 fzfi 13877 . . . . . . . . . . . . . . . . . . . . 21 (1...𝑛) ∈ Fin
112 mptfi 9295 . . . . . . . . . . . . . . . . . . . . 21 ((1...𝑛) ∈ Fin → (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ Fin)
113 rnfi 9279 . . . . . . . . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝑛) ↦ 𝐴) = (𝑘 ∈ (1...𝑛) ↦ 𝐴)
117116rnmptss 7070 . . . . . . . . . . . . . . . . . . . 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 32916 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ (toCaraSiga‘𝑀))
120107, 119eqeltrd 2838 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → 𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀))
121108, 77elcarsg 32905 . . . . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝑒 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀𝑒))
124 ineq1 4165 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 𝐸 → (𝑒 𝑘 ∈ (1...𝑛)𝐴) = (𝐸 𝑘 ∈ (1...𝑛)𝐴))
125124fveq2d 6846 . . . . . . . . . . . . . . . . . 18 (𝑒 = 𝐸 → (𝑀‘(𝑒 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)))
126 difeq1 4075 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 𝐸 → (𝑒 𝑘 ∈ (1...𝑛)𝐴) = (𝐸 𝑘 ∈ (1...𝑛)𝐴))
127126fveq2d 6846 . . . . . . . . . . . . . . . . . 18 (𝑒 = 𝐸 → (𝑀‘(𝑒 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)))
128125, 127oveq12d 7375 . . . . . . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . . . 17 (𝑒 = 𝐸 → (𝑀𝑒) = (𝑀𝐸))
130128, 129eqeq12d 2752 . . . . . . . . . . . . . . . 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 3577 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 13145 . . . . . . . . . . . . . . 15 (((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ* ∧ (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞) → ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 +∞) = +∞)
13581, 83, 134syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 +∞) = +∞)
136135adantr 481 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 +∞) = +∞)
13798, 133, 1363eqtr3d 2784 . . . . . . . . . . . 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 2948 . . . . . . . . . . . 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 2950 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞)
143 xaddass 13168 . . . . . . . . . 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 1386 . . . . . . . . 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 13157 . . . . . . . . . . 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 7373 . . . . . . . . 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 13160 . . . . . . . . . 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 2780 . . . . . . . 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 7372 . . . . . . . 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 4172 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐸 𝑘 ∈ (1...𝑛)𝐴) = (𝐸 ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)))
153152fveq2d 6846 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 ran (𝑘 ∈ (1...𝑛) ↦ 𝐴))))
154 mptss 5996 . . . . . . . . . . . . 13 ((1...𝑛) ⊆ ℕ → (𝑘 ∈ (1...𝑛) ↦ 𝐴) ⊆ (𝑘 ∈ ℕ ↦ 𝐴))
155 rnss 5894 . . . . . . . . . . . . 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 31503 . . . . . . . . . . . . 13 (Disj 𝑘 ∈ ℕ 𝐴Disj 𝑦 ∈ ran (𝑘 ∈ ℕ ↦ 𝐴)𝑦)
15962, 158syl 17 . . . . . . . . . . . 12 (𝜑Disj 𝑦 ∈ ran (𝑘 ∈ ℕ ↦ 𝐴)𝑦)
160159adantr 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → Disj 𝑦 ∈ ran (𝑘 ∈ ℕ ↦ 𝐴)𝑦)
161 disjss1 5076 . . . . . . . . . . 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 32917 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 ran (𝑘 ∈ (1...𝑛) ↦ 𝐴))) = Σ*𝑦 ∈ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)(𝑀‘(𝐸𝑦)))
164 ineq2 4166 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝐸𝑦) = (𝐸𝐴))
165164fveq2d 6846 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑀‘(𝐸𝑦)) = (𝑀‘(𝐸𝐴)))
166111elexi 3464 . . . . . . . . . . 11 (1...𝑛) ∈ V
167166a1i 11 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1...𝑛) ∈ V)
16899, 102, 22syl2anc 584 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑀‘(𝐸𝐴)) ∈ (0[,]+∞))
169 inss2 4189 . . . . . . . . . . . . . . 15 (𝐸𝐴) ⊆ 𝐴
170 sseq2 3970 . . . . . . . . . . . . . . 15 (𝐴 = ∅ → ((𝐸𝐴) ⊆ 𝐴 ↔ (𝐸𝐴) ⊆ ∅))
171169, 170mpbii 232 . . . . . . . . . . . . . 14 (𝐴 = ∅ → (𝐸𝐴) ⊆ ∅)
172 ss0 4358 . . . . . . . . . . . . . 14 ((𝐸𝐴) ⊆ ∅ → (𝐸𝐴) = ∅)
173171, 172syl 17 . . . . . . . . . . . . 13 (𝐴 = ∅ → (𝐸𝐴) = ∅)
174173adantl 482 . . . . . . . . . . . 12 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝐸𝐴) = ∅)
175174fveq2d 6846 . . . . . . . . . . 11 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝑀‘(𝐸𝐴)) = (𝑀‘∅))
176109ad2antrr 724 . . . . . . . . . . 11 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝑀‘∅) = 0)
177175, 176eqtrd 2776 . . . . . . . . . 10 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝑀‘(𝐸𝐴)) = 0)
17862adantr 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → Disj 𝑘 ∈ ℕ 𝐴)
179 disjss1 5076 . . . . . . . . . . 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 32667 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → Σ*𝑦 ∈ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)(𝑀‘(𝐸𝑦)) = Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝐸𝐴)))
182153, 163, 1813eqtrd 2780 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝐸𝐴)))
183150, 151, 1823eqtr3rd 2785 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝐸𝐴)) = ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))))
18417adantr 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ (0[,]+∞))
1853, 184sselid 3942 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
186185xnegcld 13219 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
18715adantr 481 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝑀𝐸) ∈ ℝ*)
188 iunss1 4968 . . . . . . . . . . . 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 4101 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐸 𝑘 ∈ ℕ 𝐴) ⊆ (𝐸 𝑘 ∈ (1...𝑛)𝐴))
191743adant1r 1177 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))
192108, 77, 190, 84, 191carsgmon 32914 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)))
193 xleneg 13137 . . . . . . . . . 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 477 . . . . . . . . 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 836 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ≤ -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)))
196 xleadd2a 13173 . . . . . . . 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 1373 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))) ≤ ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))))
198183, 197eqbrtrd 5127 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝐸𝐴)) ≤ ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))))
19976, 22, 198esumgect 32689 . . . . 5 (𝜑 → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸𝐴)) ≤ ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))))
20012, 27, 20, 71, 199xrletrd 13081 . . . 4 (𝜑 → (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) ≤ ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))))
2012, 200eqbrtrrid 5141 . . 3 (𝜑 → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ≤ ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))))
202 xleadd1a 13172 . . 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 1373 . 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 31885 . . 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 1371 . 2 (𝜑 → (((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) +𝑒 (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) = (𝑀𝐸))
206203, 205breqtrd 5131 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  Vcvv 3445  cdif 3907  cin 3909  wss 3910  c0 4282  𝒫 cpw 4560   cuni 4865   ciun 4954  Disj wdisj 5070   class class class wbr 5105  cmpt 5188  ran crn 5634  wf 6492  cfv 6496  (class class class)co 7357  ωcom 7802  cdom 8881  Fincfn 8883  0cc0 11051  1c1 11052  +∞cpnf 11186  -∞cmnf 11187  *cxr 11188  cle 11190  cn 12153  -𝑒cxne 13030   +𝑒 cxad 13031  [,]cicc 13267  ...cfz 13424  Σ*cesum 32626  toCaraSigaccarsg 32901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-ac2 10399  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-acn 9878  df-ac 10052  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-shft 14952  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-sum 15571  df-ef 15950  df-sin 15952  df-cos 15953  df-pi 15955  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-ordt 17383  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-ps 18455  df-tsr 18456  df-plusf 18496  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-subrg 20220  df-abv 20276  df-lmod 20324  df-scaf 20325  df-sra 20633  df-rgmod 20634  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lp 22487  df-perf 22488  df-cn 22578  df-cnp 22579  df-haus 22666  df-tx 22913  df-hmeo 23106  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-tmd 23423  df-tgp 23424  df-tsms 23478  df-trg 23511  df-xms 23673  df-ms 23674  df-tms 23675  df-nm 23938  df-ngp 23939  df-nrg 23941  df-nlm 23942  df-ii 24240  df-cncf 24241  df-limc 25230  df-dv 25231  df-log 25912  df-esum 32627  df-carsg 32902
This theorem is referenced by:  carsgclctunlem3  32920
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