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Theorem carsgclctunlem2 31476
Description: Lemma for carsgclctun 31478. (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 12807 . . . . 5 (0[,]+∞) ⊆ ℝ*
4 carsgval.2 . . . . . 6 (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))
5 nnex 11632 . . . . . . . 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 30213 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝐸𝐴) ∈ 𝒫 𝑂)
106, 9elpwiuncl 30215 . . . . . 6 (𝜑 𝑘 ∈ ℕ (𝐸𝐴) ∈ 𝒫 𝑂)
114, 10ffvelrnd 6844 . . . . 5 (𝜑 → (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) ∈ (0[,]+∞))
123, 11sseldi 3962 . . . 4 (𝜑 → (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) ∈ ℝ*)
132, 12eqeltrrid 2915 . . 3 (𝜑 → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
144, 7ffvelrnd 6844 . . . . 5 (𝜑 → (𝑀𝐸) ∈ (0[,]+∞))
153, 14sseldi 3962 . . . 4 (𝜑 → (𝑀𝐸) ∈ ℝ*)
167elpwdifcl 30214 . . . . . . 7 (𝜑 → (𝐸 𝑘 ∈ ℕ 𝐴) ∈ 𝒫 𝑂)
174, 16ffvelrnd 6844 . . . . . 6 (𝜑 → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ (0[,]+∞))
183, 17sseldi 3962 . . . . 5 (𝜑 → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
1918xnegcld 12681 . . . 4 (𝜑 → -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
2015, 19xaddcld 12682 . . 3 (𝜑 → ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) ∈ ℝ*)
214adantr 481 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
2221, 9ffvelrnd 6844 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑀‘(𝐸𝐴)) ∈ (0[,]+∞))
2322ralrimiva 3179 . . . . . . 7 (𝜑 → ∀𝑘 ∈ ℕ (𝑀‘(𝐸𝐴)) ∈ (0[,]+∞))
24 nfcv 2974 . . . . . . . 8 𝑘
2524esumcl 31188 . . . . . . 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, 26sseldi 3962 . . . . 5 (𝜑 → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸𝐴)) ∈ ℝ*)
289ralrimiva 3179 . . . . . . . . 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 13337 . . . . . . . . . 10 ℕ ≼ ω
33 mptct 9948 . . . . . . . . . 10 (ℕ ≼ ω → (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ≼ ω)
34 rnct 9935 . . . . . . . . . 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 2818 . . . . . . . . . 10 (𝑘 ∈ ℕ ↦ (𝐸𝐴)) = (𝑘 ∈ ℕ ↦ (𝐸𝐴))
3837rnmptss 6878 . . . . . . . . 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 6975 . . . . . . . . . 10 (ℕ ∈ V → (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ∈ V)
41 rnexg 7603 . . . . . . . . . 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 3989 . . . . . . . . . . . 12 (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) → (𝑥 ⊆ 𝒫 𝑂 ↔ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ⊆ 𝒫 𝑂))
4543, 443anbi23d 1430 . . . . . . . . . . 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 31192 . . . . . . . . . . . 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 346 . . . . . . . . . 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 3565 . . . . . . . . 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 1454 . . . . . . 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 485 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ (𝐸𝐴) = ∅) → (𝐸𝐴) = ∅)
5857fveq2d 6667 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ (𝐸𝐴) = ∅) → (𝑀‘(𝐸𝐴)) = (𝑀‘∅))
59 carsgsiga.1 . . . . . . . . 9 (𝜑 → (𝑀‘∅) = 0)
6059ad2antrr 722 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ (𝐸𝐴) = ∅) → (𝑀‘∅) = 0)
6158, 60eqtrd 2853 . . . . . . 7 (((𝜑𝑘 ∈ ℕ) ∧ (𝐸𝐴) = ∅) → (𝑀‘(𝐸𝐴)) = 0)
62 carsgclctunlem2.1 . . . . . . . . 9 (𝜑Disj 𝑘 ∈ ℕ 𝐴)
63 disjin 30264 . . . . . . . . 9 (Disj 𝑘 ∈ ℕ 𝐴Disj 𝑘 ∈ ℕ (𝐴𝐸))
6462, 63syl 17 . . . . . . . 8 (𝜑Disj 𝑘 ∈ ℕ (𝐴𝐸))
65 incom 4175 . . . . . . . . . 10 (𝐴𝐸) = (𝐸𝐴)
6665rgenw 3147 . . . . . . . . 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 219 . . . . . . 7 (𝜑Disj 𝑘 ∈ ℕ (𝐸𝐴))
7056, 6, 22, 9, 61, 69esumrnmpt2 31226 . . . . . 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 4106 . . . . . . . 8 (𝜑 → (𝐸 𝑘 ∈ ℕ 𝐴) ⊆ 𝐸)
74 carsgsiga.3 . . . . . . . 8 ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))
7572, 4, 73, 7, 74carsgmon 31471 . . . . . . 7 (𝜑 → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀𝐸))
7614, 17, 75xrge0subcld 30413 . . . . . 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 30213 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐸 𝑘 ∈ (1...𝑛)𝐴) ∈ 𝒫 𝑂)
8077, 79ffvelrnd 6844 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ (0[,]+∞))
813, 80sseldi 3962 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
82 xrge0neqmnf 12828 . . . . . . . . . . 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 30214 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐸 𝑘 ∈ (1...𝑛)𝐴) ∈ 𝒫 𝑂)
8577, 84ffvelrnd 6844 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ (0[,]+∞))
863, 85sseldi 3962 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
87 xrge0neqmnf 12828 . . . . . . . . . . 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 12681 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
90 xnegneg 12595 . . . . . . . . . . . . . . . . 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 12588 . . . . . . . . . . . . . . . . 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 12591 . . . . . . . . . . . . . . . 16 -𝑒-∞ = +∞
9694, 95syl6eq 2869 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → -𝑒-𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = +∞)
9792, 96eqtr3d 2855 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = +∞)
9897oveq2d 7161 . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
100 fz1ssnn 12926 . . . . . . . . . . . . . . . . . . . . . . 23 (1...𝑛) ⊆ ℕ
101100a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
102101sselda 3964 . . . . . . . . . . . . . . . . . . . . 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 3179 . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → 𝑂𝑉)
10959adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → (𝑀‘∅) = 0)
110513adant1r 1169 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))
111 fzfi 13328 . . . . . . . . . . . . . . . . . . . . 21 (1...𝑛) ∈ Fin
112 mptfi 8811 . . . . . . . . . . . . . . . . . . . . 21 ((1...𝑛) ∈ Fin → (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ Fin)
113 rnfi 8795 . . . . . . . . . . . . . . . . . . . . 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 2818 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝑛) ↦ 𝐴) = (𝑘 ∈ (1...𝑛) ↦ 𝐴)
117116rnmptss 6878 . . . . . . . . . . . . . . . . . . . 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 31473 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ (toCaraSiga‘𝑀))
120107, 119eqeltrd 2910 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → 𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀))
121108, 77elcarsg 31462 . . . . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . . . . 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 4178 . . . . . . . . . . . . . . . . . . 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 4089 . . . . . . . . . . . . . . . . . . 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 7163 . . . . . . . . . . . . . . . . 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 2834 . . . . . . . . . . . . . . . 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 3615 . . . . . . . . . . . . . . 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 12607 . . . . . . . . . . . . . . 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 2861 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → (𝑀𝐸) = +∞)
138 carsgclctunlem2.4 . . . . . . . . . . . . . 14 (𝜑 → (𝑀𝐸) ≠ +∞)
139138ad2antrr 722 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → (𝑀𝐸) ≠ +∞)
140139neneqd 3018 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → ¬ (𝑀𝐸) = +∞)
141137, 140pm2.65da 813 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ¬ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞)
142141neqned 3020 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞)
143 xaddass 12630 . . . . . . . . . 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 1378 . . . . . . . . 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 12619 . . . . . . . . . . 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 7161 . . . . . . . . 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 12622 . . . . . . . . . 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 2857 . . . . . . . 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 7160 . . . . . . . 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 4186 . . . . . . . . . 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 30263 . . . . . . . . . . . . 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 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 31474 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 ran (𝑘 ∈ (1...𝑛) ↦ 𝐴))) = Σ*𝑦 ∈ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)(𝑀‘(𝐸𝑦)))
164 ineq2 4180 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝐸𝑦) = (𝐸𝐴))
165164fveq2d 6667 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑀‘(𝐸𝑦)) = (𝑀‘(𝐸𝐴)))
166111elexi 3511 . . . . . . . . . . 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 4203 . . . . . . . . . . . . . . 15 (𝐸𝐴) ⊆ 𝐴
170 sseq2 3990 . . . . . . . . . . . . . . 15 (𝐴 = ∅ → ((𝐸𝐴) ⊆ 𝐴 ↔ (𝐸𝐴) ⊆ ∅))
171169, 170mpbii 234 . . . . . . . . . . . . . 14 (𝐴 = ∅ → (𝐸𝐴) ⊆ ∅)
172 ss0 4349 . . . . . . . . . . . . . 14 ((𝐸𝐴) ⊆ ∅ → (𝐸𝐴) = ∅)
173171, 172syl 17 . . . . . . . . . . . . 13 (𝐴 = ∅ → (𝐸𝐴) = ∅)
174173adantl 482 . . . . . . . . . . . 12 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝐸𝐴) = ∅)
175174fveq2d 6667 . . . . . . . . . . 11 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝑀‘(𝐸𝐴)) = (𝑀‘∅))
176109ad2antrr 722 . . . . . . . . . . 11 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝑀‘∅) = 0)
177175, 176eqtrd 2853 . . . . . . . . . 10 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝑀‘(𝐸𝐴)) = 0)
17862adantr 481 . . . . . . . . . . 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 31226 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → Σ*𝑦 ∈ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)(𝑀‘(𝐸𝑦)) = Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝐸𝐴)))
182153, 163, 1813eqtrd 2857 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝐸𝐴)))
183150, 151, 1823eqtr3rd 2862 . . . . . . 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, 184sseldi 3962 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
186185xnegcld 12681 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
18715adantr 481 . . . . . . . 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 4115 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐸 𝑘 ∈ ℕ 𝐴) ⊆ (𝐸 𝑘 ∈ (1...𝑛)𝐴))
191743adant1r 1169 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))
192108, 77, 190, 84, 191carsgmon 31471 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)))
193 xleneg 12599 . . . . . . . . . 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 833 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ≤ -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)))
196 xleadd2a 12635 . . . . . . . 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 1365 . . . . . . 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 31248 . . . . 5 (𝜑 → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸𝐴)) ≤ ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))))
20012, 27, 20, 71, 199xrletrd 12543 . . . 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 12634 . . 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 1365 . 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 30608 . . 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 1363 . 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135  Vcvv 3492  cdif 3930  cin 3932  wss 3933  c0 4288  𝒫 cpw 4535   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 7145  ωcom 7569  cdom 8495  Fincfn 8497  0cc0 10525  1c1 10526  +∞cpnf 10660  -∞cmnf 10661  *cxr 10662  cle 10664  cn 11626  -𝑒cxne 12492   +𝑒 cxad 12493  [,]cicc 12729  ...cfz 12880  Σ*cesum 31185  toCaraSigaccarsg 31458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-ac2 9873  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-addf 10604  ax-mulf 10605
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  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 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-acn 9359  df-ac 9530  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ioc 12731  df-ico 12732  df-icc 12733  df-fz 12881  df-fzo 13022  df-fl 13150  df-mod 13226  df-seq 13358  df-exp 13418  df-fac 13622  df-bc 13651  df-hash 13679  df-shft 14414  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-limsup 14816  df-clim 14833  df-rlim 14834  df-sum 15031  df-ef 15409  df-sin 15411  df-cos 15412  df-pi 15414  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-mulr 16567  df-starv 16568  df-sca 16569  df-vsca 16570  df-ip 16571  df-tset 16572  df-ple 16573  df-ds 16575  df-unif 16576  df-hom 16577  df-cco 16578  df-rest 16684  df-topn 16685  df-0g 16703  df-gsum 16704  df-topgen 16705  df-pt 16706  df-prds 16709  df-ordt 16762  df-xrs 16763  df-qtop 16768  df-imas 16769  df-xps 16771  df-mre 16845  df-mrc 16846  df-acs 16848  df-ps 17798  df-tsr 17799  df-plusf 17839  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-mhm 17944  df-submnd 17945  df-grp 18044  df-minusg 18045  df-sbg 18046  df-mulg 18163  df-subg 18214  df-cntz 18385  df-cmn 18837  df-abl 18838  df-mgp 19169  df-ur 19181  df-ring 19228  df-cring 19229  df-subrg 19462  df-abv 19517  df-lmod 19565  df-scaf 19566  df-sra 19873  df-rgmod 19874  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-fbas 20470  df-fg 20471  df-cnfld 20474  df-top 21430  df-topon 21447  df-topsp 21469  df-bases 21482  df-cld 21555  df-ntr 21556  df-cls 21557  df-nei 21634  df-lp 21672  df-perf 21673  df-cn 21763  df-cnp 21764  df-haus 21851  df-tx 22098  df-hmeo 22291  df-fil 22382  df-fm 22474  df-flim 22475  df-flf 22476  df-tmd 22608  df-tgp 22609  df-tsms 22662  df-trg 22695  df-xms 22857  df-ms 22858  df-tms 22859  df-nm 23119  df-ngp 23120  df-nrg 23122  df-nlm 23123  df-ii 23412  df-cncf 23413  df-limc 24391  df-dv 24392  df-log 25067  df-esum 31186  df-carsg 31459
This theorem is referenced by:  carsgclctunlem3  31477
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