Theorem carsggect 34674

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 5269	. . . 4 ⊢ ∅ ∈ V
3	2	a1i 11	. . 3 ⊢ (𝜑 → ∅ ∈ V)
4		carsggect.0	. . 3 ⊢ (𝜑 → ¬ ∅ ∈ 𝐴)
5		padct 33029	. . 3 ⊢ ((𝐴 ≼ ω ∧ ∅ ∈ V ∧ ¬ ∅ ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)))
6	1, 3, 4, 5	syl3anc 1396	. 2 ⊢ (𝜑 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)))
7		nfv 1942	. . . . 5 ⊢ Ⅎ𝑧(𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)))
8		simpr1 1211	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝑓:ℕ⟶(𝐴 ∪ {∅}))
9	8	feqmptd 6949	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝑓 = (𝑘 ∈ ℕ ↦ (𝑓‘𝑘)))
10	9	rneqd 5928	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran 𝑓 = ran (𝑘 ∈ ℕ ↦ (𝑓‘𝑘)))
11	7, 10	esumeq1d 34391	. . . 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 6894	. . . . . . . . . 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 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝐴 ⊆ (toCaraSiga‘𝑀))
16		carsgval.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑂 ∈ 𝑉)
17	16	adantr 485	. . . . . . . . . . . 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 485	. . . . . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝑀‘∅) = 0)
22	17, 19, 21	0elcarsg 34663	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∅ ∈ (toCaraSiga‘𝑀))
23	22	snssd 4751	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → {∅} ⊆ (toCaraSiga‘𝑀))
24	15, 23	unssd 4144	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝐴 ∪ {∅}) ⊆ (toCaraSiga‘𝑀))
25	13, 24	ssexd 5294	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝐴 ∪ {∅}) ∈ V)
26	19	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
27	16, 18	carsgcl 34660	. . . . . . . . . . . . 13 ⊢ (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂)
28	14, 27	sstrd 3946	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ 𝒫 𝑂)
29	28	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝐴 ⊆ 𝒫 𝑂)
30		0elpw 5326	. . . . . . . . . . . . 13 ⊢ ∅ ∈ 𝒫 𝑂
31	30	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∅ ∈ 𝒫 𝑂)
32	31	snssd 4751	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → {∅} ⊆ 𝒫 𝑂)
33	29, 32	unssd 4144	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
34	33	sselda 3936	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → 𝑧 ∈ 𝒫 𝑂)
35	26, 34	ffvelcdmd 7080	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → (𝑀‘𝑧) ∈ (0[,]+∞))
36	8	frnd 6714	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran 𝑓 ⊆ (𝐴 ∪ {∅}))
37	7, 25, 35, 36	esummono 34410	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) ≤ Σ*𝑧 ∈ (𝐴 ∪ {∅})(𝑀‘𝑧))
38		ctex 8959	. . . . . . . . . 10 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)
39	1, 38	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ V)
40	39	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝐴 ∈ V)
41	13, 23	ssexd 5294	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → {∅} ∈ V)
42	19	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ 𝐴) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
43	29	sselda 3936	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝒫 𝑂)
44	42, 43	ffvelcdmd 7080	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ 𝐴) → (𝑀‘𝑧) ∈ (0[,]+∞))
45		elsni 4605	. . . . . . . . . . 11 ⊢ (𝑧 ∈ {∅} → 𝑧 = ∅)
46	45	adantl 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ {∅}) → 𝑧 = ∅)
47	46	fveq2d 6885	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ {∅}) → (𝑀‘𝑧) = (𝑀‘∅))
48	21	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ {∅}) → (𝑀‘∅) = 0)
49	47, 48	eqtrd 2796	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ {∅}) → (𝑀‘𝑧) = 0)
50	40, 41, 44, 49	esumpad 34411	. . . . . . 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 5136	. . . . . 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 3946	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran 𝑓 ⊆ (toCaraSiga‘𝑀))
53		ssexg 5289	. . . . . . . 8 ⊢ ((ran 𝑓 ⊆ (toCaraSiga‘𝑀) ∧ (toCaraSiga‘𝑀) ∈ V) → ran 𝑓 ∈ V)
54	52, 12, 53	sylancl 597	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran 𝑓 ∈ V)
55	19	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ ran 𝑓) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
56	36, 33	sstrd 3946	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran 𝑓 ⊆ 𝒫 𝑂)
57	56	sselda 3936	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ ran 𝑓) → 𝑧 ∈ 𝒫 𝑂)
58	55, 57	ffvelcdmd 7080	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ ran 𝑓) → (𝑀‘𝑧) ∈ (0[,]+∞))
59		simpr2 1212	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝐴 ⊆ ran 𝑓)
60	7, 54, 58, 59	esummono 34410	. . . . . 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 520	. . . . 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 13456	. . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ*
63	58	ralrimiva 3155	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∀𝑧 ∈ ran 𝑓(𝑀‘𝑧) ∈ (0[,]+∞))
64		nfcv 2923	. . . . . . . . 9 ⊢ Ⅎ𝑧ran 𝑓
65	64	esumcl 34386	. . . . . . . 8 ⊢ ((ran 𝑓 ∈ V ∧ ∀𝑧 ∈ ran 𝑓(𝑀‘𝑧) ∈ (0[,]+∞)) → Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) ∈ (0[,]+∞))
66	54, 63, 65	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) ∈ (0[,]+∞))
67	62, 66	sselid 3934	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) ∈ ℝ*)
68	44	ralrimiva 3155	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∀𝑧 ∈ 𝐴 (𝑀‘𝑧) ∈ (0[,]+∞))
69		nfcv 2923	. . . . . . . . 9 ⊢ Ⅎ𝑧𝐴
70	69	esumcl 34386	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ∀𝑧 ∈ 𝐴 (𝑀‘𝑧) ∈ (0[,]+∞)) → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ∈ (0[,]+∞))
71	40, 68, 70	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ∈ (0[,]+∞))
72	62, 71	sselid 3934	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ∈ ℝ*)
73		xrletri3 13178	. . . . . 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 595	. . . . 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 260	. . . 4 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) = Σ*𝑧 ∈ 𝐴(𝑀‘𝑧))
76		fveq2 6881	. . . . 5 ⊢ (𝑧 = (𝑓‘𝑘) → (𝑀‘𝑧) = (𝑀‘(𝑓‘𝑘)))
77		nnex 12238	. . . . . 6 ⊢ ℕ ∈ V
78	77	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ℕ ∈ V)
79	19	adantr 485	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
80	33	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
81	8	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → 𝑓:ℕ⟶(𝐴 ∪ {∅}))
82		simpr 489	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
83	81, 82	ffvelcdmd 7080	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ (𝐴 ∪ {∅}))
84	80, 83	sseldd 3937	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ 𝒫 𝑂)
85	79, 84	ffvelcdmd 7080	. . . . 5 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑀‘(𝑓‘𝑘)) ∈ (0[,]+∞))
86		simpr 489	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓‘𝑘) = ∅) → (𝑓‘𝑘) = ∅)
87	86	fveq2d 6885	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓‘𝑘) = ∅) → (𝑀‘(𝑓‘𝑘)) = (𝑀‘∅))
88	21	ad2antrr 738	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓‘𝑘) = ∅) → (𝑀‘∅) = 0)
89	87, 88	eqtrd 2796	. . . . 5 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓‘𝑘) = ∅) → (𝑀‘(𝑓‘𝑘)) = 0)
90		cnvimass 6084	. . . . . . 7 ⊢ (◡𝑓 “ 𝐴) ⊆ dom 𝑓
91	90, 8	fssdm 6725	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (◡𝑓 “ 𝐴) ⊆ ℕ)
92		ffun 6708	. . . . . . . . . . 11 ⊢ (𝑓:ℕ⟶(𝐴 ∪ {∅}) → Fun 𝑓)
93	8, 92	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Fun 𝑓)
94	93	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑓 “ 𝐴))) → Fun 𝑓)
95		difpreima 7060	. . . . . . . . . . . . 13 ⊢ (Fun 𝑓 → (◡𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) = ((◡𝑓 “ (𝐴 ∪ {∅})) ∖ (◡𝑓 “ 𝐴)))
96	8, 92, 95	3syl 19	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (◡𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) = ((◡𝑓 “ (𝐴 ∪ {∅})) ∖ (◡𝑓 “ 𝐴)))
97		fimacnv 6728	. . . . . . . . . . . . . 14 ⊢ (𝑓:ℕ⟶(𝐴 ∪ {∅}) → (◡𝑓 “ (𝐴 ∪ {∅})) = ℕ)
98	8, 97	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (◡𝑓 “ (𝐴 ∪ {∅})) = ℕ)
99	98	difeq1d 4079	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ((◡𝑓 “ (𝐴 ∪ {∅})) ∖ (◡𝑓 “ 𝐴)) = (ℕ ∖ (◡𝑓 “ 𝐴)))
100	96, 99	eqtrd 2796	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (◡𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) = (ℕ ∖ (◡𝑓 “ 𝐴)))
101		uncom 4111	. . . . . . . . . . . . . . . 16 ⊢ ({∅} ∪ 𝐴) = (𝐴 ∪ {∅})
102	101	difeq1i 4076	. . . . . . . . . . . . . . 15 ⊢ (({∅} ∪ 𝐴) ∖ 𝐴) = ((𝐴 ∪ {∅}) ∖ 𝐴)
103		difun2 4441	. . . . . . . . . . . . . . 15 ⊢ (({∅} ∪ 𝐴) ∖ 𝐴) = ({∅} ∖ 𝐴)
104	102, 103	eqtr3i 2786	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∪ {∅}) ∖ 𝐴) = ({∅} ∖ 𝐴)
105		difss 4089	. . . . . . . . . . . . . 14 ⊢ ({∅} ∖ 𝐴) ⊆ {∅}
106	104, 105	eqsstri 3982	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅}
107	106	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅})
108		sspreima 7063	. . . . . . . . . . . 12 ⊢ ((Fun 𝑓 ∧ ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅}) → (◡𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) ⊆ (◡𝑓 “ {∅}))
109	93, 107, 108	syl2anc 595	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (◡𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) ⊆ (◡𝑓 “ {∅}))
110	100, 109	eqsstrrd 3971	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (ℕ ∖ (◡𝑓 “ 𝐴)) ⊆ (◡𝑓 “ {∅}))
111	110	sselda 3936	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑓 “ 𝐴))) → 𝑘 ∈ (◡𝑓 “ {∅}))
112		fvimacnvi 7047	. . . . . . . . 9 ⊢ ((Fun 𝑓 ∧ 𝑘 ∈ (◡𝑓 “ {∅})) → (𝑓‘𝑘) ∈ {∅})
113	94, 111, 112	syl2anc 595	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑓 “ 𝐴))) → (𝑓‘𝑘) ∈ {∅})
114		elsni 4605	. . . . . . . 8 ⊢ ((𝑓‘𝑘) ∈ {∅} → (𝑓‘𝑘) = ∅)
115	113, 114	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑓 “ 𝐴))) → (𝑓‘𝑘) = ∅)
116	115	ralrimiva 3155	. . . . . 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 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Disj 𝑦 ∈ 𝐴 𝑦)
119		simpr3 1213	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Fun (◡𝑓 ↾ 𝐴))
120		fresf1o 32942	. . . . . . . . . 10 ⊢ ((Fun 𝑓 ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→𝐴)
121	93, 59, 119, 120	syl3anc 1396	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→𝐴)
122		simpr 489	. . . . . . . . 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 32902	. . . . . . . 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 6900	. . . . . . . . . 10 ⊢ (𝑘 ∈ (◡𝑓 “ 𝐴) → ((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑘) = (𝑓‘𝑘))
125	124	adantl 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ (◡𝑓 “ 𝐴)) → ((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑘) = (𝑓‘𝑘))
126	125	disjeq2dv 5080	. . . . . . . 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 284	. . . . . . 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 235	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Disj 𝑘 ∈ (◡𝑓 “ 𝐴)(𝑓‘𝑘))
129		disjss3 5107	. . . . . . 7 ⊢ (((◡𝑓 “ 𝐴) ⊆ ℕ ∧ ∀𝑘 ∈ (ℕ ∖ (◡𝑓 “ 𝐴))(𝑓‘𝑘) = ∅) → (Disj 𝑘 ∈ (◡𝑓 “ 𝐴)(𝑓‘𝑘) ↔ Disj 𝑘 ∈ ℕ (𝑓‘𝑘)))
130	129	biimpa 481	. . . . . 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 850	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Disj 𝑘 ∈ ℕ (𝑓‘𝑘))
132	76, 78, 85, 84, 89, 131	esumrnmpt2 34424	. . . 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 2805	. . 3 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑘 ∈ ℕ(𝑀‘(𝑓‘𝑘)) = Σ*𝑧 ∈ 𝐴(𝑀‘𝑧))
134		uniiun 5022	. . . . . . 7 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
135	28	sselda 3936	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝒫 𝑂)
136	39, 135	elpwiuncl 32839	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝑥 ∈ 𝒫 𝑂)
137	134, 136	eqeltrid 2865	. . . . . 6 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝒫 𝑂)
138	137	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∪ 𝐴 ∈ 𝒫 𝑂)
139	19, 138	ffvelcdmd 7080	. . . 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 1194	. . . . . . . . 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 6881	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑀‘𝑦) = (𝑀‘𝑧))
143		nfcv 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑧𝑥
144		nfcv 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝑥
145		nfcv 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑧(𝑀‘𝑦)
146		nfcv 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑀‘𝑧)
147	142, 143, 144, 145, 146	cbvesum 34398	. . . . . . . . 9 ⊢ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦) = Σ*𝑧 ∈ 𝑥(𝑀‘𝑧)
148	141, 147	breqtrdi 5151	. . . . . . . 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 6705	. . . . . . . . . 10 ⊢ (𝑓:ℕ⟶(𝐴 ∪ {∅}) → 𝑓 Fn ℕ)
150		fz1ssnn 13582	. . . . . . . . . . 11 ⊢ (1...𝑛) ⊆ ℕ
151		fnssres 6658	. . . . . . . . . . 11 ⊢ ((𝑓 Fn ℕ ∧ (1...𝑛) ⊆ ℕ) → (𝑓 ↾ (1...𝑛)) Fn (1...𝑛))
152	150, 151	mpan2 703	. . . . . . . . . 10 ⊢ (𝑓 Fn ℕ → (𝑓 ↾ (1...𝑛)) Fn (1...𝑛))
153	8, 149, 152	3syl 19	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝑓 ↾ (1...𝑛)) Fn (1...𝑛))
154		fzfi 14007	. . . . . . . . . 10 ⊢ (1...𝑛) ∈ Fin
155		fnfi 9161	. . . . . . . . . 10 ⊢ (((𝑓 ↾ (1...𝑛)) Fn (1...𝑛) ∧ (1...𝑛) ∈ Fin) → (𝑓 ↾ (1...𝑛)) ∈ Fin)
156	154, 155	mpan2 703	. . . . . . . . 9 ⊢ ((𝑓 ↾ (1...𝑛)) Fn (1...𝑛) → (𝑓 ↾ (1...𝑛)) ∈ Fin)
157		rnfi 9296	. . . . . . . . 9 ⊢ ((𝑓 ↾ (1...𝑛)) ∈ Fin → ran (𝑓 ↾ (1...𝑛)) ∈ Fin)
158	153, 156, 157	3syl 19	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran (𝑓 ↾ (1...𝑛)) ∈ Fin)
159		resss 6000	. . . . . . . . . . 11 ⊢ (𝑓 ↾ (1...𝑛)) ⊆ 𝑓
160		rnss 5929	. . . . . . . . . . 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 3946	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ (toCaraSiga‘𝑀))
164	162, 36	sstrd 3946	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ (𝐴 ∪ {∅}))
165		nfcv 2923	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧𝑦
166		nfcv 2923	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦𝑧
167		id 23	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
168	165, 166, 167	cbvdisj 5085	. . . . . . . . . . . 12 ⊢ (Disj 𝑦 ∈ 𝐴 𝑦 ↔ Disj 𝑧 ∈ 𝐴 𝑧)
169		disjun0 32906	. . . . . . . . . . . 12 ⊢ (Disj 𝑧 ∈ 𝐴 𝑧 → Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
170	168, 169	sylbi 220	. . . . . . . . . . 11 ⊢ (Disj 𝑦 ∈ 𝐴 𝑦 → Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
171	117, 170	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
172	171	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
173		disjss1 5081	. . . . . . . . 9 ⊢ (ran (𝑓 ↾ (1...𝑛)) ⊆ (𝐴 ∪ {∅}) → (Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧 → Disj 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))𝑧))
174	164, 172, 173	sylc 66	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Disj 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))𝑧)
175		pwidg 4581	. . . . . . . . 9 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ 𝒫 𝑂)
176	17, 175	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝑂 ∈ 𝒫 𝑂)
177	17, 19, 21, 148, 158, 163, 174, 176	carsgclctunlem1 34673	. . . . . . 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 485	. . . . . 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 4881	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∪ ran (𝑓 ↾ (1...𝑛)) ⊆ ∪ (𝐴 ∪ {∅}))
180		uniun 4894	. . . . . . . . . . . 12 ⊢ ∪ (𝐴 ∪ {∅}) = (∪ 𝐴 ∪ ∪ {∅})
181	2	unisn 4890	. . . . . . . . . . . . 13 ⊢ ∪ {∅} = ∅
182	181	uneq2i 4118	. . . . . . . . . . . 12 ⊢ (∪ 𝐴 ∪ ∪ {∅}) = (∪ 𝐴 ∪ ∅)
183		un0 4350	. . . . . . . . . . . 12 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴
184	180, 182, 183	3eqtri 2788	. . . . . . . . . . 11 ⊢ ∪ (𝐴 ∪ {∅}) = ∪ 𝐴
185	179, 184	sseqtrdi 3976	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∪ ran (𝑓 ↾ (1...𝑛)) ⊆ ∪ 𝐴)
186	185	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → ∪ ran (𝑓 ↾ (1...𝑛)) ⊆ ∪ 𝐴)
187		uniss 4879	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝒫 𝑂 → ∪ 𝐴 ⊆ ∪ 𝒫 𝑂)
188		unipw 5431	. . . . . . . . . . . 12 ⊢ ∪ 𝒫 𝑂 = 𝑂
189	187, 188	sseqtrdi 3976	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝒫 𝑂 → ∪ 𝐴 ⊆ 𝑂)
190	28, 189	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝐴 ⊆ 𝑂)
191	190	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → ∪ 𝐴 ⊆ 𝑂)
192	186, 191	sstrd 3946	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → ∪ ran (𝑓 ↾ (1...𝑛)) ⊆ 𝑂)
193		sseqin2 4175	. . . . . . . 8 ⊢ (∪ ran (𝑓 ↾ (1...𝑛)) ⊆ 𝑂 ↔ (𝑂 ∩ ∪ ran (𝑓 ↾ (1...𝑛))) = ∪ ran (𝑓 ↾ (1...𝑛)))
194	192, 193	sylib 221	. . . . . . 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 6885	. . . . . 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 1942	. . . . . . . 8 ⊢ Ⅎ𝑧((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ)
197	164	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑓 ↾ (1...𝑛)) ⊆ (𝐴 ∪ {∅}))
198	28	ad2antrr 738	. . . . . . . . . . . . . . 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 4751	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → {∅} ⊆ 𝒫 𝑂)
201	198, 200	unssd 4144	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
202	197, 201	sstrd 3946	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑓 ↾ (1...𝑛)) ⊆ 𝒫 𝑂)
203	202	sselda 3936	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → 𝑧 ∈ 𝒫 𝑂)
204	203	elpwid 4570	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → 𝑧 ⊆ 𝑂)
205		sseqin2 4175	. . . . . . . . . . 11 ⊢ (𝑧 ⊆ 𝑂 ↔ (𝑂 ∩ 𝑧) = 𝑧)
206	204, 205	sylib 221	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → (𝑂 ∩ 𝑧) = 𝑧)
207	206	fveq2d 6885	. . . . . . . . 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 3155	. . . . . . . 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 34393	. . . . . . 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 5977	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝑓 ↾ (1...𝑛)) = ((𝑘 ∈ ℕ ↦ (𝑓‘𝑘)) ↾ (1...𝑛)))
211	210	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑓 ↾ (1...𝑛)) = ((𝑘 ∈ ℕ ↦ (𝑓‘𝑘)) ↾ (1...𝑛)))
212		resmpt 6039	. . . . . . . . . . . 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	eqtrdi 2812	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑓 ↾ (1...𝑛)) = (𝑘 ∈ (1...𝑛) ↦ (𝑓‘𝑘)))
215	214	eqcomd 2767	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ (1...𝑛) ↦ (𝑓‘𝑘)) = (𝑓 ↾ (1...𝑛)))
216	215	rneqd 5928	. . . . . . . 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 34391	. . . . . . 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 738	. . . . . . . . 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 3936	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
222	84	adantlr 727	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ 𝒫 𝑂)
223	221, 222	syldan 602	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑓‘𝑘) ∈ 𝒫 𝑂)
224	219, 223	ffvelcdmd 7080	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑀‘(𝑓‘𝑘)) ∈ (0[,]+∞))
225		simpr 489	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓‘𝑘) = ∅) → (𝑓‘𝑘) = ∅)
226	225	fveq2d 6885	. . . . . . . . 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 742	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓‘𝑘) = ∅) → (𝑀‘∅) = 0)
228	226, 227	eqtrd 2796	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓‘𝑘) = ∅) → (𝑀‘(𝑓‘𝑘)) = 0)
229		disjss1 5081	. . . . . . . . . . 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 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Disj 𝑘 ∈ (1...𝑛)(𝑓‘𝑘))
232	231	adantr 485	. . . . . . . 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 34424	. . . . . . 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 2802	. . . . . 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 2804	. . . . 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 1194	. . . . . . 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 34670	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝑀‘∪ ran (𝑓 ↾ (1...𝑛))) ≤ (𝑀‘∪ 𝐴))
239	238	adantr 485	. . . . 5 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑀‘∪ ran (𝑓 ↾ (1...𝑛))) ≤ (𝑀‘∪ 𝐴))
240	235, 239	eqbrtrrd 5134	. . . 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 34446	. . 3 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑘 ∈ ℕ(𝑀‘(𝑓‘𝑘)) ≤ (𝑀‘∪ 𝐴))
242	133, 241	eqbrtrrd 5134	. 2 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ≤ (𝑀‘∪ 𝐴))
243	6, 242	exlimddv 1963	1 ⊢ (𝜑 → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ≤ (𝑀‘∪ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568  ∃wex 1807   ∈ wcel 2141  ∀wral 3077  Vcvv 3453   ∖ cdif 3901   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4561  {csn 4588  ∪ cuni 4871  ∪ ciun 4955  Disj wdisj 5075   class class class wbr 5108   ↦ cmpt 5191  ^◡ccnv 5660  ran crn 5662   ↾ cres 5663   “ cima 5664  Fun wfun 6530   Fn wfn 6531  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410  ωcom 7861   ≼ cdom 8940  Fincfn 8942  0cc0 11099  1c1 11100  +∞cpnf 11239  ℝ^*cxr 11241   ≤ cle 11243  ℕcn 12232  [,]cicc 13374  ...cfz 13534  Σ^*cesum 34383  toCaraSigaccarsg 34657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9609  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177  ax-addf 11178  ax-mulf 11179

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-disj 5076  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-om 7862  df-1st 7985  df-2nd 7986  df-supp 8156  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-er 8693  df-map 8825  df-pm 8826  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fsupp 9321  df-fi 9370  df-sup 9401  df-inf 9402  df-oi 9471  df-dju 9886  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-7 12307  df-8 12308  df-9 12309  df-n0 12504  df-z 12591  df-dec 12711  df-uz 12862  df-q 12972  df-rp 13016  df-xneg 13136  df-xadd 13137  df-xmul 13138  df-ioo 13375  df-ioc 13376  df-ico 13377  df-icc 13378  df-fz 13535  df-fzo 13682  df-fl 13824  df-mod 13902  df-seq 14037  df-exp 14097  df-fac 14309  df-bc 14338  df-hash 14366  df-shft 15103  df-cj 15149  df-re 15150  df-im 15151  df-sqrt 15285  df-abs 15286  df-limsup 15521  df-clim 15538  df-rlim 15539  df-sum 15737  df-ef 16120  df-sin 16122  df-cos 16123  df-pi 16125  df-struct 17206  df-sets 17223  df-slot 17241  df-ndx 17253  df-base 17269  df-ress 17290  df-plusg 17322  df-mulr 17323  df-starv 17324  df-sca 17325  df-vsca 17326  df-ip 17327  df-tset 17328  df-ple 17329  df-ds 17331  df-unif 17332  df-hom 17333  df-cco 17334  df-rest 17474  df-topn 17475  df-0g 17493  df-gsum 17494  df-topgen 17495  df-pt 17496  df-prds 17499  df-ordt 17554  df-xrs 17555  df-qtop 17560  df-imas 17561  df-xps 17563  df-mre 17637  df-mrc 17638  df-acs 17640  df-ps 18621  df-tsr 18622  df-plusf 18696  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-mhm 18840  df-submnd 18841  df-grp 19002  df-minusg 19003  df-sbg 19004  df-mulg 19133  df-subg 19188  df-cntz 19386  df-cmn 19851  df-abl 19852  df-mgp 20216  df-rng 20230  df-ur 20263  df-ring 20316  df-cring 20317  df-subrng 20630  df-subrg 20654  df-abv 20891  df-lmod 20962  df-scaf 20963  df-sra 21273  df-rgmod 21274  df-psmet 21493  df-xmet 21494  df-met 21495  df-bl 21496  df-mopn 21497  df-fbas 21498  df-fg 21499  df-cnfld 21502  df-top 23030  df-topon 23047  df-topsp 23069  df-bases 23082  df-cld 23155  df-ntr 23156  df-cls 23157  df-nei 23234  df-lp 23272  df-perf 23273  df-cn 23363  df-cnp 23364  df-haus 23451  df-tx 23698  df-hmeo 23891  df-fil 23982  df-fm 24074  df-flim 24075  df-flf 24076  df-tmd 24208  df-tgp 24209  df-tsms 24263  df-trg 24296  df-xms 24456  df-ms 24457  df-tms 24458  df-nm 24718  df-ngp 24719  df-nrg 24721  df-nlm 24722  df-ii 25015  df-cncf 25016  df-limc 26004  df-dv 26005  df-log 26697  df-esum 34384  df-carsg 34658

This theorem is referenced by:  omsmeas  34679