meaiuninclem - Mathbox for Glauco Siliprandi

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Theorem meaiuninclem 47120

Description: Measures are continuous from below (bounded case): if 𝐸 is a sequence of increasing measurable sets (with uniformly bounded measure) then the measure of the union is the union of the measure. This is Proposition 112C (e) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

**Hypotheses**
Ref	Expression
meaiuninclem.m	⊢ (𝜑 → 𝑀 ∈ Meas)
meaiuninclem.n	⊢ (𝜑 → 𝑁 ∈ ℤ)
meaiuninclem.z	⊢ 𝑍 = (ℤ_≥‘𝑁)
meaiuninclem.e	⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀)
meaiuninclem.i	⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1)))
meaiuninclem.b	⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
meaiuninclem.s	⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛)))
meaiuninclem.f	⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖)))

**Assertion**
Ref	Expression
meaiuninclem	⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))

Distinct variable groups: 𝑖,𝐸,𝑛,𝑥 𝑖,𝐹,𝑛,𝑥 𝑖,𝑀,𝑛,𝑥 𝑖,𝑁,𝑛,𝑥 𝑆,𝑛,𝑥 𝑖,𝑍,𝑛,𝑥 𝜑,𝑖,𝑛,𝑥 Allowed substitution hint: 𝑆(𝑖)

Proof of Theorem meaiuninclem Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		meaiuninclem.z	. . 3 ⊢ 𝑍 = (ℤ_≥‘𝑁)
2		meaiuninclem.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
3		0xr 11256	. . . . . . 7 ⊢ 0 ∈ ℝ^*
4	3	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 0 ∈ ℝ^*)
5		pnfxr 11263	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
6	5	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → +∞ ∈ ℝ^*)
7		meaiuninclem.m	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ Meas)
8	7	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑀 ∈ Meas)
9		eqid 2769	. . . . . . 7 ⊢ dom 𝑀 = dom 𝑀
10		meaiuninclem.e	. . . . . . . 8 ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀)
11	10	ffvelcdmda 7080	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ dom 𝑀)
12	8, 9, 11	meaxrcl 47101	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) ∈ ℝ^*)
13	8, 11	meage0 47115	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 0 ≤ (𝑀‘(𝐸‘𝑛)))
14		meaiuninclem.b	. . . . . . . 8 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
15	14	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
16		simp1 1152	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → (𝜑 ∧ 𝑛 ∈ 𝑍))
17		simp2 1153	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑥 ∈ ℝ)
18		simp3 1154	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
19	16	simprd 500	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑛 ∈ 𝑍)
20		rspa 3260	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
21	18, 19, 20	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
22	12	3ad2ant1 1149	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ℝ ∧ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → (𝑀‘(𝐸‘𝑛)) ∈ ℝ^*)
23		rexr 11255	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
24	23	3ad2ant2 1150	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ℝ ∧ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑥 ∈ ℝ^*)
25	5	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ℝ ∧ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → +∞ ∈ ℝ^*)
26		simp3 1154	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ℝ ∧ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
27		ltpnf 13145	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞)
28	27	3ad2ant2 1150	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ℝ ∧ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑥 < +∞)
29	22, 24, 25, 26, 28	xrlelttrd 13185	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ℝ ∧ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → (𝑀‘(𝐸‘𝑛)) < +∞)
30	16, 17, 21, 29	syl3anc 1396	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → (𝑀‘(𝐸‘𝑛)) < +∞)
31	30	3exp 1135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ ℝ → (∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 → (𝑀‘(𝐸‘𝑛)) < +∞)))
32	31	rexlimdv 3170	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 → (𝑀‘(𝐸‘𝑛)) < +∞))
33	15, 32	mpd 16	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) < +∞)
34	4, 6, 12, 13, 33	elicod 13422	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) ∈ (0[,)+∞))
35		meaiuninclem.s	. . . . 5 ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛)))
36	34, 35	fmptd 7110	. . . 4 ⊢ (𝜑 → 𝑆:𝑍⟶(0[,)+∞))
37		rge0ssre 13483	. . . . 5 ⊢ (0[,)+∞) ⊆ ℝ
38	37	a1i 11	. . . 4 ⊢ (𝜑 → (0[,)+∞) ⊆ ℝ)
39	36, 38	fssd 6724	. . 3 ⊢ (𝜑 → 𝑆:𝑍⟶ℝ)
40	1	peano2uzs 12926	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → (𝑛 + 1) ∈ 𝑍)
41	40	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑛 + 1) ∈ 𝑍)
42	10	ffvelcdmda 7080	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝐸‘(𝑛 + 1)) ∈ dom 𝑀)
43	41, 42	syldan 602	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘(𝑛 + 1)) ∈ dom 𝑀)
44		meaiuninclem.i	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1)))
45	8, 9, 11, 43, 44	meassle 47103	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) ≤ (𝑀‘(𝐸‘(𝑛 + 1))))
46	35	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))))
47		fvexd 6897	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) ∈ V)
48	46, 47	fvmpt2d 7004	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑆‘𝑛) = (𝑀‘(𝐸‘𝑛)))
49		2fveq3 6887	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑀‘(𝐸‘𝑛)) = (𝑀‘(𝐸‘𝑚)))
50	49	cbvmptv 5219	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) = (𝑚 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑚)))
51	35, 50	eqtri 2792	. . . . . 6 ⊢ 𝑆 = (𝑚 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑚)))
52		2fveq3 6887	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (𝑀‘(𝐸‘𝑚)) = (𝑀‘(𝐸‘(𝑛 + 1))))
53		fvexd 6897	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘(𝑛 + 1))) ∈ V)
54	51, 52, 41, 53	fvmptd3 7014	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑆‘(𝑛 + 1)) = (𝑀‘(𝐸‘(𝑛 + 1))))
55	48, 54	breq12d 5126	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑆‘𝑛) ≤ (𝑆‘(𝑛 + 1)) ↔ (𝑀‘(𝐸‘𝑛)) ≤ (𝑀‘(𝐸‘(𝑛 + 1)))))
56	45, 55	mpbird 260	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑆‘𝑛) ≤ (𝑆‘(𝑛 + 1)))
57	48	eqcomd 2775	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) = (𝑆‘𝑛))
58	57	breq1d 5123	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ (𝑆‘𝑛) ≤ 𝑥))
59	58	ralbidva 3192	. . . . . . 7 ⊢ (𝜑 → (∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ∀𝑛 ∈ 𝑍 (𝑆‘𝑛) ≤ 𝑥))
60	59	biimpd 232	. . . . . 6 ⊢ (𝜑 → (∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 → ∀𝑛 ∈ 𝑍 (𝑆‘𝑛) ≤ 𝑥))
61	60	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 → ∀𝑛 ∈ 𝑍 (𝑆‘𝑛) ≤ 𝑥))
62	61	reximdva 3184	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑆‘𝑛) ≤ 𝑥))
63	14, 62	mpd 16	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑆‘𝑛) ≤ 𝑥)
64	1, 2, 39, 56, 63	climsup 15721	. 2 ⊢ (𝜑 → 𝑆 ⇝ sup(ran 𝑆, ℝ, < ))
65		nfv 1941	. . . . . 6 ⊢ Ⅎ𝑛𝜑
66		nfv 1941	. . . . . 6 ⊢ Ⅎ𝑥𝜑
67		id 23	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ 𝑍)
68		fvex 6895	. . . . . . . . . . . . 13 ⊢ (𝐸‘𝑛) ∈ V
69	68	difexi 5301	. . . . . . . . . . . 12 ⊢ ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖)) ∈ V
70	69	a1i 11	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑍 → ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖)) ∈ V)
71		meaiuninclem.f	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖)))
72	71	fvmpt2 7002	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ 𝑍 ∧ ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖)) ∈ V) → (𝐹‘𝑛) = ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖)))
73	67, 70, 72	syl2anc 595	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → (𝐹‘𝑛) = ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖)))
74	73	adantl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) = ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖)))
75	7, 9	dmmeasal 47092	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝑀 ∈ SAlg)
76	75	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom 𝑀 ∈ SAlg)
77		fzoct 46025	. . . . . . . . . . . 12 ⊢ (𝑁..^𝑛) ≼ ω
78	77	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑁..^𝑛) ≼ ω)
79	10	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑁..^𝑛)) → 𝐸:𝑍⟶dom 𝑀)
80		fzossuz 46022	. . . . . . . . . . . . . . . 16 ⊢ (𝑁..^𝑛) ⊆ (ℤ_≥‘𝑁)
81	1	eqcomi 2778	. . . . . . . . . . . . . . . 16 ⊢ (ℤ_≥‘𝑁) = 𝑍
82	80, 81	sseqtri 3993	. . . . . . . . . . . . . . 15 ⊢ (𝑁..^𝑛) ⊆ 𝑍
83	82	sseli 3941	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (𝑁..^𝑛) → 𝑖 ∈ 𝑍)
84	83	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑁..^𝑛)) → 𝑖 ∈ 𝑍)
85	79, 84	ffvelcdmd 7081	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑁..^𝑛)) → (𝐸‘𝑖) ∈ dom 𝑀)
86	85	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑖 ∈ (𝑁..^𝑛)) → (𝐸‘𝑖) ∈ dom 𝑀)
87	76, 78, 86	saliuncl 46963	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖) ∈ dom 𝑀)
88		saldifcl2 46968	. . . . . . . . . 10 ⊢ ((dom 𝑀 ∈ SAlg ∧ (𝐸‘𝑛) ∈ dom 𝑀 ∧ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖) ∈ dom 𝑀) → ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖)) ∈ dom 𝑀)
89	76, 11, 87, 88	syl3anc 1396	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖)) ∈ dom 𝑀)
90	74, 89	eqeltrd 2869	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ dom 𝑀)
91	8, 9, 90	meaxrcl 47101	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐹‘𝑛)) ∈ ℝ^*)
92	8, 90	meage0 47115	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 0 ≤ (𝑀‘(𝐹‘𝑛)))
93		difssd 4099	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖)) ⊆ (𝐸‘𝑛))
94	74, 93	eqsstrd 3979	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ⊆ (𝐸‘𝑛))
95	8, 9, 90, 11, 94	meassle 47103	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐹‘𝑛)) ≤ (𝑀‘(𝐸‘𝑛)))
96	91, 12, 6, 95, 33	xrlelttrd 13185	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐹‘𝑛)) < +∞)
97	4, 6, 91, 92, 96	elicod 13422	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐹‘𝑛)) ∈ (0[,)+∞))
98		2fveq3 6887	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑖 → (𝑀‘(𝐸‘𝑛)) = (𝑀‘(𝐸‘𝑖)))
99	98	breq1d 5123	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑖 → ((𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ (𝑀‘(𝐸‘𝑖)) ≤ 𝑥))
100	99	cbvralvw 3249	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ∀𝑖 ∈ 𝑍 (𝑀‘(𝐸‘𝑖)) ≤ 𝑥)
101	100	bilani 509	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → ∀𝑖 ∈ 𝑍 (𝑀‘(𝐸‘𝑖)) ≤ 𝑥)
102		eleq1w 2852	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑖 → (𝑛 ∈ 𝑍 ↔ 𝑖 ∈ 𝑍))
103	102	anbi2d 641	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑖 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑖 ∈ 𝑍)))
104		oveq2 7419	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑖 → (𝑁...𝑛) = (𝑁...𝑖))
105	104	sumeq1d 15751	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑖 → Σ𝑚 ∈ (𝑁...𝑛)(𝑀‘(𝐹‘𝑚)) = Σ𝑚 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑚)))
106	98, 105	eqeq12d 2785	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑖 → ((𝑀‘(𝐸‘𝑛)) = Σ𝑚 ∈ (𝑁...𝑛)(𝑀‘(𝐹‘𝑚)) ↔ (𝑀‘(𝐸‘𝑖)) = Σ𝑚 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑚))))
107	103, 106	imbi12d 347	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑖 → (((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) = Σ𝑚 ∈ (𝑁...𝑛)(𝑀‘(𝐹‘𝑚))) ↔ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝑀‘(𝐸‘𝑖)) = Σ𝑚 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑚)))))
108		eleq1w 2852	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑛 → (𝑚 ∈ 𝑍 ↔ 𝑛 ∈ 𝑍))
109	108	anbi2d 641	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑛 → ((𝜑 ∧ 𝑚 ∈ 𝑍) ↔ (𝜑 ∧ 𝑛 ∈ 𝑍)))
110		oveq2 7419	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑛 → (𝑁...𝑚) = (𝑁...𝑛))
111	110	iuneq1d 4988	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑛 → ∪ 𝑖 ∈ (𝑁...𝑚)(𝐹‘𝑖) = ∪ 𝑖 ∈ (𝑁...𝑛)(𝐹‘𝑖))
112	110	iuneq1d 4988	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑛 → ∪ 𝑖 ∈ (𝑁...𝑚)(𝐸‘𝑖) = ∪ 𝑖 ∈ (𝑁...𝑛)(𝐸‘𝑖))
113	111, 112	eqeq12d 2785	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑛 → (∪ 𝑖 ∈ (𝑁...𝑚)(𝐹‘𝑖) = ∪ 𝑖 ∈ (𝑁...𝑚)(𝐸‘𝑖) ↔ ∪ 𝑖 ∈ (𝑁...𝑛)(𝐹‘𝑖) = ∪ 𝑖 ∈ (𝑁...𝑛)(𝐸‘𝑖)))
114	109, 113	imbi12d 347	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑛 → (((𝜑 ∧ 𝑚 ∈ 𝑍) → ∪ 𝑖 ∈ (𝑁...𝑚)(𝐹‘𝑖) = ∪ 𝑖 ∈ (𝑁...𝑚)(𝐸‘𝑖)) ↔ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∪ 𝑖 ∈ (𝑁...𝑛)(𝐹‘𝑖) = ∪ 𝑖 ∈ (𝑁...𝑛)(𝐸‘𝑖))))
115		fveq2 6882	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑛 → (𝐹‘𝑖) = (𝐹‘𝑛))
116	115	cbviunv 5007	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ∪ 𝑖 ∈ (𝑁...𝑚)(𝐹‘𝑖) = ∪ 𝑛 ∈ (𝑁...𝑚)(𝐹‘𝑛)
117	116	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ∪ 𝑖 ∈ (𝑁...𝑚)(𝐹‘𝑖) = ∪ 𝑛 ∈ (𝑁...𝑚)(𝐹‘𝑛))
118	65, 1, 10, 71	iundjiun 47100	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)(𝐹‘𝑛) = ∪ 𝑛 ∈ (𝑁...𝑚)(𝐸‘𝑛) ∧ ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛) = ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∧ Disj 𝑛 ∈ 𝑍 (𝐹‘𝑛)))
119	118	simplld 779	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)(𝐹‘𝑛) = ∪ 𝑛 ∈ (𝑁...𝑚)(𝐸‘𝑛))
120	119	adantr 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)(𝐹‘𝑛) = ∪ 𝑛 ∈ (𝑁...𝑚)(𝐸‘𝑛))
121		simpr 489	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ 𝑍)
122		rspa 3260	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)(𝐹‘𝑛) = ∪ 𝑛 ∈ (𝑁...𝑚)(𝐸‘𝑛) ∧ 𝑚 ∈ 𝑍) → ∪ 𝑛 ∈ (𝑁...𝑚)(𝐹‘𝑛) = ∪ 𝑛 ∈ (𝑁...𝑚)(𝐸‘𝑛))
123	120, 121, 122	syl2anc 595	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ∪ 𝑛 ∈ (𝑁...𝑚)(𝐹‘𝑛) = ∪ 𝑛 ∈ (𝑁...𝑚)(𝐸‘𝑛))
124		fveq2 6882	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑖 → (𝐸‘𝑛) = (𝐸‘𝑖))
125	124	cbviunv 5007	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ∪ 𝑛 ∈ (𝑁...𝑚)(𝐸‘𝑛) = ∪ 𝑖 ∈ (𝑁...𝑚)(𝐸‘𝑖)
126	125	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ∪ 𝑛 ∈ (𝑁...𝑚)(𝐸‘𝑛) = ∪ 𝑖 ∈ (𝑁...𝑚)(𝐸‘𝑖))
127	117, 123, 126	3eqtrd 2808	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ∪ 𝑖 ∈ (𝑁...𝑚)(𝐹‘𝑖) = ∪ 𝑖 ∈ (𝑁...𝑚)(𝐸‘𝑖))
128	114, 127	chvarvv 2016	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∪ 𝑖 ∈ (𝑁...𝑛)(𝐹‘𝑖) = ∪ 𝑖 ∈ (𝑁...𝑛)(𝐸‘𝑖))
129	67, 1	eleqtrdi 2879	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ_≥‘𝑁))
130	129	adantl 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ (ℤ_≥‘𝑁))
131		fvoveq1 7434	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑖 → (𝐸‘(𝑛 + 1)) = (𝐸‘(𝑖 + 1)))
132	124, 131	sseq12d 3978	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑖 → ((𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1)) ↔ (𝐸‘𝑖) ⊆ (𝐸‘(𝑖 + 1))))
133	103, 132	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑖 → (((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1))) ↔ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐸‘𝑖) ⊆ (𝐸‘(𝑖 + 1)))))
134	133, 44	chvarvv 2016	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐸‘𝑖) ⊆ (𝐸‘(𝑖 + 1)))
135	84, 134	syldan 602	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑁..^𝑛)) → (𝐸‘𝑖) ⊆ (𝐸‘(𝑖 + 1)))
136	135	adantlr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑖 ∈ (𝑁..^𝑛)) → (𝐸‘𝑖) ⊆ (𝐸‘(𝑖 + 1)))
137	130, 136	iunincfi 45738	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∪ 𝑖 ∈ (𝑁...𝑛)(𝐸‘𝑖) = (𝐸‘𝑛))
138	128, 137	eqtr2d 2805	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) = ∪ 𝑖 ∈ (𝑁...𝑛)(𝐹‘𝑖))
139	138	fveq2d 6886	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) = (𝑀‘∪ 𝑖 ∈ (𝑁...𝑛)(𝐹‘𝑖)))
140		nfv 1941	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑛 ∈ 𝑍)
141		elfzuz 13548	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (𝑁...𝑛) → 𝑖 ∈ (ℤ_≥‘𝑁))
142	141, 81	eleqtrdi 2879	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (𝑁...𝑛) → 𝑖 ∈ 𝑍)
143	142	adantl 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑁...𝑛)) → 𝑖 ∈ 𝑍)
144		fveq2 6882	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑖 → (𝐹‘𝑛) = (𝐹‘𝑖))
145	144	eleq1d 2854	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑖 → ((𝐹‘𝑛) ∈ dom 𝑀 ↔ (𝐹‘𝑖) ∈ dom 𝑀))
146	103, 145	imbi12d 347	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑖 → (((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ dom 𝑀) ↔ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹‘𝑖) ∈ dom 𝑀)))
147	146, 90	chvarvv 2016	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹‘𝑖) ∈ dom 𝑀)
148	143, 147	syldan 602	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑁...𝑛)) → (𝐹‘𝑖) ∈ dom 𝑀)
149	148	adantlr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑖 ∈ (𝑁...𝑛)) → (𝐹‘𝑖) ∈ dom 𝑀)
150		fzct 46020	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁...𝑛) ≼ ω
151	150	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑁...𝑛) ≼ ω)
152	143	ssd 45726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑁...𝑛) ⊆ 𝑍)
153	118	simprd 500	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → Disj 𝑛 ∈ 𝑍 (𝐹‘𝑛))
154	144	cbvdisjv 5091	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Disj 𝑛 ∈ 𝑍 (𝐹‘𝑛) ↔ Disj 𝑖 ∈ 𝑍 (𝐹‘𝑖))
155	153, 154	sylib 221	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → Disj 𝑖 ∈ 𝑍 (𝐹‘𝑖))
156		disjss1 5086	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁...𝑛) ⊆ 𝑍 → (Disj 𝑖 ∈ 𝑍 (𝐹‘𝑖) → Disj 𝑖 ∈ (𝑁...𝑛)(𝐹‘𝑖)))
157	152, 155, 156	sylc 66	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Disj 𝑖 ∈ (𝑁...𝑛)(𝐹‘𝑖))
158	157	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → Disj 𝑖 ∈ (𝑁...𝑛)(𝐹‘𝑖))
159	140, 8, 9, 149, 151, 158	meadjiun 47106	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘∪ 𝑖 ∈ (𝑁...𝑛)(𝐹‘𝑖)) = (Σ^{^}‘(𝑖 ∈ (𝑁...𝑛) ↦ (𝑀‘(𝐹‘𝑖)))))
160		fzfid 14009	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑁...𝑛) ∈ Fin)
161		2fveq3 6887	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑖 → (𝑀‘(𝐹‘𝑛)) = (𝑀‘(𝐹‘𝑖)))
162	161	eleq1d 2854	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑖 → ((𝑀‘(𝐹‘𝑛)) ∈ (0[,)+∞) ↔ (𝑀‘(𝐹‘𝑖)) ∈ (0[,)+∞)))
163	103, 162	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑖 → (((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐹‘𝑛)) ∈ (0[,)+∞)) ↔ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝑀‘(𝐹‘𝑖)) ∈ (0[,)+∞))))
164	163, 97	chvarvv 2016	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝑀‘(𝐹‘𝑖)) ∈ (0[,)+∞))
165	143, 164	syldan 602	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑁...𝑛)) → (𝑀‘(𝐹‘𝑖)) ∈ (0[,)+∞))
166	165	adantlr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑖 ∈ (𝑁...𝑛)) → (𝑀‘(𝐹‘𝑖)) ∈ (0[,)+∞))
167	160, 166	sge0fsummpt 47030	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (Σ^{^}‘(𝑖 ∈ (𝑁...𝑛) ↦ (𝑀‘(𝐹‘𝑖)))) = Σ𝑖 ∈ (𝑁...𝑛)(𝑀‘(𝐹‘𝑖)))
168		2fveq3 6887	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑚 → (𝑀‘(𝐹‘𝑖)) = (𝑀‘(𝐹‘𝑚)))
169	168	cbvsumv 15747	. . . . . . . . . . . . . . . . . . 19 ⊢ Σ𝑖 ∈ (𝑁...𝑛)(𝑀‘(𝐹‘𝑖)) = Σ𝑚 ∈ (𝑁...𝑛)(𝑀‘(𝐹‘𝑚))
170	169	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → Σ𝑖 ∈ (𝑁...𝑛)(𝑀‘(𝐹‘𝑖)) = Σ𝑚 ∈ (𝑁...𝑛)(𝑀‘(𝐹‘𝑚)))
171	167, 170	eqtrd 2804	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (Σ^{^}‘(𝑖 ∈ (𝑁...𝑛) ↦ (𝑀‘(𝐹‘𝑖)))) = Σ𝑚 ∈ (𝑁...𝑛)(𝑀‘(𝐹‘𝑚)))
172	139, 159, 171	3eqtrd 2808	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) = Σ𝑚 ∈ (𝑁...𝑛)(𝑀‘(𝐹‘𝑚)))
173	107, 172	chvarvv 2016	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝑀‘(𝐸‘𝑖)) = Σ𝑚 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑚)))
174		2fveq3 6887	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑛 → (𝑀‘(𝐹‘𝑚)) = (𝑀‘(𝐹‘𝑛)))
175	174	cbvsumv 15747	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑚 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑚)) = Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛))
176	175	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → Σ𝑚 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑚)) = Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛)))
177	173, 176	eqtrd 2804	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝑀‘(𝐸‘𝑖)) = Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛)))
178	177	breq1d 5123	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑀‘(𝐸‘𝑖)) ≤ 𝑥 ↔ Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛)) ≤ 𝑥))
179	178	ralbidva 3192	. . . . . . . . . . . 12 ⊢ (𝜑 → (∀𝑖 ∈ 𝑍 (𝑀‘(𝐸‘𝑖)) ≤ 𝑥 ↔ ∀𝑖 ∈ 𝑍 Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛)) ≤ 𝑥))
180	179	biimpd 232	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑖 ∈ 𝑍 (𝑀‘(𝐸‘𝑖)) ≤ 𝑥 → ∀𝑖 ∈ 𝑍 Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛)) ≤ 𝑥))
181	180	imp 411	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑖 ∈ 𝑍 (𝑀‘(𝐸‘𝑖)) ≤ 𝑥) → ∀𝑖 ∈ 𝑍 Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛)) ≤ 𝑥)
182	101, 181	syldan 602	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → ∀𝑖 ∈ 𝑍 Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛)) ≤ 𝑥)
183	182	ex 417	. . . . . . . 8 ⊢ (𝜑 → (∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 → ∀𝑖 ∈ 𝑍 Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛)) ≤ 𝑥))
184	183	reximdv 3186	. . . . . . 7 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛)) ≤ 𝑥))
185	14, 184	mpd 16	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛)) ≤ 𝑥)
186	65, 66, 2, 1, 97, 185	sge0reuzb 47088	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐹‘𝑛)))) = sup(ran (𝑖 ∈ 𝑍 ↦ Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛))), ℝ, < ))
187	98	cbvmptv 5219	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) = (𝑖 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑖)))
188	35, 187	eqtri 2792	. . . . . . . . 9 ⊢ 𝑆 = (𝑖 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑖)))
189	188	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑆 = (𝑖 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑖))))
190	177	mpteq2dva 5208	. . . . . . . 8 ⊢ (𝜑 → (𝑖 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑖))) = (𝑖 ∈ 𝑍 ↦ Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛))))
191	189, 190	eqtrd 2804	. . . . . . 7 ⊢ (𝜑 → 𝑆 = (𝑖 ∈ 𝑍 ↦ Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛))))
192	191	rneqd 5929	. . . . . 6 ⊢ (𝜑 → ran 𝑆 = ran (𝑖 ∈ 𝑍 ↦ Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛))))
193	192	supeq1d 9406	. . . . 5 ⊢ (𝜑 → sup(ran 𝑆, ℝ, < ) = sup(ran (𝑖 ∈ 𝑍 ↦ Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛))), ℝ, < ))
194	186, 193	eqtr4d 2807	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐹‘𝑛)))) = sup(ran 𝑆, ℝ, < ))
195	194	eqcomd 2775	. . 3 ⊢ (𝜑 → sup(ran 𝑆, ℝ, < ) = (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐹‘𝑛)))))
196	1	uzct 45709	. . . . . 6 ⊢ 𝑍 ≼ ω
197	196	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑍 ≼ ω)
198	65, 7, 9, 90, 197, 153	meadjiun 47106	. . . 4 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛)) = (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐹‘𝑛)))))
199	198	eqcomd 2775	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐹‘𝑛)))) = (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛)))
200	118	simplrd 781	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛) = ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
201	200	fveq2d 6886	. . 3 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛)) = (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
202	195, 199, 201	3eqtrd 2808	. 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ, < ) = (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
203	64, 202	breqtrd 5141	1 ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ∀wral 3085 ∃wrex 3095 Vcvv 3463 ∖ cdif 3910 ⊆ wss 3913 ∪ ciun 4960 Disj wdisj 5080 class class class wbr 5113 ↦ cmpt 5196 dom cdm 5662 ran crn 5663 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 ωcom 7862 ≼ cdom 8941 supcsup 9400 ℝcr 11099 0cc0 11100 1c1 11101 + caddc 11103 +∞cpnf 11240 ℝ^*cxr 11242 < clt 11243 ≤ cle 11244 ℤcz 12591 ℤ_≥cuz 12862 [,)cico 13374 ...cfz 13535 ..^cfzo 13682 ⇝ cli 15535 Σcsu 15737 SAlgcsalg 46948 Σ^{^}csumge0 47002 Meascmea 47089

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9610 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-disj 5081 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-oadd 8457 df-omul 8458 df-er 8694 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9402 df-oi 9472 df-card 9925 df-acn 9928 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-n0 12505 df-z 12592 df-uz 12863 df-rp 13017 df-xadd 13138 df-ico 13378 df-icc 13379 df-fz 13536 df-fzo 13683 df-seq 14038 df-exp 14098 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15539 df-sum 15738 df-salg 46949 df-sumge0 47003 df-mea 47090

This theorem is referenced by: meaiuninc 47121

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