Theorem carsggect 31576

Description: The outer measure is countably superadditive on Caratheodory measurable sets. (Contributed by Thierry Arnoux, 31-May-2020.)

Hypotheses

Ref	Expression
carsgval.1	⊢ (𝜑 → 𝑂 ∈ 𝑉)
carsgval.2	⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))
carsgsiga.1	⊢ (𝜑 → (𝑀‘∅) = 0)
carsgsiga.2	⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))
carsggect.0	⊢ (𝜑 → ¬ ∅ ∈ 𝐴)
carsggect.1	⊢ (𝜑 → 𝐴 ≼ ω)
carsggect.2	⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))
carsggect.3	⊢ (𝜑 → Disj 𝑦 ∈ 𝐴 𝑦)
carsggect.4	⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))

Assertion

Ref	Expression
carsggect	⊢ (𝜑 → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ≤ (𝑀‘∪ 𝐴))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑀,𝑦   𝑥,𝑂,𝑦   𝜑,𝑥,𝑦   𝑧,𝐴   𝑧,𝑀   𝑧,𝑂,𝑥,𝑦   𝜑,𝑧

Allowed substitution hints:   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem carsggect

Dummy variables 𝑓 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		carsggect.1	. . 3 ⊢ (𝜑 → 𝐴 ≼ ω)
2		0ex 5211	. . . 4 ⊢ ∅ ∈ V
3	2	a1i 11	. . 3 ⊢ (𝜑 → ∅ ∈ V)
4		carsggect.0	. . 3 ⊢ (𝜑 → ¬ ∅ ∈ 𝐴)
5		padct 30455	. . 3 ⊢ ((𝐴 ≼ ω ∧ ∅ ∈ V ∧ ¬ ∅ ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)))
6	1, 3, 4, 5	syl3anc 1367	. 2 ⊢ (𝜑 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)))
7		nfv 1915	. . . . 5 ⊢ Ⅎ𝑧(𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)))
8		simpr1 1190	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝑓:ℕ⟶(𝐴 ∪ {∅}))
9	8	feqmptd 6733	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝑓 = (𝑘 ∈ ℕ ↦ (𝑓‘𝑘)))
10	9	rneqd 5808	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran 𝑓 = ran (𝑘 ∈ ℕ ↦ (𝑓‘𝑘)))
11	7, 10	esumeq1d 31294	. . . 4 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) = Σ*𝑧 ∈ ran (𝑘 ∈ ℕ ↦ (𝑓‘𝑘))(𝑀‘𝑧))
12		fvex 6683	. . . . . . . . . 10 ⊢ (toCaraSiga‘𝑀) ∈ V
13	12	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (toCaraSiga‘𝑀) ∈ V)
14		carsggect.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))
15	14	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝐴 ⊆ (toCaraSiga‘𝑀))
16		carsgval.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑂 ∈ 𝑉)
17	16	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝑂 ∈ 𝑉)
18		carsgval.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))
19	18	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
20		carsgsiga.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀‘∅) = 0)
21	20	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝑀‘∅) = 0)
22	17, 19, 21	0elcarsg 31565	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∅ ∈ (toCaraSiga‘𝑀))
23	22	snssd 4742	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → {∅} ⊆ (toCaraSiga‘𝑀))
24	15, 23	unssd 4162	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝐴 ∪ {∅}) ⊆ (toCaraSiga‘𝑀))
25	13, 24	ssexd 5228	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝐴 ∪ {∅}) ∈ V)
26	19	adantr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
27	16, 18	carsgcl 31562	. . . . . . . . . . . . 13 ⊢ (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂)
28	14, 27	sstrd 3977	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ 𝒫 𝑂)
29	28	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝐴 ⊆ 𝒫 𝑂)
30		0elpw 5256	. . . . . . . . . . . . 13 ⊢ ∅ ∈ 𝒫 𝑂
31	30	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∅ ∈ 𝒫 𝑂)
32	31	snssd 4742	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → {∅} ⊆ 𝒫 𝑂)
33	29, 32	unssd 4162	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
34	33	sselda 3967	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → 𝑧 ∈ 𝒫 𝑂)
35	26, 34	ffvelrnd 6852	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → (𝑀‘𝑧) ∈ (0[,]+∞))
36	8	frnd 6521	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran 𝑓 ⊆ (𝐴 ∪ {∅}))
37	7, 25, 35, 36	esummono 31313	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) ≤ Σ*𝑧 ∈ (𝐴 ∪ {∅})(𝑀‘𝑧))
38		ctex 8524	. . . . . . . . . 10 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)
39	1, 38	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ V)
40	39	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝐴 ∈ V)
41	13, 23	ssexd 5228	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → {∅} ∈ V)
42	19	adantr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ 𝐴) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
43	29	sselda 3967	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝒫 𝑂)
44	42, 43	ffvelrnd 6852	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ 𝐴) → (𝑀‘𝑧) ∈ (0[,]+∞))
45		elsni 4584	. . . . . . . . . . 11 ⊢ (𝑧 ∈ {∅} → 𝑧 = ∅)
46	45	adantl 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ {∅}) → 𝑧 = ∅)
47	46	fveq2d 6674	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ {∅}) → (𝑀‘𝑧) = (𝑀‘∅))
48	21	adantr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ {∅}) → (𝑀‘∅) = 0)
49	47, 48	eqtrd 2856	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ {∅}) → (𝑀‘𝑧) = 0)
50	40, 41, 44, 49	esumpad 31314	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ (𝐴 ∪ {∅})(𝑀‘𝑧) = Σ*𝑧 ∈ 𝐴(𝑀‘𝑧))
51	37, 50	breqtrd 5092	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) ≤ Σ*𝑧 ∈ 𝐴(𝑀‘𝑧))
52	36, 24	sstrd 3977	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran 𝑓 ⊆ (toCaraSiga‘𝑀))
53		ssexg 5227	. . . . . . . 8 ⊢ ((ran 𝑓 ⊆ (toCaraSiga‘𝑀) ∧ (toCaraSiga‘𝑀) ∈ V) → ran 𝑓 ∈ V)
54	52, 12, 53	sylancl 588	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran 𝑓 ∈ V)
55	19	adantr 483	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ ran 𝑓) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
56	36, 33	sstrd 3977	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran 𝑓 ⊆ 𝒫 𝑂)
57	56	sselda 3967	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ ran 𝑓) → 𝑧 ∈ 𝒫 𝑂)
58	55, 57	ffvelrnd 6852	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ ran 𝑓) → (𝑀‘𝑧) ∈ (0[,]+∞))
59		simpr2 1191	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝐴 ⊆ ran 𝑓)
60	7, 54, 58, 59	esummono 31313	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ≤ Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧))
61	51, 60	jca 514	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) ≤ Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ∧ Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ≤ Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧)))
62		iccssxr 12820	. . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ*
63	58	ralrimiva 3182	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∀𝑧 ∈ ran 𝑓(𝑀‘𝑧) ∈ (0[,]+∞))
64		nfcv 2977	. . . . . . . . 9 ⊢ Ⅎ𝑧ran 𝑓
65	64	esumcl 31289	. . . . . . . 8 ⊢ ((ran 𝑓 ∈ V ∧ ∀𝑧 ∈ ran 𝑓(𝑀‘𝑧) ∈ (0[,]+∞)) → Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) ∈ (0[,]+∞))
66	54, 63, 65	syl2anc 586	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) ∈ (0[,]+∞))
67	62, 66	sseldi 3965	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) ∈ ℝ*)
68	44	ralrimiva 3182	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∀𝑧 ∈ 𝐴 (𝑀‘𝑧) ∈ (0[,]+∞))
69		nfcv 2977	. . . . . . . . 9 ⊢ Ⅎ𝑧𝐴
70	69	esumcl 31289	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ∀𝑧 ∈ 𝐴 (𝑀‘𝑧) ∈ (0[,]+∞)) → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ∈ (0[,]+∞))
71	40, 68, 70	syl2anc 586	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ∈ (0[,]+∞))
72	62, 71	sseldi 3965	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ∈ ℝ*)
73		xrletri3 12548	. . . . . 6 ⊢ ((Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) ∈ ℝ* ∧ Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ∈ ℝ*) → (Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) = Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ↔ (Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) ≤ Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ∧ Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ≤ Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧))))
74	67, 72, 73	syl2anc 586	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) = Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ↔ (Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) ≤ Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ∧ Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ≤ Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧))))
75	61, 74	mpbird 259	. . . 4 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) = Σ*𝑧 ∈ 𝐴(𝑀‘𝑧))
76		fveq2 6670	. . . . 5 ⊢ (𝑧 = (𝑓‘𝑘) → (𝑀‘𝑧) = (𝑀‘(𝑓‘𝑘)))
77		nnex 11644	. . . . . 6 ⊢ ℕ ∈ V
78	77	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ℕ ∈ V)
79	19	adantr 483	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
80	33	adantr 483	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
81	8	adantr 483	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → 𝑓:ℕ⟶(𝐴 ∪ {∅}))
82		simpr 487	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
83	81, 82	ffvelrnd 6852	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ (𝐴 ∪ {∅}))
84	80, 83	sseldd 3968	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ 𝒫 𝑂)
85	79, 84	ffvelrnd 6852	. . . . 5 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑀‘(𝑓‘𝑘)) ∈ (0[,]+∞))
86		simpr 487	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓‘𝑘) = ∅) → (𝑓‘𝑘) = ∅)
87	86	fveq2d 6674	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓‘𝑘) = ∅) → (𝑀‘(𝑓‘𝑘)) = (𝑀‘∅))
88	21	ad2antrr 724	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓‘𝑘) = ∅) → (𝑀‘∅) = 0)
89	87, 88	eqtrd 2856	. . . . 5 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓‘𝑘) = ∅) → (𝑀‘(𝑓‘𝑘)) = 0)
90		cnvimass 5949	. . . . . . 7 ⊢ (◡𝑓 “ 𝐴) ⊆ dom 𝑓
91	90, 8	fssdm 6530	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (◡𝑓 “ 𝐴) ⊆ ℕ)
92		ffun 6517	. . . . . . . . . . 11 ⊢ (𝑓:ℕ⟶(𝐴 ∪ {∅}) → Fun 𝑓)
93	8, 92	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Fun 𝑓)
94	93	adantr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑓 “ 𝐴))) → Fun 𝑓)
95		difpreima 6835	. . . . . . . . . . . . 13 ⊢ (Fun 𝑓 → (◡𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) = ((◡𝑓 “ (𝐴 ∪ {∅})) ∖ (◡𝑓 “ 𝐴)))
96	8, 92, 95	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (◡𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) = ((◡𝑓 “ (𝐴 ∪ {∅})) ∖ (◡𝑓 “ 𝐴)))
97		fimacnv 6839	. . . . . . . . . . . . . 14 ⊢ (𝑓:ℕ⟶(𝐴 ∪ {∅}) → (◡𝑓 “ (𝐴 ∪ {∅})) = ℕ)
98	8, 97	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (◡𝑓 “ (𝐴 ∪ {∅})) = ℕ)
99	98	difeq1d 4098	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ((◡𝑓 “ (𝐴 ∪ {∅})) ∖ (◡𝑓 “ 𝐴)) = (ℕ ∖ (◡𝑓 “ 𝐴)))
100	96, 99	eqtrd 2856	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (◡𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) = (ℕ ∖ (◡𝑓 “ 𝐴)))
101		uncom 4129	. . . . . . . . . . . . . . . 16 ⊢ ({∅} ∪ 𝐴) = (𝐴 ∪ {∅})
102	101	difeq1i 4095	. . . . . . . . . . . . . . 15 ⊢ (({∅} ∪ 𝐴) ∖ 𝐴) = ((𝐴 ∪ {∅}) ∖ 𝐴)
103		difun2 4429	. . . . . . . . . . . . . . 15 ⊢ (({∅} ∪ 𝐴) ∖ 𝐴) = ({∅} ∖ 𝐴)
104	102, 103	eqtr3i 2846	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∪ {∅}) ∖ 𝐴) = ({∅} ∖ 𝐴)
105		difss 4108	. . . . . . . . . . . . . 14 ⊢ ({∅} ∖ 𝐴) ⊆ {∅}
106	104, 105	eqsstri 4001	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅}
107	106	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅})
108		sspreima 30392	. . . . . . . . . . . 12 ⊢ ((Fun 𝑓 ∧ ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅}) → (◡𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) ⊆ (◡𝑓 “ {∅}))
109	93, 107, 108	syl2anc 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (◡𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) ⊆ (◡𝑓 “ {∅}))
110	100, 109	eqsstrrd 4006	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (ℕ ∖ (◡𝑓 “ 𝐴)) ⊆ (◡𝑓 “ {∅}))
111	110	sselda 3967	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑓 “ 𝐴))) → 𝑘 ∈ (◡𝑓 “ {∅}))
112		fvimacnvi 6822	. . . . . . . . 9 ⊢ ((Fun 𝑓 ∧ 𝑘 ∈ (◡𝑓 “ {∅})) → (𝑓‘𝑘) ∈ {∅})
113	94, 111, 112	syl2anc 586	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑓 “ 𝐴))) → (𝑓‘𝑘) ∈ {∅})
114		elsni 4584	. . . . . . . 8 ⊢ ((𝑓‘𝑘) ∈ {∅} → (𝑓‘𝑘) = ∅)
115	113, 114	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑓 “ 𝐴))) → (𝑓‘𝑘) = ∅)
116	115	ralrimiva 3182	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∀𝑘 ∈ (ℕ ∖ (◡𝑓 “ 𝐴))(𝑓‘𝑘) = ∅)
117		carsggect.3	. . . . . . . 8 ⊢ (𝜑 → Disj 𝑦 ∈ 𝐴 𝑦)
118	117	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Disj 𝑦 ∈ 𝐴 𝑦)
119		simpr3 1192	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Fun (◡𝑓 ↾ 𝐴))
120		fresf1o 30376	. . . . . . . . . 10 ⊢ ((Fun 𝑓 ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→𝐴)
121	93, 59, 119, 120	syl3anc 1367	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→𝐴)
122		simpr 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑦 = ((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑘)) → 𝑦 = ((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑘))
123	121, 122	disjrdx 30341	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (Disj 𝑘 ∈ (◡𝑓 “ 𝐴)((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑘) ↔ Disj 𝑦 ∈ 𝐴 𝑦))
124		fvres 6689	. . . . . . . . . 10 ⊢ (𝑘 ∈ (◡𝑓 “ 𝐴) → ((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑘) = (𝑓‘𝑘))
125	124	adantl 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ (◡𝑓 “ 𝐴)) → ((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑘) = (𝑓‘𝑘))
126	125	disjeq2dv 5036	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (Disj 𝑘 ∈ (◡𝑓 “ 𝐴)((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑘) ↔ Disj 𝑘 ∈ (◡𝑓 “ 𝐴)(𝑓‘𝑘)))
127	123, 126	bitr3d 283	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (Disj 𝑦 ∈ 𝐴 𝑦 ↔ Disj 𝑘 ∈ (◡𝑓 “ 𝐴)(𝑓‘𝑘)))
128	118, 127	mpbid 234	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Disj 𝑘 ∈ (◡𝑓 “ 𝐴)(𝑓‘𝑘))
129		disjss3 5065	. . . . . . 7 ⊢ (((◡𝑓 “ 𝐴) ⊆ ℕ ∧ ∀𝑘 ∈ (ℕ ∖ (◡𝑓 “ 𝐴))(𝑓‘𝑘) = ∅) → (Disj 𝑘 ∈ (◡𝑓 “ 𝐴)(𝑓‘𝑘) ↔ Disj 𝑘 ∈ ℕ (𝑓‘𝑘)))
130	129	biimpa 479	. . . . . 6 ⊢ ((((◡𝑓 “ 𝐴) ⊆ ℕ ∧ ∀𝑘 ∈ (ℕ ∖ (◡𝑓 “ 𝐴))(𝑓‘𝑘) = ∅) ∧ Disj 𝑘 ∈ (◡𝑓 “ 𝐴)(𝑓‘𝑘)) → Disj 𝑘 ∈ ℕ (𝑓‘𝑘))
131	91, 116, 128, 130	syl21anc 835	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Disj 𝑘 ∈ ℕ (𝑓‘𝑘))
132	76, 78, 85, 84, 89, 131	esumrnmpt2 31327	. . . 4 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ ran (𝑘 ∈ ℕ ↦ (𝑓‘𝑘))(𝑀‘𝑧) = Σ*𝑘 ∈ ℕ(𝑀‘(𝑓‘𝑘)))
133	11, 75, 132	3eqtr3rd 2865	. . 3 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑘 ∈ ℕ(𝑀‘(𝑓‘𝑘)) = Σ*𝑧 ∈ 𝐴(𝑀‘𝑧))
134		uniiun 4982	. . . . . . 7 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
135	28	sselda 3967	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝒫 𝑂)
136	39, 135	elpwiuncl 30288	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝑥 ∈ 𝒫 𝑂)
137	134, 136	eqeltrid 2917	. . . . . 6 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝒫 𝑂)
138	137	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∪ 𝐴 ∈ 𝒫 𝑂)
139	19, 138	ffvelrnd 6852	. . . 4 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝑀‘∪ 𝐴) ∈ (0[,]+∞))
140		carsgsiga.2	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))
141	140	3adant1r 1173	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))
142		fveq2 6670	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑀‘𝑦) = (𝑀‘𝑧))
143		nfcv 2977	. . . . . . . . . 10 ⊢ Ⅎ𝑧𝑥
144		nfcv 2977	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝑥
145		nfcv 2977	. . . . . . . . . 10 ⊢ Ⅎ𝑧(𝑀‘𝑦)
146		nfcv 2977	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑀‘𝑧)
147	142, 143, 144, 145, 146	cbvesum 31301	. . . . . . . . 9 ⊢ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦) = Σ*𝑧 ∈ 𝑥(𝑀‘𝑧)
148	141, 147	breqtrdi 5107	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑧 ∈ 𝑥(𝑀‘𝑧))
149		ffn 6514	. . . . . . . . . 10 ⊢ (𝑓:ℕ⟶(𝐴 ∪ {∅}) → 𝑓 Fn ℕ)
150		fz1ssnn 12939	. . . . . . . . . . 11 ⊢ (1...𝑛) ⊆ ℕ
151		fnssres 6470	. . . . . . . . . . 11 ⊢ ((𝑓 Fn ℕ ∧ (1...𝑛) ⊆ ℕ) → (𝑓 ↾ (1...𝑛)) Fn (1...𝑛))
152	150, 151	mpan2 689	. . . . . . . . . 10 ⊢ (𝑓 Fn ℕ → (𝑓 ↾ (1...𝑛)) Fn (1...𝑛))
153	8, 149, 152	3syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝑓 ↾ (1...𝑛)) Fn (1...𝑛))
154		fzfi 13341	. . . . . . . . . 10 ⊢ (1...𝑛) ∈ Fin
155		fnfi 8796	. . . . . . . . . 10 ⊢ (((𝑓 ↾ (1...𝑛)) Fn (1...𝑛) ∧ (1...𝑛) ∈ Fin) → (𝑓 ↾ (1...𝑛)) ∈ Fin)
156	154, 155	mpan2 689	. . . . . . . . 9 ⊢ ((𝑓 ↾ (1...𝑛)) Fn (1...𝑛) → (𝑓 ↾ (1...𝑛)) ∈ Fin)
157		rnfi 8807	. . . . . . . . 9 ⊢ ((𝑓 ↾ (1...𝑛)) ∈ Fin → ran (𝑓 ↾ (1...𝑛)) ∈ Fin)
158	153, 156, 157	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran (𝑓 ↾ (1...𝑛)) ∈ Fin)
159		resss 5878	. . . . . . . . . . 11 ⊢ (𝑓 ↾ (1...𝑛)) ⊆ 𝑓
160		rnss 5809	. . . . . . . . . . 11 ⊢ ((𝑓 ↾ (1...𝑛)) ⊆ 𝑓 → ran (𝑓 ↾ (1...𝑛)) ⊆ ran 𝑓)
161	159, 160	ax-mp 5	. . . . . . . . . 10 ⊢ ran (𝑓 ↾ (1...𝑛)) ⊆ ran 𝑓
162	161	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ ran 𝑓)
163	162, 52	sstrd 3977	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ (toCaraSiga‘𝑀))
164	162, 36	sstrd 3977	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ (𝐴 ∪ {∅}))
165		nfcv 2977	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧𝑦
166		nfcv 2977	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦𝑧
167		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
168	165, 166, 167	cbvdisj 5041	. . . . . . . . . . . 12 ⊢ (Disj 𝑦 ∈ 𝐴 𝑦 ↔ Disj 𝑧 ∈ 𝐴 𝑧)
169		disjun0 30345	. . . . . . . . . . . 12 ⊢ (Disj 𝑧 ∈ 𝐴 𝑧 → Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
170	168, 169	sylbi 219	. . . . . . . . . . 11 ⊢ (Disj 𝑦 ∈ 𝐴 𝑦 → Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
171	117, 170	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
172	171	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
173		disjss1 5037	. . . . . . . . 9 ⊢ (ran (𝑓 ↾ (1...𝑛)) ⊆ (𝐴 ∪ {∅}) → (Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧 → Disj 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))𝑧))
174	164, 172, 173	sylc 65	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Disj 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))𝑧)
175		pwidg 4561	. . . . . . . . 9 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ 𝒫 𝑂)
176	17, 175	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝑂 ∈ 𝒫 𝑂)
177	17, 19, 21, 148, 158, 163, 174, 176	carsgclctunlem1 31575	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝑀‘(𝑂 ∩ ∪ ran (𝑓 ↾ (1...𝑛)))) = Σ*𝑧 ∈ ran (𝑓 ↾ (1...𝑛))(𝑀‘(𝑂 ∩ 𝑧)))
178	177	adantr 483	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝑂 ∩ ∪ ran (𝑓 ↾ (1...𝑛)))) = Σ*𝑧 ∈ ran (𝑓 ↾ (1...𝑛))(𝑀‘(𝑂 ∩ 𝑧)))
179	164	unissd 4848	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∪ ran (𝑓 ↾ (1...𝑛)) ⊆ ∪ (𝐴 ∪ {∅}))
180		uniun 4861	. . . . . . . . . . . 12 ⊢ ∪ (𝐴 ∪ {∅}) = (∪ 𝐴 ∪ ∪ {∅})
181	2	unisn 4858	. . . . . . . . . . . . 13 ⊢ ∪ {∅} = ∅
182	181	uneq2i 4136	. . . . . . . . . . . 12 ⊢ (∪ 𝐴 ∪ ∪ {∅}) = (∪ 𝐴 ∪ ∅)
183		un0 4344	. . . . . . . . . . . 12 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴
184	180, 182, 183	3eqtri 2848	. . . . . . . . . . 11 ⊢ ∪ (𝐴 ∪ {∅}) = ∪ 𝐴
185	179, 184	sseqtrdi 4017	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∪ ran (𝑓 ↾ (1...𝑛)) ⊆ ∪ 𝐴)
186	185	adantr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → ∪ ran (𝑓 ↾ (1...𝑛)) ⊆ ∪ 𝐴)
187		uniss 4846	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝒫 𝑂 → ∪ 𝐴 ⊆ ∪ 𝒫 𝑂)
188		unipw 5343	. . . . . . . . . . . 12 ⊢ ∪ 𝒫 𝑂 = 𝑂
189	187, 188	sseqtrdi 4017	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝒫 𝑂 → ∪ 𝐴 ⊆ 𝑂)
190	28, 189	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝐴 ⊆ 𝑂)
191	190	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → ∪ 𝐴 ⊆ 𝑂)
192	186, 191	sstrd 3977	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → ∪ ran (𝑓 ↾ (1...𝑛)) ⊆ 𝑂)
193		sseqin2 4192	. . . . . . . 8 ⊢ (∪ ran (𝑓 ↾ (1...𝑛)) ⊆ 𝑂 ↔ (𝑂 ∩ ∪ ran (𝑓 ↾ (1...𝑛))) = ∪ ran (𝑓 ↾ (1...𝑛)))
194	192, 193	sylib 220	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑂 ∩ ∪ ran (𝑓 ↾ (1...𝑛))) = ∪ ran (𝑓 ↾ (1...𝑛)))
195	194	fveq2d 6674	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝑂 ∩ ∪ ran (𝑓 ↾ (1...𝑛)))) = (𝑀‘∪ ran (𝑓 ↾ (1...𝑛))))
196		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑧((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ)
197	164	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑓 ↾ (1...𝑛)) ⊆ (𝐴 ∪ {∅}))
198	28	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ 𝒫 𝑂)
199	30	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → ∅ ∈ 𝒫 𝑂)
200	199	snssd 4742	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → {∅} ⊆ 𝒫 𝑂)
201	198, 200	unssd 4162	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
202	197, 201	sstrd 3977	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑓 ↾ (1...𝑛)) ⊆ 𝒫 𝑂)
203	202	sselda 3967	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → 𝑧 ∈ 𝒫 𝑂)
204	203	elpwid 4550	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → 𝑧 ⊆ 𝑂)
205		sseqin2 4192	. . . . . . . . . . 11 ⊢ (𝑧 ⊆ 𝑂 ↔ (𝑂 ∩ 𝑧) = 𝑧)
206	204, 205	sylib 220	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → (𝑂 ∩ 𝑧) = 𝑧)
207	206	fveq2d 6674	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → (𝑀‘(𝑂 ∩ 𝑧)) = (𝑀‘𝑧))
208	207	ralrimiva 3182	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → ∀𝑧 ∈ ran (𝑓 ↾ (1...𝑛))(𝑀‘(𝑂 ∩ 𝑧)) = (𝑀‘𝑧))
209	196, 208	esumeq2d 31296	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → Σ*𝑧 ∈ ran (𝑓 ↾ (1...𝑛))(𝑀‘(𝑂 ∩ 𝑧)) = Σ*𝑧 ∈ ran (𝑓 ↾ (1...𝑛))(𝑀‘𝑧))
210	9	reseq1d 5852	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝑓 ↾ (1...𝑛)) = ((𝑘 ∈ ℕ ↦ (𝑓‘𝑘)) ↾ (1...𝑛)))
211	210	adantr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑓 ↾ (1...𝑛)) = ((𝑘 ∈ ℕ ↦ (𝑓‘𝑘)) ↾ (1...𝑛)))
212		resmpt 5905	. . . . . . . . . . . 12 ⊢ ((1...𝑛) ⊆ ℕ → ((𝑘 ∈ ℕ ↦ (𝑓‘𝑘)) ↾ (1...𝑛)) = (𝑘 ∈ (1...𝑛) ↦ (𝑓‘𝑘)))
213	150, 212	ax-mp 5	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ↦ (𝑓‘𝑘)) ↾ (1...𝑛)) = (𝑘 ∈ (1...𝑛) ↦ (𝑓‘𝑘))
214	211, 213	syl6eq 2872	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑓 ↾ (1...𝑛)) = (𝑘 ∈ (1...𝑛) ↦ (𝑓‘𝑘)))
215	214	eqcomd 2827	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ (1...𝑛) ↦ (𝑓‘𝑘)) = (𝑓 ↾ (1...𝑛)))
216	215	rneqd 5808	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑘 ∈ (1...𝑛) ↦ (𝑓‘𝑘)) = ran (𝑓 ↾ (1...𝑛)))
217	196, 216	esumeq1d 31294	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → Σ*𝑧 ∈ ran (𝑘 ∈ (1...𝑛) ↦ (𝑓‘𝑘))(𝑀‘𝑧) = Σ*𝑧 ∈ ran (𝑓 ↾ (1...𝑛))(𝑀‘𝑧))
218	154	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin)
219	19	ad2antrr 724	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
220	150	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
221	220	sselda 3967	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
222	84	adantlr 713	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ 𝒫 𝑂)
223	221, 222	syldan 593	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑓‘𝑘) ∈ 𝒫 𝑂)
224	219, 223	ffvelrnd 6852	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑀‘(𝑓‘𝑘)) ∈ (0[,]+∞))
225		simpr 487	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓‘𝑘) = ∅) → (𝑓‘𝑘) = ∅)
226	225	fveq2d 6674	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓‘𝑘) = ∅) → (𝑀‘(𝑓‘𝑘)) = (𝑀‘∅))
227	21	ad3antrrr 728	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓‘𝑘) = ∅) → (𝑀‘∅) = 0)
228	226, 227	eqtrd 2856	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓‘𝑘) = ∅) → (𝑀‘(𝑓‘𝑘)) = 0)
229		disjss1 5037	. . . . . . . . . . 11 ⊢ ((1...𝑛) ⊆ ℕ → (Disj 𝑘 ∈ ℕ (𝑓‘𝑘) → Disj 𝑘 ∈ (1...𝑛)(𝑓‘𝑘)))
230	150, 229	ax-mp 5	. . . . . . . . . 10 ⊢ (Disj 𝑘 ∈ ℕ (𝑓‘𝑘) → Disj 𝑘 ∈ (1...𝑛)(𝑓‘𝑘))
231	131, 230	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Disj 𝑘 ∈ (1...𝑛)(𝑓‘𝑘))
232	231	adantr 483	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → Disj 𝑘 ∈ (1...𝑛)(𝑓‘𝑘))
233	76, 218, 224, 223, 228, 232	esumrnmpt2 31327	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → Σ*𝑧 ∈ ran (𝑘 ∈ (1...𝑛) ↦ (𝑓‘𝑘))(𝑀‘𝑧) = Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝑓‘𝑘)))
234	209, 217, 233	3eqtr2d 2862	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → Σ*𝑧 ∈ ran (𝑓 ↾ (1...𝑛))(𝑀‘(𝑂 ∩ 𝑧)) = Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝑓‘𝑘)))
235	178, 195, 234	3eqtr3d 2864	. . . . 5 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑀‘∪ ran (𝑓 ↾ (1...𝑛))) = Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝑓‘𝑘)))
236		carsggect.4	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))
237	236	3adant1r 1173	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))
238	17, 19, 185, 138, 237	carsgmon 31572	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝑀‘∪ ran (𝑓 ↾ (1...𝑛))) ≤ (𝑀‘∪ 𝐴))
239	238	adantr 483	. . . . 5 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑀‘∪ ran (𝑓 ↾ (1...𝑛))) ≤ (𝑀‘∪ 𝐴))
240	235, 239	eqbrtrrd 5090	. . . 4 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝑓‘𝑘)) ≤ (𝑀‘∪ 𝐴))
241	139, 85, 240	esumgect 31349	. . 3 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑘 ∈ ℕ(𝑀‘(𝑓‘𝑘)) ≤ (𝑀‘∪ 𝐴))
242	133, 241	eqbrtrrd 5090	. 2 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ≤ (𝑀‘∪ 𝐴))
243	6, 242	exlimddv 1936	1 ⊢ (𝜑 → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ≤ (𝑀‘∪ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∀wral 3138  Vcvv 3494   ∖ cdif 3933   ∪ cun 3934   ∩ cin 3935   ⊆ wss 3936  ∅c0 4291  𝒫 cpw 4539  {csn 4567  ∪ cuni 4838  ∪ ciun 4919  Disj wdisj 5031   class class class wbr 5066   ↦ cmpt 5146  ^◡ccnv 5554  ran crn 5556   ↾ cres 5557   “ cima 5558  Fun wfun 6349   Fn wfn 6350  ⟶wf 6351  –1-1-onto→wf1o 6354  ‘cfv 6355  (class class class)co 7156  ωcom 7580   ≼ cdom 8507  Fincfn 8509  0cc0 10537  1c1 10538  +∞cpnf 10672  ℝ^*cxr 10674   ≤ cle 10676  ℕcn 11638  [,]cicc 12742  ...cfz 12893  Σ^*cesum 31286  toCaraSigaccarsg 31559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615  ax-addf 10616  ax-mulf 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-iin 4922  df-disj 5032  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7409  df-om 7581  df-1st 7689  df-2nd 7690  df-supp 7831  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-er 8289  df-map 8408  df-pm 8409  df-ixp 8462  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-fsupp 8834  df-fi 8875  df-sup 8906  df-inf 8907  df-oi 8974  df-dju 9330  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-z 11983  df-dec 12100  df-uz 12245  df-q 12350  df-rp 12391  df-xneg 12508  df-xadd 12509  df-xmul 12510  df-ioo 12743  df-ioc 12744  df-ico 12745  df-icc 12746  df-fz 12894  df-fzo 13035  df-fl 13163  df-mod 13239  df-seq 13371  df-exp 13431  df-fac 13635  df-bc 13664  df-hash 13692  df-shft 14426  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-limsup 14828  df-clim 14845  df-rlim 14846  df-sum 15043  df-ef 15421  df-sin 15423  df-cos 15424  df-pi 15426  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-starv 16580  df-sca 16581  df-vsca 16582  df-ip 16583  df-tset 16584  df-ple 16585  df-ds 16587  df-unif 16588  df-hom 16589  df-cco 16590  df-rest 16696  df-topn 16697  df-0g 16715  df-gsum 16716  df-topgen 16717  df-pt 16718  df-prds 16721  df-ordt 16774  df-xrs 16775  df-qtop 16780  df-imas 16781  df-xps 16783  df-mre 16857  df-mrc 16858  df-acs 16860  df-ps 17810  df-tsr 17811  df-plusf 17851  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-mhm 17956  df-submnd 17957  df-grp 18106  df-minusg 18107  df-sbg 18108  df-mulg 18225  df-subg 18276  df-cntz 18447  df-cmn 18908  df-abl 18909  df-mgp 19240  df-ur 19252  df-ring 19299  df-cring 19300  df-subrg 19533  df-abv 19588  df-lmod 19636  df-scaf 19637  df-sra 19944  df-rgmod 19945  df-psmet 20537  df-xmet 20538  df-met 20539  df-bl 20540  df-mopn 20541  df-fbas 20542  df-fg 20543  df-cnfld 20546  df-top 21502  df-topon 21519  df-topsp 21541  df-bases 21554  df-cld 21627  df-ntr 21628  df-cls 21629  df-nei 21706  df-lp 21744  df-perf 21745  df-cn 21835  df-cnp 21836  df-haus 21923  df-tx 22170  df-hmeo 22363  df-fil 22454  df-fm 22546  df-flim 22547  df-flf 22548  df-tmd 22680  df-tgp 22681  df-tsms 22735  df-trg 22768  df-xms 22930  df-ms 22931  df-tms 22932  df-nm 23192  df-ngp 23193  df-nrg 23195  df-nlm 23196  df-ii 23485  df-cncf 23486  df-limc 24464  df-dv 24465  df-log 25140  df-esum 31287  df-carsg 31560

This theorem is referenced by:  omsmeas  31581