Theorem carsggect 34304

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 5257	. . . 4 ⊢ ∅ ∈ V
3	2	a1i 11	. . 3 ⊢ (𝜑 → ∅ ∈ V)
4		carsggect.0	. . 3 ⊢ (𝜑 → ¬ ∅ ∈ 𝐴)
5		padct 32695	. . 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 1914	. . . . 5 ⊢ Ⅎ𝑧(𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)))
8		simpr1 1195	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝑓:ℕ⟶(𝐴 ∪ {∅}))
9	8	feqmptd 6912	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝑓 = (𝑘 ∈ ℕ ↦ (𝑓‘𝑘)))
10	9	rneqd 5892	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran 𝑓 = ran (𝑘 ∈ ℕ ↦ (𝑓‘𝑘)))
11	7, 10	esumeq1d 34020	. . . 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 6854	. . . . . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝐴 ⊆ (toCaraSiga‘𝑀))
16		carsgval.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑂 ∈ 𝑉)
17	16	adantr 480	. . . . . . . . . . . 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 480	. . . . . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝑀‘∅) = 0)
22	17, 19, 21	0elcarsg 34293	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∅ ∈ (toCaraSiga‘𝑀))
23	22	snssd 4769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → {∅} ⊆ (toCaraSiga‘𝑀))
24	15, 23	unssd 4151	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝐴 ∪ {∅}) ⊆ (toCaraSiga‘𝑀))
25	13, 24	ssexd 5274	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝐴 ∪ {∅}) ∈ V)
26	19	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
27	16, 18	carsgcl 34290	. . . . . . . . . . . . 13 ⊢ (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂)
28	14, 27	sstrd 3954	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ 𝒫 𝑂)
29	28	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝐴 ⊆ 𝒫 𝑂)
30		0elpw 5306	. . . . . . . . . . . . 13 ⊢ ∅ ∈ 𝒫 𝑂
31	30	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∅ ∈ 𝒫 𝑂)
32	31	snssd 4769	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → {∅} ⊆ 𝒫 𝑂)
33	29, 32	unssd 4151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
34	33	sselda 3943	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → 𝑧 ∈ 𝒫 𝑂)
35	26, 34	ffvelcdmd 7040	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ (𝐴 ∪ {∅})) → (𝑀‘𝑧) ∈ (0[,]+∞))
36	8	frnd 6679	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran 𝑓 ⊆ (𝐴 ∪ {∅}))
37	7, 25, 35, 36	esummono 34039	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) ≤ Σ*𝑧 ∈ (𝐴 ∪ {∅})(𝑀‘𝑧))
38		ctex 8913	. . . . . . . . . 10 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V)
39	1, 38	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ V)
40	39	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝐴 ∈ V)
41	13, 23	ssexd 5274	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → {∅} ∈ V)
42	19	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ 𝐴) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
43	29	sselda 3943	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝒫 𝑂)
44	42, 43	ffvelcdmd 7040	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ 𝐴) → (𝑀‘𝑧) ∈ (0[,]+∞))
45		elsni 4602	. . . . . . . . . . 11 ⊢ (𝑧 ∈ {∅} → 𝑧 = ∅)
46	45	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ {∅}) → 𝑧 = ∅)
47	46	fveq2d 6845	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ {∅}) → (𝑀‘𝑧) = (𝑀‘∅))
48	21	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ {∅}) → (𝑀‘∅) = 0)
49	47, 48	eqtrd 2764	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ {∅}) → (𝑀‘𝑧) = 0)
50	40, 41, 44, 49	esumpad 34040	. . . . . . 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 5128	. . . . . 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 3954	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran 𝑓 ⊆ (toCaraSiga‘𝑀))
53		ssexg 5273	. . . . . . . 8 ⊢ ((ran 𝑓 ⊆ (toCaraSiga‘𝑀) ∧ (toCaraSiga‘𝑀) ∈ V) → ran 𝑓 ∈ V)
54	52, 12, 53	sylancl 586	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran 𝑓 ∈ V)
55	19	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ ran 𝑓) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
56	36, 33	sstrd 3954	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran 𝑓 ⊆ 𝒫 𝑂)
57	56	sselda 3943	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ ran 𝑓) → 𝑧 ∈ 𝒫 𝑂)
58	55, 57	ffvelcdmd 7040	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑧 ∈ ran 𝑓) → (𝑀‘𝑧) ∈ (0[,]+∞))
59		simpr2 1196	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → 𝐴 ⊆ ran 𝑓)
60	7, 54, 58, 59	esummono 34039	. . . . . 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 511	. . . . 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 13370	. . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ*
63	58	ralrimiva 3125	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∀𝑧 ∈ ran 𝑓(𝑀‘𝑧) ∈ (0[,]+∞))
64		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑧ran 𝑓
65	64	esumcl 34015	. . . . . . . 8 ⊢ ((ran 𝑓 ∈ V ∧ ∀𝑧 ∈ ran 𝑓(𝑀‘𝑧) ∈ (0[,]+∞)) → Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) ∈ (0[,]+∞))
66	54, 63, 65	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) ∈ (0[,]+∞))
67	62, 66	sselid 3941	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) ∈ ℝ*)
68	44	ralrimiva 3125	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∀𝑧 ∈ 𝐴 (𝑀‘𝑧) ∈ (0[,]+∞))
69		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑧𝐴
70	69	esumcl 34015	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ ∀𝑧 ∈ 𝐴 (𝑀‘𝑧) ∈ (0[,]+∞)) → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ∈ (0[,]+∞))
71	40, 68, 70	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ∈ (0[,]+∞))
72	62, 71	sselid 3941	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ∈ ℝ*)
73		xrletri3 13093	. . . . . 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 584	. . . . 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 257	. . . 4 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ ran 𝑓(𝑀‘𝑧) = Σ*𝑧 ∈ 𝐴(𝑀‘𝑧))
76		fveq2 6841	. . . . 5 ⊢ (𝑧 = (𝑓‘𝑘) → (𝑀‘𝑧) = (𝑀‘(𝑓‘𝑘)))
77		nnex 12171	. . . . . 6 ⊢ ℕ ∈ V
78	77	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ℕ ∈ V)
79	19	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
80	33	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
81	8	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → 𝑓:ℕ⟶(𝐴 ∪ {∅}))
82		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
83	81, 82	ffvelcdmd 7040	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ (𝐴 ∪ {∅}))
84	80, 83	sseldd 3944	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ 𝒫 𝑂)
85	79, 84	ffvelcdmd 7040	. . . . 5 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) → (𝑀‘(𝑓‘𝑘)) ∈ (0[,]+∞))
86		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓‘𝑘) = ∅) → (𝑓‘𝑘) = ∅)
87	86	fveq2d 6845	. . . . . 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 2764	. . . . 5 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ ℕ) ∧ (𝑓‘𝑘) = ∅) → (𝑀‘(𝑓‘𝑘)) = 0)
90		cnvimass 6043	. . . . . . 7 ⊢ (◡𝑓 “ 𝐴) ⊆ dom 𝑓
91	90, 8	fssdm 6690	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (◡𝑓 “ 𝐴) ⊆ ℕ)
92		ffun 6674	. . . . . . . . . . 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 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑓 “ 𝐴))) → Fun 𝑓)
95		difpreima 7020	. . . . . . . . . . . . 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 6693	. . . . . . . . . . . . . 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 4084	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ((◡𝑓 “ (𝐴 ∪ {∅})) ∖ (◡𝑓 “ 𝐴)) = (ℕ ∖ (◡𝑓 “ 𝐴)))
100	96, 99	eqtrd 2764	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (◡𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) = (ℕ ∖ (◡𝑓 “ 𝐴)))
101		uncom 4117	. . . . . . . . . . . . . . . 16 ⊢ ({∅} ∪ 𝐴) = (𝐴 ∪ {∅})
102	101	difeq1i 4081	. . . . . . . . . . . . . . 15 ⊢ (({∅} ∪ 𝐴) ∖ 𝐴) = ((𝐴 ∪ {∅}) ∖ 𝐴)
103		difun2 4440	. . . . . . . . . . . . . . 15 ⊢ (({∅} ∪ 𝐴) ∖ 𝐴) = ({∅} ∖ 𝐴)
104	102, 103	eqtr3i 2754	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∪ {∅}) ∖ 𝐴) = ({∅} ∖ 𝐴)
105		difss 4095	. . . . . . . . . . . . . 14 ⊢ ({∅} ∖ 𝐴) ⊆ {∅}
106	104, 105	eqsstri 3990	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅}
107	106	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅})
108		sspreima 7023	. . . . . . . . . . . 12 ⊢ ((Fun 𝑓 ∧ ((𝐴 ∪ {∅}) ∖ 𝐴) ⊆ {∅}) → (◡𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) ⊆ (◡𝑓 “ {∅}))
109	93, 107, 108	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (◡𝑓 “ ((𝐴 ∪ {∅}) ∖ 𝐴)) ⊆ (◡𝑓 “ {∅}))
110	100, 109	eqsstrrd 3979	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (ℕ ∖ (◡𝑓 “ 𝐴)) ⊆ (◡𝑓 “ {∅}))
111	110	sselda 3943	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑓 “ 𝐴))) → 𝑘 ∈ (◡𝑓 “ {∅}))
112		fvimacnvi 7007	. . . . . . . . 9 ⊢ ((Fun 𝑓 ∧ 𝑘 ∈ (◡𝑓 “ {∅})) → (𝑓‘𝑘) ∈ {∅})
113	94, 111, 112	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ (ℕ ∖ (◡𝑓 “ 𝐴))) → (𝑓‘𝑘) ∈ {∅})
114		elsni 4602	. . . . . . . 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 3125	. . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Disj 𝑦 ∈ 𝐴 𝑦)
119		simpr3 1197	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Fun (◡𝑓 ↾ 𝐴))
120		fresf1o 32607	. . . . . . . . . 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 484	. . . . . . . . 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 32572	. . . . . . . 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 6860	. . . . . . . . . 10 ⊢ (𝑘 ∈ (◡𝑓 “ 𝐴) → ((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑘) = (𝑓‘𝑘))
125	124	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑘 ∈ (◡𝑓 “ 𝐴)) → ((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑘) = (𝑓‘𝑘))
126	125	disjeq2dv 5074	. . . . . . . 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 281	. . . . . . 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 232	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Disj 𝑘 ∈ (◡𝑓 “ 𝐴)(𝑓‘𝑘))
129		disjss3 5101	. . . . . . 7 ⊢ (((◡𝑓 “ 𝐴) ⊆ ℕ ∧ ∀𝑘 ∈ (ℕ ∖ (◡𝑓 “ 𝐴))(𝑓‘𝑘) = ∅) → (Disj 𝑘 ∈ (◡𝑓 “ 𝐴)(𝑓‘𝑘) ↔ Disj 𝑘 ∈ ℕ (𝑓‘𝑘)))
130	129	biimpa 476	. . . . . 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 837	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Disj 𝑘 ∈ ℕ (𝑓‘𝑘))
132	76, 78, 85, 84, 89, 131	esumrnmpt2 34053	. . . 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 2773	. . 3 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑘 ∈ ℕ(𝑀‘(𝑓‘𝑘)) = Σ*𝑧 ∈ 𝐴(𝑀‘𝑧))
134		uniiun 5017	. . . . . . 7 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
135	28	sselda 3943	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝒫 𝑂)
136	39, 135	elpwiuncl 32508	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝑥 ∈ 𝒫 𝑂)
137	134, 136	eqeltrid 2832	. . . . . 6 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝒫 𝑂)
138	137	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∪ 𝐴 ∈ 𝒫 𝑂)
139	19, 138	ffvelcdmd 7040	. . . 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 1178	. . . . . . . . 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 6841	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑀‘𝑦) = (𝑀‘𝑧))
143		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑧𝑥
144		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝑥
145		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑧(𝑀‘𝑦)
146		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑀‘𝑧)
147	142, 143, 144, 145, 146	cbvesum 34027	. . . . . . . . 9 ⊢ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦) = Σ*𝑧 ∈ 𝑥(𝑀‘𝑧)
148	141, 147	breqtrdi 5143	. . . . . . . 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 6671	. . . . . . . . . 10 ⊢ (𝑓:ℕ⟶(𝐴 ∪ {∅}) → 𝑓 Fn ℕ)
150		fz1ssnn 13495	. . . . . . . . . . 11 ⊢ (1...𝑛) ⊆ ℕ
151		fnssres 6624	. . . . . . . . . . 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 13916	. . . . . . . . . 10 ⊢ (1...𝑛) ∈ Fin
155		fnfi 9120	. . . . . . . . . 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 9268	. . . . . . . . 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 5962	. . . . . . . . . . 11 ⊢ (𝑓 ↾ (1...𝑛)) ⊆ 𝑓
160		rnss 5893	. . . . . . . . . . 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 3954	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ (toCaraSiga‘𝑀))
164	162, 36	sstrd 3954	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ran (𝑓 ↾ (1...𝑛)) ⊆ (𝐴 ∪ {∅}))
165		nfcv 2891	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧𝑦
166		nfcv 2891	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦𝑧
167		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
168	165, 166, 167	cbvdisj 5079	. . . . . . . . . . . 12 ⊢ (Disj 𝑦 ∈ 𝐴 𝑦 ↔ Disj 𝑧 ∈ 𝐴 𝑧)
169		disjun0 32576	. . . . . . . . . . . 12 ⊢ (Disj 𝑧 ∈ 𝐴 𝑧 → Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
170	168, 169	sylbi 217	. . . . . . . . . . 11 ⊢ (Disj 𝑦 ∈ 𝐴 𝑦 → Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
171	117, 170	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
172	171	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Disj 𝑧 ∈ (𝐴 ∪ {∅})𝑧)
173		disjss1 5075	. . . . . . . . 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 4579	. . . . . . . . 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 34303	. . . . . . 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 480	. . . . . 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 4877	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∪ ran (𝑓 ↾ (1...𝑛)) ⊆ ∪ (𝐴 ∪ {∅}))
180		uniun 4890	. . . . . . . . . . . 12 ⊢ ∪ (𝐴 ∪ {∅}) = (∪ 𝐴 ∪ ∪ {∅})
181	2	unisn 4886	. . . . . . . . . . . . 13 ⊢ ∪ {∅} = ∅
182	181	uneq2i 4124	. . . . . . . . . . . 12 ⊢ (∪ 𝐴 ∪ ∪ {∅}) = (∪ 𝐴 ∪ ∅)
183		un0 4353	. . . . . . . . . . . 12 ⊢ (∪ 𝐴 ∪ ∅) = ∪ 𝐴
184	180, 182, 183	3eqtri 2756	. . . . . . . . . . 11 ⊢ ∪ (𝐴 ∪ {∅}) = ∪ 𝐴
185	179, 184	sseqtrdi 3984	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → ∪ ran (𝑓 ↾ (1...𝑛)) ⊆ ∪ 𝐴)
186	185	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → ∪ ran (𝑓 ↾ (1...𝑛)) ⊆ ∪ 𝐴)
187		uniss 4875	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝒫 𝑂 → ∪ 𝐴 ⊆ ∪ 𝒫 𝑂)
188		unipw 5405	. . . . . . . . . . . 12 ⊢ ∪ 𝒫 𝑂 = 𝑂
189	187, 188	sseqtrdi 3984	. . . . . . . . . . 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 3954	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → ∪ ran (𝑓 ↾ (1...𝑛)) ⊆ 𝑂)
193		sseqin2 4182	. . . . . . . 8 ⊢ (∪ ran (𝑓 ↾ (1...𝑛)) ⊆ 𝑂 ↔ (𝑂 ∩ ∪ ran (𝑓 ↾ (1...𝑛))) = ∪ ran (𝑓 ↾ (1...𝑛)))
194	192, 193	sylib 218	. . . . . . 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 6845	. . . . . 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 1914	. . . . . . . 8 ⊢ Ⅎ𝑧((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ)
197	164	adantr 480	. . . . . . . . . . . . . 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 4769	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → {∅} ⊆ 𝒫 𝑂)
201	198, 200	unssd 4151	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝐴 ∪ {∅}) ⊆ 𝒫 𝑂)
202	197, 201	sstrd 3954	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → ran (𝑓 ↾ (1...𝑛)) ⊆ 𝒫 𝑂)
203	202	sselda 3943	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → 𝑧 ∈ 𝒫 𝑂)
204	203	elpwid 4568	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → 𝑧 ⊆ 𝑂)
205		sseqin2 4182	. . . . . . . . . . 11 ⊢ (𝑧 ⊆ 𝑂 ↔ (𝑂 ∩ 𝑧) = 𝑧)
206	204, 205	sylib 218	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑧 ∈ ran (𝑓 ↾ (1...𝑛))) → (𝑂 ∩ 𝑧) = 𝑧)
207	206	fveq2d 6845	. . . . . . . . 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 3125	. . . . . . . 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 34022	. . . . . . 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 5939	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝑓 ↾ (1...𝑛)) = ((𝑘 ∈ ℕ ↦ (𝑓‘𝑘)) ↾ (1...𝑛)))
211	210	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑓 ↾ (1...𝑛)) = ((𝑘 ∈ ℕ ↦ (𝑓‘𝑘)) ↾ (1...𝑛)))
212		resmpt 5998	. . . . . . . . . . . 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 2780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑓 ↾ (1...𝑛)) = (𝑘 ∈ (1...𝑛) ↦ (𝑓‘𝑘)))
215	214	eqcomd 2735	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ (1...𝑛) ↦ (𝑓‘𝑘)) = (𝑓 ↾ (1...𝑛)))
216	215	rneqd 5892	. . . . . . . 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 34020	. . . . . . 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 3943	. . . . . . . . . 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 591	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑓‘𝑘) ∈ 𝒫 𝑂)
224	219, 223	ffvelcdmd 7040	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑀‘(𝑓‘𝑘)) ∈ (0[,]+∞))
225		simpr 484	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓‘𝑘) = ∅) → (𝑓‘𝑘) = ∅)
226	225	fveq2d 6845	. . . . . . . . 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 2764	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ (𝑓‘𝑘) = ∅) → (𝑀‘(𝑓‘𝑘)) = 0)
229		disjss1 5075	. . . . . . . . . . 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 480	. . . . . . . 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 34053	. . . . . . 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 2770	. . . . . 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 2772	. . . . 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 1178	. . . . . . 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 34300	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → (𝑀‘∪ ran (𝑓 ↾ (1...𝑛))) ≤ (𝑀‘∪ 𝐴))
239	238	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) ∧ 𝑛 ∈ ℕ) → (𝑀‘∪ ran (𝑓 ↾ (1...𝑛))) ≤ (𝑀‘∪ 𝐴))
240	235, 239	eqbrtrrd 5126	. . . 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 34075	. . 3 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑘 ∈ ℕ(𝑀‘(𝑓‘𝑘)) ≤ (𝑀‘∪ 𝐴))
242	133, 241	eqbrtrrd 5126	. 2 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶(𝐴 ∪ {∅}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴))) → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ≤ (𝑀‘∪ 𝐴))
243	6, 242	exlimddv 1935	1 ⊢ (𝜑 → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ≤ (𝑀‘∪ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  Vcvv 3444   ∖ cdif 3908   ∪ cun 3909   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  𝒫 cpw 4559  {csn 4585  ∪ cuni 4867  ∪ ciun 4951  Disj wdisj 5069   class class class wbr 5102   ↦ cmpt 5183  ^◡ccnv 5630  ran crn 5632   ↾ cres 5633   “ cima 5634  Fun wfun 6494   Fn wfn 6495  ⟶wf 6496  –1-1-onto→wf1o 6499  ‘cfv 6500  (class class class)co 7370  ωcom 7823   ≼ cdom 8894  Fincfn 8896  0cc0 11047  1c1 11048  +∞cpnf 11184  ℝ^*cxr 11186   ≤ cle 11188  ℕcn 12165  [,]cicc 13288  ...cfz 13447  Σ^*cesum 34012  toCaraSigaccarsg 34287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7692  ax-inf2 9573  ax-cnex 11103  ax-resscn 11104  ax-1cn 11105  ax-icn 11106  ax-addcl 11107  ax-addrcl 11108  ax-mulcl 11109  ax-mulrcl 11110  ax-mulcom 11111  ax-addass 11112  ax-mulass 11113  ax-distr 11114  ax-i2m1 11115  ax-1ne0 11116  ax-1rid 11117  ax-rnegex 11118  ax-rrecex 11119  ax-cnre 11120  ax-pre-lttri 11121  ax-pre-lttrn 11122  ax-pre-ltadd 11123  ax-pre-mulgt0 11124  ax-pre-sup 11125  ax-addf 11126  ax-mulf 11127

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-disj 5070  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6263  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6453  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-of 7634  df-om 7824  df-1st 7948  df-2nd 7949  df-supp 8118  df-frecs 8238  df-wrecs 8269  df-recs 8318  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8649  df-map 8779  df-pm 8780  df-ixp 8849  df-en 8897  df-dom 8898  df-sdom 8899  df-fin 8900  df-fsupp 9290  df-fi 9339  df-sup 9370  df-inf 9371  df-oi 9440  df-dju 9833  df-card 9871  df-pnf 11189  df-mnf 11190  df-xr 11191  df-ltxr 11192  df-le 11193  df-sub 11386  df-neg 11387  df-div 11815  df-nn 12166  df-2 12228  df-3 12229  df-4 12230  df-5 12231  df-6 12232  df-7 12233  df-8 12234  df-9 12235  df-n0 12422  df-z 12509  df-dec 12629  df-uz 12773  df-q 12887  df-rp 12931  df-xneg 13051  df-xadd 13052  df-xmul 13053  df-ioo 13289  df-ioc 13290  df-ico 13291  df-icc 13292  df-fz 13448  df-fzo 13595  df-fl 13733  df-mod 13811  df-seq 13946  df-exp 14006  df-fac 14218  df-bc 14247  df-hash 14275  df-shft 15011  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-limsup 15415  df-clim 15432  df-rlim 15433  df-sum 15631  df-ef 16011  df-sin 16013  df-cos 16014  df-pi 16016  df-struct 17095  df-sets 17112  df-slot 17130  df-ndx 17142  df-base 17158  df-ress 17179  df-plusg 17211  df-mulr 17212  df-starv 17213  df-sca 17214  df-vsca 17215  df-ip 17216  df-tset 17217  df-ple 17218  df-ds 17220  df-unif 17221  df-hom 17222  df-cco 17223  df-rest 17363  df-topn 17364  df-0g 17382  df-gsum 17383  df-topgen 17384  df-pt 17385  df-prds 17388  df-ordt 17442  df-xrs 17443  df-qtop 17448  df-imas 17449  df-xps 17451  df-mre 17525  df-mrc 17526  df-acs 17528  df-ps 18509  df-tsr 18510  df-plusf 18550  df-mgm 18551  df-sgrp 18630  df-mnd 18646  df-mhm 18694  df-submnd 18695  df-grp 18852  df-minusg 18853  df-sbg 18854  df-mulg 18984  df-subg 19039  df-cntz 19233  df-cmn 19698  df-abl 19699  df-mgp 20063  df-rng 20075  df-ur 20104  df-ring 20157  df-cring 20158  df-subrng 20468  df-subrg 20492  df-abv 20731  df-lmod 20802  df-scaf 20803  df-sra 21114  df-rgmod 21115  df-psmet 21290  df-xmet 21291  df-met 21292  df-bl 21293  df-mopn 21294  df-fbas 21295  df-fg 21296  df-cnfld 21299  df-top 22816  df-topon 22833  df-topsp 22855  df-bases 22868  df-cld 22941  df-ntr 22942  df-cls 22943  df-nei 23020  df-lp 23058  df-perf 23059  df-cn 23149  df-cnp 23150  df-haus 23237  df-tx 23484  df-hmeo 23677  df-fil 23768  df-fm 23860  df-flim 23861  df-flf 23862  df-tmd 23994  df-tgp 23995  df-tsms 24049  df-trg 24082  df-xms 24243  df-ms 24244  df-tms 24245  df-nm 24505  df-ngp 24506  df-nrg 24508  df-nlm 24509  df-ii 24805  df-cncf 24806  df-limc 25802  df-dv 25803  df-log 26500  df-esum 34013  df-carsg 34288

This theorem is referenced by:  omsmeas  34309