Theorem carsggect 31951

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 5185	. . . 4 ⊢ ∅ ∈ V
3	2	a1i 11	. . 3 ⊢ (𝜑 → ∅ ∈ V)
4		carsggect.0	. . 3 ⊢ (𝜑 → ¬ ∅ ∈ 𝐴)
5		padct 30728	. . 3 ⊢ ((𝐴 ≼ ω ∧ ∅ ∈ V ∧ ¬ ∅ ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)))
6	1, 3, 4, 5	syl3anc 1373	. 2 ⊢ (𝜑 → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)))
7		nfv 1922	. . . . 5 ⊢ Ⅎ𝑧(𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)))
8		simpr1 1196	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝑓:ℕ⟶(𝐴 ∪ {∅}))
9	8	feqmptd 6758	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝑓 = (𝑘 ∈ ℕ ↦ (𝑓‘𝑘)))
10	9	rneqd 5792	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran 𝑓 = ran (𝑘 ∈ ℕ ↦ (𝑓‘𝑘)))
11	7, 10	esumeq1d 31669	. . . 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 6708	. . . . . . . . . 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 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝐴 ⊆ (toCaraSiga‘𝑀))
16		carsgval.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑂 ∈ 𝑉)
17	16	adantr 484	. . . . . . . . . . . 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 484	. . . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝑀‘∅) = 0)
22	17, 19, 21	0elcarsg 31940	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∅ ∈ (toCaraSiga‘𝑀))
23	22	snssd 4708	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → {∅} ⊆ (toCaraSiga‘𝑀))
24	15, 23	unssd 4086	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝐴 ∪ {∅}) ⊆ (toCaraSiga‘𝑀))
25	13, 24	ssexd 5202	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝐴 ∪ {∅}) ∈ V)
26	19	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
27	16, 18	carsgcl 31937	. . . . . . . . . . . . 13 ⊢ (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂)
28	14, 27	sstrd 3897	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ 𝒫 𝑂)
29	28	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝐴 ⊆ 𝒫 𝑂)
30		0elpw 5232	. . . . . . . . . . . . 13 ⊢ ∅ ∈ 𝒫 𝑂
31	30	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∅ ∈ 𝒫 𝑂)
32	31	snssd 4708	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → {∅} ⊆ 𝒫 𝑂)
33	29, 32	unssd 4086	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
34	33	sselda 3887	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → 𝑧 ∈ 𝒫 𝑂)
35	26, 34	ffvelrnd 6883	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → (𝑀‘𝑧) ∈ (0[,]+∞))
36	8	frnd 6531	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran 𝑓 ⊆ (𝐴 ∪ {∅}))
37	7, 25, 35, 36	esummono 31688	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) ≤ Σ*𝑧 ∈ (𝐴 ∪ {∅})(𝑀‘𝑧))
38		ctex 8621	. . . . . . . . . 10 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)
39	1, 38	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ V)
40	39	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝐴 ∈ V)
41	13, 23	ssexd 5202	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → {∅} ∈ V)
42	19	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ 𝐴) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
43	29	sselda 3887	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝒫 𝑂)
44	42, 43	ffvelrnd 6883	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ 𝐴) → (𝑀‘𝑧) ∈ (0[,]+∞))
45		elsni 4544	. . . . . . . . . . 11 ⊢ (𝑧 ∈ {∅} → 𝑧 = ∅)
46	45	adantl 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ {∅}) → 𝑧 = ∅)
47	46	fveq2d 6699	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ {∅}) → (𝑀‘𝑧) = (𝑀‘∅))
48	21	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ {∅}) → (𝑀‘∅) = 0)
49	47, 48	eqtrd 2771	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ {∅}) → (𝑀‘𝑧) = 0)
50	40, 41, 44, 49	esumpad 31689	. . . . . . 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 5065	. . . . . 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 3897	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran 𝑓 ⊆ (toCaraSiga‘𝑀))
53		ssexg 5201	. . . . . . . 8 ⊢ ((ran 𝑓 ⊆ (toCaraSiga‘𝑀) ∧ (toCaraSiga‘𝑀) ∈ V) → ran 𝑓 ∈ V)
54	52, 12, 53	sylancl 589	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran 𝑓 ∈ V)
55	19	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ ran 𝑓) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
56	36, 33	sstrd 3897	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran 𝑓 ⊆ 𝒫 𝑂)
57	56	sselda 3887	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ ran 𝑓) → 𝑧 ∈ 𝒫 𝑂)
58	55, 57	ffvelrnd 6883	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ ran 𝑓) → (𝑀‘𝑧) ∈ (0[,]+∞))
59		simpr2 1197	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝐴 ⊆ ran 𝑓)
60	7, 54, 58, 59	esummono 31688	. . . . . 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 515	. . . . 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 12983	. . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ*
63	58	ralrimiva 3095	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∀𝑧 ∈ ran 𝑓(𝑀‘𝑧) ∈ (0[,]+∞))
64		nfcv 2897	. . . . . . . . 9 ⊢ Ⅎ𝑧ran 𝑓
65	64	esumcl 31664	. . . . . . . 8 ⊢ ((ran 𝑓 ∈ V ∧ ∀𝑧 ∈ ran 𝑓(𝑀‘𝑧) ∈ (0[,]+∞)) → Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) ∈ (0[,]+∞))
66	54, 63, 65	syl2anc 587	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) ∈ (0[,]+∞))
67	62, 66	sseldi 3885	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) ∈ ℝ*)
68	44	ralrimiva 3095	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∀𝑧 ∈ 𝐴 (𝑀‘𝑧) ∈ (0[,]+∞))
69		nfcv 2897	. . . . . . . . 9 ⊢ Ⅎ𝑧𝐴
70	69	esumcl 31664	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ∀𝑧 ∈ 𝐴 (𝑀‘𝑧) ∈ (0[,]+∞)) → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ∈ (0[,]+∞))
71	40, 68, 70	syl2anc 587	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ∈ (0[,]+∞))
72	62, 71	sseldi 3885	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ∈ ℝ*)
73		xrletri3 12709	. . . . . 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 587	. . . . 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 6695	. . . . 5 ⊢ (𝑧 = (𝑓‘𝑘) → (𝑀‘𝑧) = (𝑀‘(𝑓‘𝑘)))
77		nnex 11801	. . . . . 6 ⊢ ℕ ∈ V
78	77	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ℕ ∈ V)
79	19	adantr 484	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
80	33	adantr 484	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
81	8	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → 𝑓:ℕ⟶(𝐴 ∪ {∅}))
82		simpr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
83	81, 82	ffvelrnd 6883	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ (𝐴 ∪ {∅}))
84	80, 83	sseldd 3888	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ 𝒫 𝑂)
85	79, 84	ffvelrnd 6883	. . . . 5 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑀‘(𝑓‘𝑘)) ∈ (0[,]+∞))
86		simpr 488	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓‘𝑘) = ∅) → (𝑓‘𝑘) = ∅)
87	86	fveq2d 6699	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓‘𝑘) = ∅) → (𝑀‘(𝑓‘𝑘)) = (𝑀‘∅))
88	21	ad2antrr 726	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓‘𝑘) = ∅) → (𝑀‘∅) = 0)
89	87, 88	eqtrd 2771	. . . . 5 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓‘𝑘) = ∅) → (𝑀‘(𝑓‘𝑘)) = 0)
90		cnvimass 5934	. . . . . . 7 ⊢ (◡𝑓 “ 𝐴) ⊆ dom 𝑓
91	90, 8	fssdm 6543	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (◡𝑓 “ 𝐴) ⊆ ℕ)
92		ffun 6526	. . . . . . . . . . 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 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑓 “ 𝐴))) → Fun 𝑓)
95		difpreima 6863	. . . . . . . . . . . . 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 6545	. . . . . . . . . . . . . 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 4022	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ((◡𝑓 “ (𝐴 ∪ {∅})) ∖ (◡𝑓 “ 𝐴)) = (ℕ ∖ (◡𝑓 “ 𝐴)))
100	96, 99	eqtrd 2771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (◡𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) = (ℕ ∖ (◡𝑓 “ 𝐴)))
101		uncom 4053	. . . . . . . . . . . . . . . 16 ⊢ ({∅} ∪ 𝐴) = (𝐴 ∪ {∅})
102	101	difeq1i 4019	. . . . . . . . . . . . . . 15 ⊢ (({∅} ∪ 𝐴) ∖ 𝐴) = ((𝐴 ∪ {∅}) ∖ 𝐴)
103		difun2 4381	. . . . . . . . . . . . . . 15 ⊢ (({∅} ∪ 𝐴) ∖ 𝐴) = ({∅} ∖ 𝐴)
104	102, 103	eqtr3i 2761	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∪ {∅}) ∖ 𝐴) = ({∅} ∖ 𝐴)
105		difss 4032	. . . . . . . . . . . . . 14 ⊢ ({∅} ∖ 𝐴) ⊆ {∅}
106	104, 105	eqsstri 3921	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅}
107	106	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅})
108		sspreima 6866	. . . . . . . . . . . 12 ⊢ ((Fun 𝑓 ∧ ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅}) → (◡𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) ⊆ (◡𝑓 “ {∅}))
109	93, 107, 108	syl2anc 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (◡𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) ⊆ (◡𝑓 “ {∅}))
110	100, 109	eqsstrrd 3926	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (ℕ ∖ (◡𝑓 “ 𝐴)) ⊆ (◡𝑓 “ {∅}))
111	110	sselda 3887	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑓 “ 𝐴))) → 𝑘 ∈ (◡𝑓 “ {∅}))
112		fvimacnvi 6850	. . . . . . . . 9 ⊢ ((Fun 𝑓 ∧ 𝑘 ∈ (◡𝑓 “ {∅})) → (𝑓‘𝑘) ∈ {∅})
113	94, 111, 112	syl2anc 587	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑓 “ 𝐴))) → (𝑓‘𝑘) ∈ {∅})
114		elsni 4544	. . . . . . . 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 3095	. . . . . 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Disj 𝑦 ∈ 𝐴 𝑦)
119		simpr3 1198	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Fun (◡𝑓 ↾ 𝐴))
120		fresf1o 30639	. . . . . . . . . 10 ⊢ ((Fun 𝑓 ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→𝐴)
121	93, 59, 119, 120	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→𝐴)
122		simpr 488	. . . . . . . . 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 30603	. . . . . . . 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 6714	. . . . . . . . . 10 ⊢ (𝑘 ∈ (◡𝑓 “ 𝐴) → ((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑘) = (𝑓‘𝑘))
125	124	adantl 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ (◡𝑓 “ 𝐴)) → ((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑘) = (𝑓‘𝑘))
126	125	disjeq2dv 5009	. . . . . . . 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 5038	. . . . . . 7 ⊢ (((◡𝑓 “ 𝐴) ⊆ ℕ ∧ ∀𝑘 ∈ (ℕ ∖ (◡𝑓 “ 𝐴))(𝑓‘𝑘) = ∅) → (Disj 𝑘 ∈ (◡𝑓 “ 𝐴)(𝑓‘𝑘) ↔ Disj 𝑘 ∈ ℕ (𝑓‘𝑘)))
130	129	biimpa 480	. . . . . 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 838	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Disj 𝑘 ∈ ℕ (𝑓‘𝑘))
132	76, 78, 85, 84, 89, 131	esumrnmpt2 31702	. . . 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 2780	. . 3 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑘 ∈ ℕ(𝑀‘(𝑓‘𝑘)) = Σ*𝑧 ∈ 𝐴(𝑀‘𝑧))
134		uniiun 4953	. . . . . . 7 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
135	28	sselda 3887	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝒫 𝑂)
136	39, 135	elpwiuncl 30549	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝑥 ∈ 𝒫 𝑂)
137	134, 136	eqeltrid 2835	. . . . . 6 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝒫 𝑂)
138	137	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∪ 𝐴 ∈ 𝒫 𝑂)
139	19, 138	ffvelrnd 6883	. . . 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 1179	. . . . . . . . 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 6695	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑀‘𝑦) = (𝑀‘𝑧))
143		nfcv 2897	. . . . . . . . . 10 ⊢ Ⅎ𝑧𝑥
144		nfcv 2897	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝑥
145		nfcv 2897	. . . . . . . . . 10 ⊢ Ⅎ𝑧(𝑀‘𝑦)
146		nfcv 2897	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑀‘𝑧)
147	142, 143, 144, 145, 146	cbvesum 31676	. . . . . . . . 9 ⊢ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦) = Σ*𝑧 ∈ 𝑥(𝑀‘𝑧)
148	141, 147	breqtrdi 5080	. . . . . . . 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 6523	. . . . . . . . . 10 ⊢ (𝑓:ℕ⟶(𝐴 ∪ {∅}) → 𝑓 Fn ℕ)
150		fz1ssnn 13108	. . . . . . . . . . 11 ⊢ (1...𝑛) ⊆ ℕ
151		fnssres 6478	. . . . . . . . . . 11 ⊢ ((𝑓 Fn ℕ ∧ (1...𝑛) ⊆ ℕ) → (𝑓 ↾ (1...𝑛)) Fn (1...𝑛))
152	150, 151	mpan2 691	. . . . . . . . . 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 13510	. . . . . . . . . 10 ⊢ (1...𝑛) ∈ Fin
155		fnfi 8835	. . . . . . . . . 10 ⊢ (((𝑓 ↾ (1...𝑛)) Fn (1...𝑛) ∧ (1...𝑛) ∈ Fin) → (𝑓 ↾ (1...𝑛)) ∈ Fin)
156	154, 155	mpan2 691	. . . . . . . . 9 ⊢ ((𝑓 ↾ (1...𝑛)) Fn (1...𝑛) → (𝑓 ↾ (1...𝑛)) ∈ Fin)
157		rnfi 8937	. . . . . . . . 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 5861	. . . . . . . . . . 11 ⊢ (𝑓 ↾ (1...𝑛)) ⊆ 𝑓
160		rnss 5793	. . . . . . . . . . 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 3897	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ (toCaraSiga‘𝑀))
164	162, 36	sstrd 3897	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ (𝐴 ∪ {∅}))
165		nfcv 2897	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧𝑦
166		nfcv 2897	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦𝑧
167		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
168	165, 166, 167	cbvdisj 5014	. . . . . . . . . . . 12 ⊢ (Disj 𝑦 ∈ 𝐴 𝑦 ↔ Disj 𝑧 ∈ 𝐴 𝑧)
169		disjun0 30607	. . . . . . . . . . . 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 17	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
172	171	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
173		disjss1 5010	. . . . . . . . 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 4521	. . . . . . . . 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 31950	. . . . . . 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 484	. . . . . 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 4815	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∪ ran (𝑓 ↾ (1...𝑛)) ⊆ ∪ (𝐴 ∪ {∅}))
180		uniun 4830	. . . . . . . . . . . 12 ⊢ ∪ (𝐴 ∪ {∅}) = (∪ 𝐴 ∪ ∪ {∅})
181	2	unisn 4827	. . . . . . . . . . . . 13 ⊢ ∪ {∅} = ∅
182	181	uneq2i 4060	. . . . . . . . . . . 12 ⊢ (∪ 𝐴 ∪ ∪ {∅}) = (∪ 𝐴 ∪ ∅)
183		un0 4291	. . . . . . . . . . . 12 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴
184	180, 182, 183	3eqtri 2763	. . . . . . . . . . 11 ⊢ ∪ (𝐴 ∪ {∅}) = ∪ 𝐴
185	179, 184	sseqtrdi 3937	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∪ ran (𝑓 ↾ (1...𝑛)) ⊆ ∪ 𝐴)
186	185	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → ∪ ran (𝑓 ↾ (1...𝑛)) ⊆ ∪ 𝐴)
187		uniss 4813	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝒫 𝑂 → ∪ 𝐴 ⊆ ∪ 𝒫 𝑂)
188		unipw 5320	. . . . . . . . . . . 12 ⊢ ∪ 𝒫 𝑂 = 𝑂
189	187, 188	sseqtrdi 3937	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝒫 𝑂 → ∪ 𝐴 ⊆ 𝑂)
190	28, 189	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝐴 ⊆ 𝑂)
191	190	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → ∪ 𝐴 ⊆ 𝑂)
192	186, 191	sstrd 3897	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → ∪ ran (𝑓 ↾ (1...𝑛)) ⊆ 𝑂)
193		sseqin2 4116	. . . . . . . 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 6699	. . . . . 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 1922	. . . . . . . 8 ⊢ Ⅎ𝑧((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ)
197	164	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑓 ↾ (1...𝑛)) ⊆ (𝐴 ∪ {∅}))
198	28	ad2antrr 726	. . . . . . . . . . . . . . 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 4708	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → {∅} ⊆ 𝒫 𝑂)
201	198, 200	unssd 4086	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
202	197, 201	sstrd 3897	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑓 ↾ (1...𝑛)) ⊆ 𝒫 𝑂)
203	202	sselda 3887	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → 𝑧 ∈ 𝒫 𝑂)
204	203	elpwid 4510	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → 𝑧 ⊆ 𝑂)
205		sseqin2 4116	. . . . . . . . . . 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 6699	. . . . . . . . 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 3095	. . . . . . . 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 31671	. . . . . . 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 5835	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝑓 ↾ (1...𝑛)) = ((𝑘 ∈ ℕ ↦ (𝑓‘𝑘)) ↾ (1...𝑛)))
211	210	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑓 ↾ (1...𝑛)) = ((𝑘 ∈ ℕ ↦ (𝑓‘𝑘)) ↾ (1...𝑛)))
212		resmpt 5890	. . . . . . . . . . . 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 2787	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑓 ↾ (1...𝑛)) = (𝑘 ∈ (1...𝑛) ↦ (𝑓‘𝑘)))
215	214	eqcomd 2742	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ (1...𝑛) ↦ (𝑓‘𝑘)) = (𝑓 ↾ (1...𝑛)))
216	215	rneqd 5792	. . . . . . . 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 31669	. . . . . . 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 726	. . . . . . . . 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 3887	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
222	84	adantlr 715	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ 𝒫 𝑂)
223	221, 222	syldan 594	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑓‘𝑘) ∈ 𝒫 𝑂)
224	219, 223	ffvelrnd 6883	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑀‘(𝑓‘𝑘)) ∈ (0[,]+∞))
225		simpr 488	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓‘𝑘) = ∅) → (𝑓‘𝑘) = ∅)
226	225	fveq2d 6699	. . . . . . . . 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 730	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓‘𝑘) = ∅) → (𝑀‘∅) = 0)
228	226, 227	eqtrd 2771	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓‘𝑘) = ∅) → (𝑀‘(𝑓‘𝑘)) = 0)
229		disjss1 5010	. . . . . . . . . . 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 484	. . . . . . . 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 31702	. . . . . . 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 2777	. . . . . 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 2779	. . . . 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 1179	. . . . . . 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 31947	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝑀‘∪ ran (𝑓 ↾ (1...𝑛))) ≤ (𝑀‘∪ 𝐴))
239	238	adantr 484	. . . . 5 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑀‘∪ ran (𝑓 ↾ (1...𝑛))) ≤ (𝑀‘∪ 𝐴))
240	235, 239	eqbrtrrd 5063	. . . 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 31724	. . 3 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑘 ∈ ℕ(𝑀‘(𝑓‘𝑘)) ≤ (𝑀‘∪ 𝐴))
242	133, 241	eqbrtrrd 5063	. 2 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ≤ (𝑀‘∪ 𝐴))
243	6, 242	exlimddv 1943	1 ⊢ (𝜑 → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ≤ (𝑀‘∪ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543  ∃wex 1787   ∈ wcel 2112  ∀wral 3051  Vcvv 3398   ∖ cdif 3850   ∪ cun 3851   ∩ cin 3852   ⊆ wss 3853  ∅c0 4223  𝒫 cpw 4499  {csn 4527  ∪ cuni 4805  ∪ ciun 4890  Disj wdisj 5004   class class class wbr 5039   ↦ cmpt 5120  ^◡ccnv 5535  ran crn 5537   ↾ cres 5538   “ cima 5539  Fun wfun 6352   Fn wfn 6353  ⟶wf 6354  –1-1-onto→wf1o 6357  ‘cfv 6358  (class class class)co 7191  ωcom 7622   ≼ cdom 8602  Fincfn 8604  0cc0 10694  1c1 10695  +∞cpnf 10829  ℝ^*cxr 10831   ≤ cle 10833  ℕcn 11795  [,]cicc 12903  ...cfz 13060  Σ^*cesum 31661  toCaraSigaccarsg 31934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501  ax-inf2 9234  ax-cnex 10750  ax-resscn 10751  ax-1cn 10752  ax-icn 10753  ax-addcl 10754  ax-addrcl 10755  ax-mulcl 10756  ax-mulrcl 10757  ax-mulcom 10758  ax-addass 10759  ax-mulass 10760  ax-distr 10761  ax-i2m1 10762  ax-1ne0 10763  ax-1rid 10764  ax-rnegex 10765  ax-rrecex 10766  ax-cnre 10767  ax-pre-lttri 10768  ax-pre-lttrn 10769  ax-pre-ltadd 10770  ax-pre-mulgt0 10771  ax-pre-sup 10772  ax-addf 10773  ax-mulf 10774

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-int 4846  df-iun 4892  df-iin 4893  df-disj 5005  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-se 5495  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-isom 6367  df-riota 7148  df-ov 7194  df-oprab 7195  df-mpo 7196  df-of 7447  df-om 7623  df-1st 7739  df-2nd 7740  df-supp 7882  df-wrecs 8025  df-recs 8086  df-rdg 8124  df-1o 8180  df-2o 8181  df-er 8369  df-map 8488  df-pm 8489  df-ixp 8557  df-en 8605  df-dom 8606  df-sdom 8607  df-fin 8608  df-fsupp 8964  df-fi 9005  df-sup 9036  df-inf 9037  df-oi 9104  df-dju 9482  df-card 9520  df-pnf 10834  df-mnf 10835  df-xr 10836  df-ltxr 10837  df-le 10838  df-sub 11029  df-neg 11030  df-div 11455  df-nn 11796  df-2 11858  df-3 11859  df-4 11860  df-5 11861  df-6 11862  df-7 11863  df-8 11864  df-9 11865  df-n0 12056  df-z 12142  df-dec 12259  df-uz 12404  df-q 12510  df-rp 12552  df-xneg 12669  df-xadd 12670  df-xmul 12671  df-ioo 12904  df-ioc 12905  df-ico 12906  df-icc 12907  df-fz 13061  df-fzo 13204  df-fl 13332  df-mod 13408  df-seq 13540  df-exp 13601  df-fac 13805  df-bc 13834  df-hash 13862  df-shft 14595  df-cj 14627  df-re 14628  df-im 14629  df-sqrt 14763  df-abs 14764  df-limsup 14997  df-clim 15014  df-rlim 15015  df-sum 15215  df-ef 15592  df-sin 15594  df-cos 15595  df-pi 15597  df-struct 16668  df-ndx 16669  df-slot 16670  df-base 16672  df-sets 16673  df-ress 16674  df-plusg 16762  df-mulr 16763  df-starv 16764  df-sca 16765  df-vsca 16766  df-ip 16767  df-tset 16768  df-ple 16769  df-ds 16771  df-unif 16772  df-hom 16773  df-cco 16774  df-rest 16881  df-topn 16882  df-0g 16900  df-gsum 16901  df-topgen 16902  df-pt 16903  df-prds 16906  df-ordt 16960  df-xrs 16961  df-qtop 16966  df-imas 16967  df-xps 16969  df-mre 17043  df-mrc 17044  df-acs 17046  df-ps 18026  df-tsr 18027  df-plusf 18067  df-mgm 18068  df-sgrp 18117  df-mnd 18128  df-mhm 18172  df-submnd 18173  df-grp 18322  df-minusg 18323  df-sbg 18324  df-mulg 18443  df-subg 18494  df-cntz 18665  df-cmn 19126  df-abl 19127  df-mgp 19459  df-ur 19471  df-ring 19518  df-cring 19519  df-subrg 19752  df-abv 19807  df-lmod 19855  df-scaf 19856  df-sra 20163  df-rgmod 20164  df-psmet 20309  df-xmet 20310  df-met 20311  df-bl 20312  df-mopn 20313  df-fbas 20314  df-fg 20315  df-cnfld 20318  df-top 21745  df-topon 21762  df-topsp 21784  df-bases 21797  df-cld 21870  df-ntr 21871  df-cls 21872  df-nei 21949  df-lp 21987  df-perf 21988  df-cn 22078  df-cnp 22079  df-haus 22166  df-tx 22413  df-hmeo 22606  df-fil 22697  df-fm 22789  df-flim 22790  df-flf 22791  df-tmd 22923  df-tgp 22924  df-tsms 22978  df-trg 23011  df-xms 23172  df-ms 23173  df-tms 23174  df-nm 23434  df-ngp 23435  df-nrg 23437  df-nlm 23438  df-ii 23728  df-cncf 23729  df-limc 24717  df-dv 24718  df-log 25399  df-esum 31662  df-carsg 31935

This theorem is referenced by:  omsmeas  31956