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Theorem carsgclctunlem2 30686
Description: Lemma for carsgclctun 30688. (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 4732 . . . . 5 𝑘 ∈ ℕ (𝐸𝐴) = (𝐸 𝑘 ∈ ℕ 𝐴)
21fveq2i 6351 . . . 4 (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) = (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))
3 iccssxr 12445 . . . . 5 (0[,]+∞) ⊆ ℝ*
4 carsgval.2 . . . . . 6 (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))
5 nnex 11214 . . . . . . . 8 ℕ ∈ V
65a1i 11 . . . . . . 7 (𝜑 → ℕ ∈ V)
7 carsgclctunlem2.3 . . . . . . . . 9 (𝜑𝐸 ∈ 𝒫 𝑂)
87adantr 472 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝐸 ∈ 𝒫 𝑂)
98elpwincl1 29660 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → (𝐸𝐴) ∈ 𝒫 𝑂)
106, 9elpwiuncl 29662 . . . . . 6 (𝜑 𝑘 ∈ ℕ (𝐸𝐴) ∈ 𝒫 𝑂)
114, 10ffvelrnd 6519 . . . . 5 (𝜑 → (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) ∈ (0[,]+∞))
123, 11sseldi 3738 . . . 4 (𝜑 → (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) ∈ ℝ*)
132, 12syl5eqelr 2840 . . 3 (𝜑 → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
144, 7ffvelrnd 6519 . . . . 5 (𝜑 → (𝑀𝐸) ∈ (0[,]+∞))
153, 14sseldi 3738 . . . 4 (𝜑 → (𝑀𝐸) ∈ ℝ*)
167elpwdifcl 29661 . . . . . . 7 (𝜑 → (𝐸 𝑘 ∈ ℕ 𝐴) ∈ 𝒫 𝑂)
174, 16ffvelrnd 6519 . . . . . 6 (𝜑 → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ (0[,]+∞))
183, 17sseldi 3738 . . . . 5 (𝜑 → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
1918xnegcld 12319 . . . 4 (𝜑 → -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
2015, 19xaddcld 12320 . . 3 (𝜑 → ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) ∈ ℝ*)
214adantr 472 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
2221, 9ffvelrnd 6519 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑀‘(𝐸𝐴)) ∈ (0[,]+∞))
2322ralrimiva 3100 . . . . . . 7 (𝜑 → ∀𝑘 ∈ ℕ (𝑀‘(𝐸𝐴)) ∈ (0[,]+∞))
24 nfcv 2898 . . . . . . . 8 𝑘
2524esumcl 30397 . . . . . . 7 ((ℕ ∈ V ∧ ∀𝑘 ∈ ℕ (𝑀‘(𝐸𝐴)) ∈ (0[,]+∞)) → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸𝐴)) ∈ (0[,]+∞))
266, 23, 25syl2anc 696 . . . . . 6 (𝜑 → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸𝐴)) ∈ (0[,]+∞))
273, 26sseldi 3738 . . . . 5 (𝜑 → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸𝐴)) ∈ ℝ*)
289ralrimiva 3100 . . . . . . . . 9 (𝜑 → ∀𝑘 ∈ ℕ (𝐸𝐴) ∈ 𝒫 𝑂)
29 dfiun3g 5529 . . . . . . . . 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 6352 . . . . . . 7 (𝜑 → (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) = (𝑀 ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))))
32 nnct 12970 . . . . . . . . . 10 ℕ ≼ ω
33 mptct 9548 . . . . . . . . . 10 (ℕ ≼ ω → (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ≼ ω)
34 rnct 9535 . . . . . . . . . 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 2756 . . . . . . . . . 10 (𝑘 ∈ ℕ ↦ (𝐸𝐴)) = (𝑘 ∈ ℕ ↦ (𝐸𝐴))
3837rnmptss 6551 . . . . . . . . 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 6644 . . . . . . . . . 10 (ℕ ∈ V → (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ∈ V)
41 rnexg 7259 . . . . . . . . . 10 ((𝑘 ∈ ℕ ↦ (𝐸𝐴)) ∈ V → ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ∈ V)
425, 40, 41mp2b 10 . . . . . . . . 9 ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ∈ V
43 breq1 4803 . . . . . . . . . . . 12 (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) → (𝑥 ≼ ω ↔ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ≼ ω))
44 sseq1 3763 . . . . . . . . . . . 12 (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) → (𝑥 ⊆ 𝒫 𝑂 ↔ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) ⊆ 𝒫 𝑂))
4543, 443anbi23d 1547 . . . . . . . . . . 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 4592 . . . . . . . . . . . . 13 (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) → 𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)))
4746fveq2d 6352 . . . . . . . . . . . 12 (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) → (𝑀 𝑥) = (𝑀 ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))))
48 esumeq1 30401 . . . . . . . . . . . 12 (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸𝐴)) → Σ*𝑦𝑥(𝑀𝑦) = Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))(𝑀𝑦))
4947, 48breq12d 4813 . . . . . . . . . . 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 333 . . . . . . . . . 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 3402 . . . . . . . . 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 1571 . . . . . . 7 (𝜑 → (𝑀 ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))) ≤ Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))(𝑀𝑦))
5531, 54eqbrtrd 4822 . . . . . 6 (𝜑 → (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) ≤ Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))(𝑀𝑦))
56 fveq2 6348 . . . . . . 7 (𝑦 = (𝐸𝐴) → (𝑀𝑦) = (𝑀‘(𝐸𝐴)))
57 simpr 479 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ (𝐸𝐴) = ∅) → (𝐸𝐴) = ∅)
5857fveq2d 6352 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ (𝐸𝐴) = ∅) → (𝑀‘(𝐸𝐴)) = (𝑀‘∅))
59 carsgsiga.1 . . . . . . . . 9 (𝜑 → (𝑀‘∅) = 0)
6059ad2antrr 764 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ (𝐸𝐴) = ∅) → (𝑀‘∅) = 0)
6158, 60eqtrd 2790 . . . . . . 7 (((𝜑𝑘 ∈ ℕ) ∧ (𝐸𝐴) = ∅) → (𝑀‘(𝐸𝐴)) = 0)
62 carsgclctunlem2.1 . . . . . . . . 9 (𝜑Disj 𝑘 ∈ ℕ 𝐴)
63 disjin 29702 . . . . . . . . 9 (Disj 𝑘 ∈ ℕ 𝐴Disj 𝑘 ∈ ℕ (𝐴𝐸))
6462, 63syl 17 . . . . . . . 8 (𝜑Disj 𝑘 ∈ ℕ (𝐴𝐸))
65 incom 3944 . . . . . . . . . 10 (𝐴𝐸) = (𝐸𝐴)
6665rgenw 3058 . . . . . . . . 9 𝑘 ∈ ℕ (𝐴𝐸) = (𝐸𝐴)
67 disjeq2 4772 . . . . . . . . 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 208 . . . . . . 7 (𝜑Disj 𝑘 ∈ ℕ (𝐸𝐴))
7056, 6, 22, 9, 61, 69esumrnmpt2 30435 . . . . . 6 (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸𝐴))(𝑀𝑦) = Σ*𝑘 ∈ ℕ(𝑀‘(𝐸𝐴)))
7155, 70breqtrd 4826 . . . . 5 (𝜑 → (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) ≤ Σ*𝑘 ∈ ℕ(𝑀‘(𝐸𝐴)))
72 carsgval.1 . . . . . . . 8 (𝜑𝑂𝑉)
73 difssd 3877 . . . . . . . 8 (𝜑 → (𝐸 𝑘 ∈ ℕ 𝐴) ⊆ 𝐸)
74 carsgsiga.3 . . . . . . . 8 ((𝜑𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))
7572, 4, 73, 7, 74carsgmon 30681 . . . . . . 7 (𝜑 → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀𝐸))
7614, 17, 75xrge0subcld 29833 . . . . . 6 (𝜑 → ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) ∈ (0[,]+∞))
774adantr 472 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
787adantr 472 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → 𝐸 ∈ 𝒫 𝑂)
7978elpwincl1 29660 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐸 𝑘 ∈ (1...𝑛)𝐴) ∈ 𝒫 𝑂)
8077, 79ffvelrnd 6519 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ (0[,]+∞))
813, 80sseldi 3738 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
82 xrge0neqmnf 12465 . . . . . . . . . . 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 29661 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐸 𝑘 ∈ (1...𝑛)𝐴) ∈ 𝒫 𝑂)
8577, 84ffvelrnd 6519 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ (0[,]+∞))
863, 85sseldi 3738 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
87 xrge0neqmnf 12465 . . . . . . . . . . 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 12319 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
90 xnegneg 12234 . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . 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 12227 . . . . . . . . . . . . . . . . 17 (-𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞ → -𝑒-𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -𝑒-∞)
9493adantl 473 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → -𝑒-𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -𝑒-∞)
95 xnegmnf 12230 . . . . . . . . . . . . . . . 16 -𝑒-∞ = +∞
9694, 95syl6eq 2806 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → -𝑒-𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = +∞)
9792, 96eqtr3d 2792 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = +∞)
9897oveq2d 6825 . . . . . . . . . . . . 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 807 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
100 fz1ssnn 12561 . . . . . . . . . . . . . . . . . . . . . . 23 (1...𝑛) ⊆ ℕ
101100a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
102101sselda 3740 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
103 carsgclctunlem2.2 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ (toCaraSiga‘𝑀))
10499, 102, 103syl2anc 696 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (toCaraSiga‘𝑀))
105104ralrimiva 3100 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀))
106 dfiun3g 5529 . . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → 𝑂𝑉)
10959adantr 472 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → (𝑀‘∅) = 0)
110513adant1r 1188 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀 𝑥) ≤ Σ*𝑦𝑥(𝑀𝑦))
111 fzfi 12961 . . . . . . . . . . . . . . . . . . . . 21 (1...𝑛) ∈ Fin
112 mptfi 8426 . . . . . . . . . . . . . . . . . . . . 21 ((1...𝑛) ∈ Fin → (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ Fin)
113 rnfi 8410 . . . . . . . . . . . . . . . . . . . . 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 2756 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝑛) ↦ 𝐴) = (𝑘 ∈ (1...𝑛) ↦ 𝐴)
117116rnmptss 6551 . . . . . . . . . . . . . . . . . . . 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 30683 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 ∈ ℕ) → ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ (toCaraSiga‘𝑀))
120107, 119eqeltrd 2835 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ ℕ) → 𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀))
121108, 77elcarsg 30672 . . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ) → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝑒 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀𝑒))
124 ineq1 3946 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 𝐸 → (𝑒 𝑘 ∈ (1...𝑛)𝐴) = (𝐸 𝑘 ∈ (1...𝑛)𝐴))
125124fveq2d 6352 . . . . . . . . . . . . . . . . . 18 (𝑒 = 𝐸 → (𝑀‘(𝑒 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)))
126 difeq1 3860 . . . . . . . . . . . . . . . . . . 19 (𝑒 = 𝐸 → (𝑒 𝑘 ∈ (1...𝑛)𝐴) = (𝐸 𝑘 ∈ (1...𝑛)𝐴))
127126fveq2d 6352 . . . . . . . . . . . . . . . . . 18 (𝑒 = 𝐸 → (𝑀‘(𝑒 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)))
128125, 127oveq12d 6827 . . . . . . . . . . . . . . . . 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 6348 . . . . . . . . . . . . . . . . 17 (𝑒 = 𝐸 → (𝑀𝑒) = (𝑀𝐸))
130128, 129eqeq12d 2771 . . . . . . . . . . . . . . . 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 3441 . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . 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 12246 . . . . . . . . . . . . . . 15 (((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ* ∧ (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞) → ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 +∞) = +∞)
13581, 83, 134syl2anc 696 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ) → ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 +∞) = +∞)
136135adantr 472 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → ((𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 +∞) = +∞)
13798, 133, 1363eqtr3d 2798 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → (𝑀𝐸) = +∞)
138 carsgclctunlem2.4 . . . . . . . . . . . . . 14 (𝜑 → (𝑀𝐸) ≠ +∞)
139138ad2antrr 764 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → (𝑀𝐸) ≠ +∞)
140139neneqd 2933 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → ¬ (𝑀𝐸) = +∞)
141137, 140pm2.65da 601 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ¬ -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = -∞)
142141neqned 2935 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞)
143 xaddass 12268 . . . . . . . . . 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 1493 . . . . . . . . 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 12258 . . . . . . . . . . 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 6825 . . . . . . . . 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 12261 . . . . . . . . . 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 2794 . . . . . . . 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 6824 . . . . . . . 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 3953 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐸 𝑘 ∈ (1...𝑛)𝐴) = (𝐸 ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)))
153152fveq2d 6352 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 ran (𝑘 ∈ (1...𝑛) ↦ 𝐴))))
154 mptss 5608 . . . . . . . . . . . . 13 ((1...𝑛) ⊆ ℕ → (𝑘 ∈ (1...𝑛) ↦ 𝐴) ⊆ (𝑘 ∈ ℕ ↦ 𝐴))
155 rnss 5505 . . . . . . . . . . . . 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 29701 . . . . . . . . . . . . 13 (Disj 𝑘 ∈ ℕ 𝐴Disj 𝑦 ∈ ran (𝑘 ∈ ℕ ↦ 𝐴)𝑦)
15962, 158syl 17 . . . . . . . . . . . 12 (𝜑Disj 𝑦 ∈ ran (𝑘 ∈ ℕ ↦ 𝐴)𝑦)
160159adantr 472 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → Disj 𝑦 ∈ ran (𝑘 ∈ ℕ ↦ 𝐴)𝑦)
161 disjss1 4774 . . . . . . . . . . 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 30684 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 ran (𝑘 ∈ (1...𝑛) ↦ 𝐴))) = Σ*𝑦 ∈ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)(𝑀‘(𝐸𝑦)))
164 ineq2 3947 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝐸𝑦) = (𝐸𝐴))
165164fveq2d 6352 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑀‘(𝐸𝑦)) = (𝑀‘(𝐸𝐴)))
166111elexi 3349 . . . . . . . . . . 11 (1...𝑛) ∈ V
167166a1i 11 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (1...𝑛) ∈ V)
16899, 102, 22syl2anc 696 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑀‘(𝐸𝐴)) ∈ (0[,]+∞))
169 inss2 3973 . . . . . . . . . . . . . . 15 (𝐸𝐴) ⊆ 𝐴
170 sseq2 3764 . . . . . . . . . . . . . . 15 (𝐴 = ∅ → ((𝐸𝐴) ⊆ 𝐴 ↔ (𝐸𝐴) ⊆ ∅))
171169, 170mpbii 223 . . . . . . . . . . . . . 14 (𝐴 = ∅ → (𝐸𝐴) ⊆ ∅)
172 ss0 4113 . . . . . . . . . . . . . 14 ((𝐸𝐴) ⊆ ∅ → (𝐸𝐴) = ∅)
173171, 172syl 17 . . . . . . . . . . . . 13 (𝐴 = ∅ → (𝐸𝐴) = ∅)
174173adantl 473 . . . . . . . . . . . 12 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝐸𝐴) = ∅)
175174fveq2d 6352 . . . . . . . . . . 11 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝑀‘(𝐸𝐴)) = (𝑀‘∅))
176109ad2antrr 764 . . . . . . . . . . 11 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝑀‘∅) = 0)
177175, 176eqtrd 2790 . . . . . . . . . 10 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝑀‘(𝐸𝐴)) = 0)
17862adantr 472 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → Disj 𝑘 ∈ ℕ 𝐴)
179 disjss1 4774 . . . . . . . . . . 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 30435 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → Σ*𝑦 ∈ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)(𝑀‘(𝐸𝑦)) = Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝐸𝐴)))
182153, 163, 1813eqtrd 2794 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) = Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝐸𝐴)))
183150, 151, 1823eqtr3rd 2799 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝐸𝐴)) = ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))))
18417adantr 472 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ (0[,]+∞))
1853, 184sseldi 3738 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
186185xnegcld 12319 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
18715adantr 472 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝑀𝐸) ∈ ℝ*)
188 iunss1 4680 . . . . . . . . . . . 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 3886 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐸 𝑘 ∈ ℕ 𝐴) ⊆ (𝐸 𝑘 ∈ (1...𝑛)𝐴))
191743adant1r 1188 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥𝑦𝑦 ∈ 𝒫 𝑂) → (𝑀𝑥) ≤ (𝑀𝑦))
192108, 77, 190, 84, 191carsgmon 30681 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)))
193 xleneg 12238 . . . . . . . . . 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 502 . . . . . . . . 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 1476 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴)) ≤ -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)))
196 xleadd2a 12273 . . . . . . . 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 1480 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ (1...𝑛)𝐴))) ≤ ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))))
198183, 197eqbrtrd 4822 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝐸𝐴)) ≤ ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))))
19976, 22, 198esumgect 30457 . . . . 5 (𝜑 → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸𝐴)) ≤ ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))))
20012, 27, 20, 71, 199xrletrd 12182 . . . 4 (𝜑 → (𝑀 𝑘 ∈ ℕ (𝐸𝐴)) ≤ ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))))
2012, 200syl5eqbrr 4836 . . 3 (𝜑 → (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴)) ≤ ((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))))
202 xleadd1a 12272 . . 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 1480 . 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 29999 . . 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 1477 . 2 (𝜑 → (((𝑀𝐸) +𝑒 -𝑒(𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) +𝑒 (𝑀‘(𝐸 𝑘 ∈ ℕ 𝐴))) = (𝑀𝐸))
206203, 205breqtrd 4826 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 196  wa 383  w3a 1072   = wceq 1628  wcel 2135  wne 2928  wral 3046  Vcvv 3336  cdif 3708  cin 3710  wss 3711  c0 4054  𝒫 cpw 4298   cuni 4584   ciun 4668  Disj wdisj 4768   class class class wbr 4800  cmpt 4877  ran crn 5263  wf 6041  cfv 6045  (class class class)co 6809  ωcom 7226  cdom 8115  Fincfn 8117  0cc0 10124  1c1 10125  +∞cpnf 10259  -∞cmnf 10260  *cxr 10261  cle 10263  cn 11208  -𝑒cxne 12132   +𝑒 cxad 12133  [,]cicc 12367  ...cfz 12515  Σ*cesum 30394  toCaraSigaccarsg 30668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110  ax-inf2 8707  ax-ac2 9473  ax-cnex 10180  ax-resscn 10181  ax-1cn 10182  ax-icn 10183  ax-addcl 10184  ax-addrcl 10185  ax-mulcl 10186  ax-mulrcl 10187  ax-mulcom 10188  ax-addass 10189  ax-mulass 10190  ax-distr 10191  ax-i2m1 10192  ax-1ne0 10193  ax-1rid 10194  ax-rnegex 10195  ax-rrecex 10196  ax-cnre 10197  ax-pre-lttri 10198  ax-pre-lttrn 10199  ax-pre-ltadd 10200  ax-pre-mulgt0 10201  ax-pre-sup 10202  ax-addf 10203  ax-mulf 10204
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-fal 1634  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-nel 3032  df-ral 3051  df-rex 3052  df-reu 3053  df-rmo 3054  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-int 4624  df-iun 4670  df-iin 4671  df-disj 4769  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-se 5222  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-isom 6054  df-riota 6770  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-of 7058  df-om 7227  df-1st 7329  df-2nd 7330  df-supp 7460  df-wrecs 7572  df-recs 7633  df-rdg 7671  df-1o 7725  df-2o 7726  df-oadd 7729  df-er 7907  df-map 8021  df-pm 8022  df-ixp 8071  df-en 8118  df-dom 8119  df-sdom 8120  df-fin 8121  df-fsupp 8437  df-fi 8478  df-sup 8509  df-inf 8510  df-oi 8576  df-card 8951  df-acn 8954  df-ac 9125  df-cda 9178  df-pnf 10264  df-mnf 10265  df-xr 10266  df-ltxr 10267  df-le 10268  df-sub 10456  df-neg 10457  df-div 10873  df-nn 11209  df-2 11267  df-3 11268  df-4 11269  df-5 11270  df-6 11271  df-7 11272  df-8 11273  df-9 11274  df-n0 11481  df-z 11566  df-dec 11682  df-uz 11876  df-q 11978  df-rp 12022  df-xneg 12135  df-xadd 12136  df-xmul 12137  df-ioo 12368  df-ioc 12369  df-ico 12370  df-icc 12371  df-fz 12516  df-fzo 12656  df-fl 12783  df-mod 12859  df-seq 12992  df-exp 13051  df-fac 13251  df-bc 13280  df-hash 13308  df-shft 14002  df-cj 14034  df-re 14035  df-im 14036  df-sqrt 14170  df-abs 14171  df-limsup 14397  df-clim 14414  df-rlim 14415  df-sum 14612  df-ef 14993  df-sin 14995  df-cos 14996  df-pi 14998  df-struct 16057  df-ndx 16058  df-slot 16059  df-base 16061  df-sets 16062  df-ress 16063  df-plusg 16152  df-mulr 16153  df-starv 16154  df-sca 16155  df-vsca 16156  df-ip 16157  df-tset 16158  df-ple 16159  df-ds 16162  df-unif 16163  df-hom 16164  df-cco 16165  df-rest 16281  df-topn 16282  df-0g 16300  df-gsum 16301  df-topgen 16302  df-pt 16303  df-prds 16306  df-ordt 16359  df-xrs 16360  df-qtop 16365  df-imas 16366  df-xps 16368  df-mre 16444  df-mrc 16445  df-acs 16447  df-ps 17397  df-tsr 17398  df-plusf 17438  df-mgm 17439  df-sgrp 17481  df-mnd 17492  df-mhm 17532  df-submnd 17533  df-grp 17622  df-minusg 17623  df-sbg 17624  df-mulg 17738  df-subg 17788  df-cntz 17946  df-cmn 18391  df-abl 18392  df-mgp 18686  df-ur 18698  df-ring 18745  df-cring 18746  df-subrg 18976  df-abv 19015  df-lmod 19063  df-scaf 19064  df-sra 19370  df-rgmod 19371  df-psmet 19936  df-xmet 19937  df-met 19938  df-bl 19939  df-mopn 19940  df-fbas 19941  df-fg 19942  df-cnfld 19945  df-top 20897  df-topon 20914  df-topsp 20935  df-bases 20948  df-cld 21021  df-ntr 21022  df-cls 21023  df-nei 21100  df-lp 21138  df-perf 21139  df-cn 21229  df-cnp 21230  df-haus 21317  df-tx 21563  df-hmeo 21756  df-fil 21847  df-fm 21939  df-flim 21940  df-flf 21941  df-tmd 22073  df-tgp 22074  df-tsms 22127  df-trg 22160  df-xms 22322  df-ms 22323  df-tms 22324  df-nm 22584  df-ngp 22585  df-nrg 22587  df-nlm 22588  df-ii 22877  df-cncf 22878  df-limc 23825  df-dv 23826  df-log 24498  df-esum 30395  df-carsg 30669
This theorem is referenced by:  carsgclctunlem3  30687
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