meaiuninclem - Mathbox for Glauco Siliprandi

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Theorem meaiuninclem 47490

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 11356	. . . . . . 7 ⊢ 0 ∈ ℝ^*
4	3	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 0 ∈ ℝ^*)
5		pnfxr 11363	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
6	5	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → +∞ ∈ ℝ^*)
7		meaiuninclem.m	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ Meas)
8	7	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑀 ∈ Meas)
9		eqid 2760	. . . . . . 7 ⊢ dom 𝑀 = dom 𝑀
10		meaiuninclem.e	. . . . . . . 8 ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀)
11	10	ffvelcdmda 7084	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ dom 𝑀)
12	8, 9, 11	meaxrcl 47471	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) ∈ ℝ^*)
13	8, 11	meage0 47485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 0 ≤ (𝑀‘(𝐸‘𝑛)))
14		meaiuninclem.b	. . . . . . . 8 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
15	14	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
16		simp1 1154	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → (𝜑 ∧ 𝑛 ∈ 𝑍))
17		simp2 1155	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑥 ∈ ℝ)
18		simp3 1156	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
19	16	simprd 501	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑛 ∈ 𝑍)
20		rspa 3251	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
21	18, 19, 20	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
22	12	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ℝ ∧ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → (𝑀‘(𝐸‘𝑛)) ∈ ℝ^*)
23		rexr 11355	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
24	23	3ad2ant2 1152	. . . . . . . . . . 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 1156	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ℝ ∧ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
27		ltpnf 13249	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞)
28	27	3ad2ant2 1152	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ℝ ∧ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑥 < +∞)
29	22, 24, 25, 26, 28	xrlelttrd 13289	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ℝ ∧ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → (𝑀‘(𝐸‘𝑛)) < +∞)
30	16, 17, 21, 29	syl3anc 1398	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → (𝑀‘(𝐸‘𝑛)) < +∞)
31	30	3exp 1137	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ ℝ → (∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 → (𝑀‘(𝐸‘𝑛)) < +∞)))
32	31	rexlimdv 3161	. . . . . . 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 13526	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) ∈ (0[,)+∞))
35		meaiuninclem.s	. . . . 5 ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛)))
36	34, 35	fmptd 7114	. . . 4 ⊢ (𝜑 → 𝑆:𝑍⟶(0[,)+∞))
37		rge0ssre 13587	. . . . 5 ⊢ (0[,)+∞) ⊆ ℝ
38	37	a1i 11	. . . 4 ⊢ (𝜑 → (0[,)+∞) ⊆ ℝ)
39	36, 38	fssd 6727	. . 3 ⊢ (𝜑 → 𝑆:𝑍⟶ℝ)
40	1	peano2uzs 13029	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → (𝑛 + 1) ∈ 𝑍)
41	40	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑛 + 1) ∈ 𝑍)
42	10	ffvelcdmda 7084	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝐸‘(𝑛 + 1)) ∈ dom 𝑀)
43	41, 42	syldan 603	. . . . 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 47473	. . . 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 6900	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) ∈ V)
48	46, 47	fvmpt2d 7007	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑆‘𝑛) = (𝑀‘(𝐸‘𝑛)))
49		2fveq3 6890	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑀‘(𝐸‘𝑛)) = (𝑀‘(𝐸‘𝑚)))
50	49	cbvmptv 5208	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) = (𝑚 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑚)))
51	35, 50	eqtri 2783	. . . . . 6 ⊢ 𝑆 = (𝑚 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑚)))
52		2fveq3 6890	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (𝑀‘(𝐸‘𝑚)) = (𝑀‘(𝐸‘(𝑛 + 1))))
53		fvexd 6900	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘(𝑛 + 1))) ∈ V)
54	51, 52, 41, 53	fvmptd3 7017	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑆‘(𝑛 + 1)) = (𝑀‘(𝐸‘(𝑛 + 1))))
55	48, 54	breq12d 5115	. . . 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 2766	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) = (𝑆‘𝑛))
58	57	breq1d 5112	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ (𝑆‘𝑛) ≤ 𝑥))
59	58	ralbidva 3183	. . . . . . 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 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 → ∀𝑛 ∈ 𝑍 (𝑆‘𝑛) ≤ 𝑥))
62	61	reximdva 3175	. . . 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 15837	. 2 ⊢ (𝜑 → 𝑆 ⇝ sup(ran 𝑆, ℝ, < ))
65		nfv 1946	. . . . . 6 ⊢ Ⅎ𝑛𝜑
66		nfv 1946	. . . . . 6 ⊢ Ⅎ𝑥𝜑
67		id 23	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ 𝑍)
68		fvex 6898	. . . . . . . . . . . . 13 ⊢ (𝐸‘𝑛) ∈ V
69	68	difexi 5291	. . . . . . . . . . . 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 7005	. . . . . . . . . . 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 596	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 → (𝐹‘𝑛) = ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖)))
74	73	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) = ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖)))
75	7, 9	dmmeasal 47462	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝑀 ∈ SAlg)
76	75	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom 𝑀 ∈ SAlg)
77		fzoct 46395	. . . . . . . . . . . 12 ⊢ (𝑁..^𝑛) ≼ ω
78	77	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑁..^𝑛) ≼ ω)
79	10	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑁..^𝑛)) → 𝐸:𝑍⟶dom 𝑀)
80		fzossuz 46392	. . . . . . . . . . . . . . . 16 ⊢ (𝑁..^𝑛) ⊆ (ℤ_≥‘𝑁)
81	1	eqcomi 2769	. . . . . . . . . . . . . . . 16 ⊢ (ℤ_≥‘𝑁) = 𝑍
82	80, 81	sseqtri 3956	. . . . . . . . . . . . . . 15 ⊢ (𝑁..^𝑛) ⊆ 𝑍
83	82	sseli 3904	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (𝑁..^𝑛) → 𝑖 ∈ 𝑍)
84	83	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑁..^𝑛)) → 𝑖 ∈ 𝑍)
85	79, 84	ffvelcdmd 7085	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑁..^𝑛)) → (𝐸‘𝑖) ∈ dom 𝑀)
86	85	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑖 ∈ (𝑁..^𝑛)) → (𝐸‘𝑖) ∈ dom 𝑀)
87	76, 78, 86	saliuncl 47333	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖) ∈ dom 𝑀)
88		saldifcl2 47338	. . . . . . . . . 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 1398	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖)) ∈ dom 𝑀)
90	74, 89	eqeltrd 2860	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ dom 𝑀)
91	8, 9, 90	meaxrcl 47471	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐹‘𝑛)) ∈ ℝ^*)
92	8, 90	meage0 47485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 0 ≤ (𝑀‘(𝐹‘𝑛)))
93		difssd 4073	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖)) ⊆ (𝐸‘𝑛))
94	74, 93	eqsstrd 3942	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ⊆ (𝐸‘𝑛))
95	8, 9, 90, 11, 94	meassle 47473	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐹‘𝑛)) ≤ (𝑀‘(𝐸‘𝑛)))
96	91, 12, 6, 95, 33	xrlelttrd 13289	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐹‘𝑛)) < +∞)
97	4, 6, 91, 92, 96	elicod 13526	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐹‘𝑛)) ∈ (0[,)+∞))
98		2fveq3 6890	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑖 → (𝑀‘(𝐸‘𝑛)) = (𝑀‘(𝐸‘𝑖)))
99	98	breq1d 5112	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑖 → ((𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ (𝑀‘(𝐸‘𝑖)) ≤ 𝑥))
100	99	cbvralvw 3240	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ∀𝑖 ∈ 𝑍 (𝑀‘(𝐸‘𝑖)) ≤ 𝑥)
101	100	bilani 510	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → ∀𝑖 ∈ 𝑍 (𝑀‘(𝐸‘𝑖)) ≤ 𝑥)
102		eleq1w 2843	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑖 → (𝑛 ∈ 𝑍 ↔ 𝑖 ∈ 𝑍))
103	102	anbi2d 642	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑖 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑖 ∈ 𝑍)))
104		oveq2 7428	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑖 → (𝑁...𝑛) = (𝑁...𝑖))
105	104	sumeq1d 15867	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑖 → Σ𝑚 ∈ (𝑁...𝑛)(𝑀‘(𝐹‘𝑚)) = Σ𝑚 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑚)))
106	98, 105	eqeq12d 2776	. . . . . . . . . . . . . . . . 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 2843	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑛 → (𝑚 ∈ 𝑍 ↔ 𝑛 ∈ 𝑍))
109	108	anbi2d 642	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑛 → ((𝜑 ∧ 𝑚 ∈ 𝑍) ↔ (𝜑 ∧ 𝑛 ∈ 𝑍)))
110		oveq2 7428	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑛 → (𝑁...𝑚) = (𝑁...𝑛))
111	110	iuneq1d 4978	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑛 → ∪ 𝑖 ∈ (𝑁...𝑚)(𝐹‘𝑖) = ∪ 𝑖 ∈ (𝑁...𝑛)(𝐹‘𝑖))
112	110	iuneq1d 4978	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑛 → ∪ 𝑖 ∈ (𝑁...𝑚)(𝐸‘𝑖) = ∪ 𝑖 ∈ (𝑁...𝑛)(𝐸‘𝑖))
113	111, 112	eqeq12d 2776	. . . . . . . . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑛 → (𝐹‘𝑖) = (𝐹‘𝑛))
116	115	cbviunv 4996	. . . . . . . . . . . . . . . . . . . . . 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 47470	. . . . . . . . . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)(𝐹‘𝑛) = ∪ 𝑛 ∈ (𝑁...𝑚)(𝐸‘𝑛))
120	119	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)(𝐹‘𝑛) = ∪ 𝑛 ∈ (𝑁...𝑚)(𝐸‘𝑛))
121		simpr 490	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ 𝑍)
122		rspa 3251	. . . . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ∪ 𝑛 ∈ (𝑁...𝑚)(𝐹‘𝑛) = ∪ 𝑛 ∈ (𝑁...𝑚)(𝐸‘𝑛))
124		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑖 → (𝐸‘𝑛) = (𝐸‘𝑖))
125	124	cbviunv 4996	. . . . . . . . . . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ∪ 𝑖 ∈ (𝑁...𝑚)(𝐹‘𝑖) = ∪ 𝑖 ∈ (𝑁...𝑚)(𝐸‘𝑖))
128	114, 127	chvarvv 2021	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∪ 𝑖 ∈ (𝑁...𝑛)(𝐹‘𝑖) = ∪ 𝑖 ∈ (𝑁...𝑛)(𝐸‘𝑖))
129	67, 1	eleqtrdi 2870	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ_≥‘𝑁))
130	129	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ (ℤ_≥‘𝑁))
131		fvoveq1 7443	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑖 → (𝐸‘(𝑛 + 1)) = (𝐸‘(𝑖 + 1)))
132	124, 131	sseq12d 3941	. . . . . . . . . . . . . . . . . . . . . . . 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 2021	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐸‘𝑖) ⊆ (𝐸‘(𝑖 + 1)))
135	84, 134	syldan 603	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑁..^𝑛)) → (𝐸‘𝑖) ⊆ (𝐸‘(𝑖 + 1)))
136	135	adantlr 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑖 ∈ (𝑁..^𝑛)) → (𝐸‘𝑖) ⊆ (𝐸‘(𝑖 + 1)))
137	130, 136	iunincfi 46109	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∪ 𝑖 ∈ (𝑁...𝑛)(𝐸‘𝑖) = (𝐸‘𝑛))
138	128, 137	eqtr2d 2796	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) = ∪ 𝑖 ∈ (𝑁...𝑛)(𝐹‘𝑖))
139	138	fveq2d 6889	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) = (𝑀‘∪ 𝑖 ∈ (𝑁...𝑛)(𝐹‘𝑖)))
140		nfv 1946	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑛 ∈ 𝑍)
141		elfzuz 13652	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (𝑁...𝑛) → 𝑖 ∈ (ℤ_≥‘𝑁))
142	141, 81	eleqtrdi 2870	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (𝑁...𝑛) → 𝑖 ∈ 𝑍)
143	142	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑁...𝑛)) → 𝑖 ∈ 𝑍)
144		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑖 → (𝐹‘𝑛) = (𝐹‘𝑖))
145	144	eleq1d 2845	. . . . . . . . . . . . . . . . . . . . . 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 2021	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹‘𝑖) ∈ dom 𝑀)
148	143, 147	syldan 603	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑁...𝑛)) → (𝐹‘𝑖) ∈ dom 𝑀)
149	148	adantlr 728	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑖 ∈ (𝑁...𝑛)) → (𝐹‘𝑖) ∈ dom 𝑀)
150		fzct 46390	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁...𝑛) ≼ ω
151	150	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑁...𝑛) ≼ ω)
152	143	ssd 46097	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑁...𝑛) ⊆ 𝑍)
153	118	simprd 501	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → Disj 𝑛 ∈ 𝑍 (𝐹‘𝑛))
154	144	cbvdisjv 5080	. . . . . . . . . . . . . . . . . . . . 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 5075	. . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → Disj 𝑖 ∈ (𝑁...𝑛)(𝐹‘𝑖))
159	140, 8, 9, 149, 151, 158	meadjiun 47476	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘∪ 𝑖 ∈ (𝑁...𝑛)(𝐹‘𝑖)) = (Σ^{^}‘(𝑖 ∈ (𝑁...𝑛) ↦ (𝑀‘(𝐹‘𝑖)))))
160		fzfid 14116	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑁...𝑛) ∈ Fin)
161		2fveq3 6890	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑖 → (𝑀‘(𝐹‘𝑛)) = (𝑀‘(𝐹‘𝑖)))
162	161	eleq1d 2845	. . . . . . . . . . . . . . . . . . . . . . 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 2021	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝑀‘(𝐹‘𝑖)) ∈ (0[,)+∞))
165	143, 164	syldan 603	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑁...𝑛)) → (𝑀‘(𝐹‘𝑖)) ∈ (0[,)+∞))
166	165	adantlr 728	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑖 ∈ (𝑁...𝑛)) → (𝑀‘(𝐹‘𝑖)) ∈ (0[,)+∞))
167	160, 166	sge0fsummpt 47400	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (Σ^{^}‘(𝑖 ∈ (𝑁...𝑛) ↦ (𝑀‘(𝐹‘𝑖)))) = Σ𝑖 ∈ (𝑁...𝑛)(𝑀‘(𝐹‘𝑖)))
168		2fveq3 6890	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑚 → (𝑀‘(𝐹‘𝑖)) = (𝑀‘(𝐹‘𝑚)))
169	168	cbvsumv 15863	. . . . . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (Σ^{^}‘(𝑖 ∈ (𝑁...𝑛) ↦ (𝑀‘(𝐹‘𝑖)))) = Σ𝑚 ∈ (𝑁...𝑛)(𝑀‘(𝐹‘𝑚)))
172	139, 159, 171	3eqtrd 2799	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) = Σ𝑚 ∈ (𝑁...𝑛)(𝑀‘(𝐹‘𝑚)))
173	107, 172	chvarvv 2021	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝑀‘(𝐸‘𝑖)) = Σ𝑚 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑚)))
174		2fveq3 6890	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑛 → (𝑀‘(𝐹‘𝑚)) = (𝑀‘(𝐹‘𝑛)))
175	174	cbvsumv 15863	. . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝑀‘(𝐸‘𝑖)) = Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛)))
178	177	breq1d 5112	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑀‘(𝐸‘𝑖)) ≤ 𝑥 ↔ Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛)) ≤ 𝑥))
179	178	ralbidva 3183	. . . . . . . . . . . 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 412	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑖 ∈ 𝑍 (𝑀‘(𝐸‘𝑖)) ≤ 𝑥) → ∀𝑖 ∈ 𝑍 Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛)) ≤ 𝑥)
182	101, 181	syldan 603	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → ∀𝑖 ∈ 𝑍 Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛)) ≤ 𝑥)
183	182	ex 418	. . . . . . . 8 ⊢ (𝜑 → (∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 → ∀𝑖 ∈ 𝑍 Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛)) ≤ 𝑥))
184	183	reximdv 3177	. . . . . . 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 47458	. . . . 5 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐹‘𝑛)))) = sup(ran (𝑖 ∈ 𝑍 ↦ Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛))), ℝ, < ))
187	98	cbvmptv 5208	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) = (𝑖 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑖)))
188	35, 187	eqtri 2783	. . . . . . . . 9 ⊢ 𝑆 = (𝑖 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑖)))
189	188	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑆 = (𝑖 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑖))))
190	177	mpteq2dva 5197	. . . . . . . 8 ⊢ (𝜑 → (𝑖 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑖))) = (𝑖 ∈ 𝑍 ↦ Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛))))
191	189, 190	eqtrd 2795	. . . . . . 7 ⊢ (𝜑 → 𝑆 = (𝑖 ∈ 𝑍 ↦ Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛))))
192	191	rneqd 5919	. . . . . 6 ⊢ (𝜑 → ran 𝑆 = ran (𝑖 ∈ 𝑍 ↦ Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛))))
193	192	supeq1d 9438	. . . . 5 ⊢ (𝜑 → sup(ran 𝑆, ℝ, < ) = sup(ran (𝑖 ∈ 𝑍 ↦ Σ𝑛 ∈ (𝑁...𝑖)(𝑀‘(𝐹‘𝑛))), ℝ, < ))
194	186, 193	eqtr4d 2798	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐹‘𝑛)))) = sup(ran 𝑆, ℝ, < ))
195	194	eqcomd 2766	. . 3 ⊢ (𝜑 → sup(ran 𝑆, ℝ, < ) = (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐹‘𝑛)))))
196	1	uzct 46080	. . . . . 6 ⊢ 𝑍 ≼ ω
197	196	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑍 ≼ ω)
198	65, 7, 9, 90, 197, 153	meadjiun 47476	. . . 4 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛)) = (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐹‘𝑛)))))
199	198	eqcomd 2766	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐹‘𝑛)))) = (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛)))
200	118	simplrd 782	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛) = ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
201	200	fveq2d 6889	. . 3 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛)) = (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
202	195, 199, 201	3eqtrd 2799	. 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ, < ) = (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
203	64, 202	breqtrd 5130	1 ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ⊆ wss 3895 ∖ cdif 4045 ∪ ciun 4950 Disj wdisj 5069 class class class wbr 5102 ↦ cmpt 5185 dom cdm 5650 ran crn 5651 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ωcom 7877 ≼ cdom 8971 supcsup 9432 ℝcr 11199 0cc0 11200 1c1 11201 + caddc 11203 +∞cpnf 11340 ℝ^*cxr 11342 < clt 11343 ≤ cle 11344 ℤcz 12693 ℤ_≥cuz 12965 [,)cico 13478 ...cfz 13639 ..^cfzo 13788 ⇝ cli 15651 Σcsu 15853 SAlgcsalg 47318 Σ^{^}csumge0 47372 Meascmea 47459

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-disj 5070 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-oadd 8480 df-omul 8481 df-er 8717 df-map 8849 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-sup 9434 df-oi 9504 df-card 10020 df-acn 10023 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-n0 12607 df-z 12694 df-uz 12966 df-rp 13121 df-xadd 13242 df-ico 13482 df-icc 13483 df-fz 13640 df-fzo 13789 df-seq 14145 df-exp 14205 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-clim 15655 df-sum 15854 df-salg 47319 df-sumge0 47373 df-mea 47460

This theorem is used by: meaiuninc 47491

W3C validator