meadjiunlem - Mathbox for Glauco Siliprandi

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Theorem meadjiunlem 47108

Description: The sum of nonnegative extended reals, restricted to the range of another function. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

**Hypotheses**
Ref	Expression
meadjiunlem.f	⊢ (𝜑 → 𝑀 ∈ Meas)
meadjiunlem.3	⊢ 𝑆 = dom 𝑀
meadjiunlem.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
meadjiunlem.g	⊢ (𝜑 → 𝐺:𝑋⟶𝑆)
meadjiunlem.y	⊢ 𝑌 = {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅}
meadjiunlem.dj	⊢ (𝜑 → Disj 𝑖 ∈ 𝑋 (𝐺‘𝑖))

**Assertion**
Ref	Expression
meadjiunlem	⊢ (𝜑 → (Σ^{^}‘(𝑀 ↾ ran 𝐺)) = (Σ^{^}‘(𝑀 ∘ 𝐺)))

Distinct variable groups: 𝑖,𝐺 𝑖,𝑋 𝑖,𝑌 𝜑,𝑖 Allowed substitution hints: 𝑆(𝑖) 𝑀(𝑖) 𝑉(𝑖)

Proof of Theorem meadjiunlem Dummy variables 𝑗 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1941	. . . 4 ⊢ Ⅎ𝑘𝜑
2		meadjiunlem.g	. . . . . 6 ⊢ (𝜑 → 𝐺:𝑋⟶𝑆)
3		meadjiunlem.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
4	2, 3	jca 520	. . . . 5 ⊢ (𝜑 → (𝐺:𝑋⟶𝑆 ∧ 𝑋 ∈ 𝑉))
5		fex 7227	. . . . 5 ⊢ ((𝐺:𝑋⟶𝑆 ∧ 𝑋 ∈ 𝑉) → 𝐺 ∈ V)
6		rnexg 7901	. . . . 5 ⊢ (𝐺 ∈ V → ran 𝐺 ∈ V)
7	4, 5, 6	3syl 19	. . . 4 ⊢ (𝜑 → ran 𝐺 ∈ V)
8		difssd 4099	. . . 4 ⊢ (𝜑 → (ran 𝐺 ∖ {∅}) ⊆ ran 𝐺)
9		meadjiunlem.f	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ Meas)
10		meadjiunlem.3	. . . . . . 7 ⊢ 𝑆 = dom 𝑀
11	9, 10	meaf 47096	. . . . . 6 ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞))
12	11	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → 𝑀:𝑆⟶(0[,]+∞))
13	2	frnd 6717	. . . . . . 7 ⊢ (𝜑 → ran 𝐺 ⊆ 𝑆)
14	13	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → ran 𝐺 ⊆ 𝑆)
15	8	sselda 3945	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ ran 𝐺)
16	14, 15	sseldd 3946	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ 𝑆)
17	12, 16	ffvelcdmd 7083	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → (𝑀‘𝑘) ∈ (0[,]+∞))
18		simpl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))) → 𝜑)
19		id 23	. . . . . . . 8 ⊢ (𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})))
20		dfin4 4239	. . . . . . . . 9 ⊢ (ran 𝐺 ∩ {∅}) = (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))
21	20	eqcomi 2778	. . . . . . . 8 ⊢ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})) = (ran 𝐺 ∩ {∅})
22	19, 21	eleqtrdi 2879	. . . . . . 7 ⊢ (𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ (ran 𝐺 ∩ {∅}))
23		elinel2 4163	. . . . . . . 8 ⊢ (𝑘 ∈ (ran 𝐺 ∩ {∅}) → 𝑘 ∈ {∅})
24		elsni 4611	. . . . . . . 8 ⊢ (𝑘 ∈ {∅} → 𝑘 = ∅)
25	23, 24	syl 18	. . . . . . 7 ⊢ (𝑘 ∈ (ran 𝐺 ∩ {∅}) → 𝑘 = ∅)
26	22, 25	syl 18	. . . . . 6 ⊢ (𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})) → 𝑘 = ∅)
27	26	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))) → 𝑘 = ∅)
28		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = ∅) → 𝑘 = ∅)
29	28	fveq2d 6888	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = ∅) → (𝑀‘𝑘) = (𝑀‘∅))
30	9	mea0 47097	. . . . . . 7 ⊢ (𝜑 → (𝑀‘∅) = 0)
31	30	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = ∅) → (𝑀‘∅) = 0)
32	29, 31	eqtrd 2804	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = ∅) → (𝑀‘𝑘) = 0)
33	18, 27, 32	syl2anc 595	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))) → (𝑀‘𝑘) = 0)
34	1, 7, 8, 17, 33	sge0ss 47055	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))) = (Σ^{^}‘(𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘))))
35	34	eqcomd 2775	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘))) = (Σ^{^}‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))))
36	11, 13	feqresmpt 6953	. . 3 ⊢ (𝜑 → (𝑀 ↾ ran 𝐺) = (𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘)))
37	36	fveq2d 6888	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑀 ↾ ran 𝐺)) = (Σ^{^}‘(𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘))))
38	2	ffvelcdmda 7082	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐺‘𝑗) ∈ 𝑆)
39	2	feqmptd 6952	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑗 ∈ 𝑋 ↦ (𝐺‘𝑗)))
40	11	feqmptd 6952	. . . . 5 ⊢ (𝜑 → 𝑀 = (𝑘 ∈ 𝑆 ↦ (𝑀‘𝑘)))
41		fveq2 6884	. . . . 5 ⊢ (𝑘 = (𝐺‘𝑗) → (𝑀‘𝑘) = (𝑀‘(𝐺‘𝑗)))
42	38, 39, 40, 41	fmptco 7128	. . . 4 ⊢ (𝜑 → (𝑀 ∘ 𝐺) = (𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗))))
43	42	fveq2d 6888	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑀 ∘ 𝐺)) = (Σ^{^}‘(𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗)))))
44		nfv 1941	. . . . 5 ⊢ Ⅎ𝑗𝜑
45		meadjiunlem.y	. . . . . 6 ⊢ 𝑌 = {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅}
46		ssrab2 4042	. . . . . . 7 ⊢ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ⊆ 𝑋
47	46	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ⊆ 𝑋)
48	45, 47	eqsstrid 3983	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
49	11	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑀:𝑆⟶(0[,]+∞))
50	2	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐺:𝑋⟶𝑆)
51	48	sselda 3945	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 ∈ 𝑋)
52	50, 51	ffvelcdmd 7083	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗) ∈ 𝑆)
53	49, 52	ffvelcdmd 7083	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑀‘(𝐺‘𝑗)) ∈ (0[,]+∞))
54		eldifi 4093	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (𝑋 ∖ 𝑌) → 𝑗 ∈ 𝑋)
55	54	ad2antlr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → 𝑗 ∈ 𝑋)
56		fveq2 6884	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘𝑗) = ∅ → (𝑀‘(𝐺‘𝑗)) = (𝑀‘∅))
57	56	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → (𝑀‘(𝐺‘𝑗)) = (𝑀‘∅))
58	9	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → 𝑀 ∈ Meas)
59	58	mea0 47097	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → (𝑀‘∅) = 0)
60	57, 59	eqtrd 2804	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → (𝑀‘(𝐺‘𝑗)) = 0)
61	60	ad4ant14 764	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) ∧ (𝐺‘𝑗) = ∅) → (𝑀‘(𝐺‘𝑗)) = 0)
62		neneq 2970	. . . . . . . . . . . . 13 ⊢ ((𝑀‘(𝐺‘𝑗)) ≠ 0 → ¬ (𝑀‘(𝐺‘𝑗)) = 0)
63	62	ad2antlr 739	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) ∧ (𝐺‘𝑗) = ∅) → ¬ (𝑀‘(𝐺‘𝑗)) = 0)
64	61, 63	pm2.65da 828	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → ¬ (𝐺‘𝑗) = ∅)
65	64	neqned 2971	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → (𝐺‘𝑗) ≠ ∅)
66	55, 65	jca 520	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → (𝑗 ∈ 𝑋 ∧ (𝐺‘𝑗) ≠ ∅))
67		fveq2 6884	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (𝐺‘𝑖) = (𝐺‘𝑗))
68	67	neeq1d 3023	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((𝐺‘𝑖) ≠ ∅ ↔ (𝐺‘𝑗) ≠ ∅))
69	68	elrab 3659	. . . . . . . . 9 ⊢ (𝑗 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ↔ (𝑗 ∈ 𝑋 ∧ (𝐺‘𝑗) ≠ ∅))
70	66, 69	sylibr 237	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → 𝑗 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅})
71	70, 45	eleqtrrdi 2880	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → 𝑗 ∈ 𝑌)
72		eldifn 4094	. . . . . . . 8 ⊢ (𝑗 ∈ (𝑋 ∖ 𝑌) → ¬ 𝑗 ∈ 𝑌)
73	72	ad2antlr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → ¬ 𝑗 ∈ 𝑌)
74	71, 73	pm2.65da 828	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) → ¬ (𝑀‘(𝐺‘𝑗)) ≠ 0)
75		nne 2968	. . . . . 6 ⊢ (¬ (𝑀‘(𝐺‘𝑗)) ≠ 0 ↔ (𝑀‘(𝐺‘𝑗)) = 0)
76	74, 75	sylib 221	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) → (𝑀‘(𝐺‘𝑗)) = 0)
77	44, 3, 48, 53, 76	sge0ss 47055	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗)))))
78	77	eqcomd 2775	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗)))))
79	3, 48	ssexd 5297	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ V)
80		nfv 1941	. . . . . . . . 9 ⊢ Ⅎ𝑖𝜑
81		eqid 2769	. . . . . . . . 9 ⊢ (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) = (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖))
82	2	ffnd 6709	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 Fn 𝑋)
83		dffn3 6721	. . . . . . . . . . . . 13 ⊢ (𝐺 Fn 𝑋 ↔ 𝐺:𝑋⟶ran 𝐺)
84	82, 83	sylib 221	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:𝑋⟶ran 𝐺)
85	84	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → 𝐺:𝑋⟶ran 𝐺)
86	48	sselda 3945	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → 𝑖 ∈ 𝑋)
87	85, 86	ffvelcdmd 7083	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → (𝐺‘𝑖) ∈ ran 𝐺)
88	45	eleq2i 2861	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝑌 ↔ 𝑖 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅})
89		rabid 3444	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ↔ (𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅))
90	88, 89	bitri 278	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ 𝑌 ↔ (𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅))
91	90	biimpi 219	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ 𝑌 → (𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅))
92	91	simprd 500	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ 𝑌 → (𝐺‘𝑖) ≠ ∅)
93	92	adantl 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → (𝐺‘𝑖) ≠ ∅)
94		nelsn 4637	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑖) ≠ ∅ → ¬ (𝐺‘𝑖) ∈ {∅})
95	93, 94	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → ¬ (𝐺‘𝑖) ∈ {∅})
96	87, 95	eldifd 3924	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → (𝐺‘𝑖) ∈ (ran 𝐺 ∖ {∅}))
97		meadjiunlem.dj	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑖 ∈ 𝑋 (𝐺‘𝑖))
98		disjss1 5086	. . . . . . . . . 10 ⊢ (𝑌 ⊆ 𝑋 → (Disj 𝑖 ∈ 𝑋 (𝐺‘𝑖) → Disj 𝑖 ∈ 𝑌 (𝐺‘𝑖)))
99	48, 97, 98	sylc 66	. . . . . . . . 9 ⊢ (𝜑 → Disj 𝑖 ∈ 𝑌 (𝐺‘𝑖))
100	80, 81, 96, 93, 99	disjf1 45830	. . . . . . . 8 ⊢ (𝜑 → (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)):𝑌–1-1→(ran 𝐺 ∖ {∅}))
101	2, 48	feqresmpt 6953	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ↾ 𝑌) = (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)))
102		f1eq1 6772	. . . . . . . . 9 ⊢ ((𝐺 ↾ 𝑌) = (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) → ((𝐺 ↾ 𝑌):𝑌–1-1→(ran 𝐺 ∖ {∅}) ↔ (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)):𝑌–1-1→(ran 𝐺 ∖ {∅})))
103	101, 102	syl 18	. . . . . . . 8 ⊢ (𝜑 → ((𝐺 ↾ 𝑌):𝑌–1-1→(ran 𝐺 ∖ {∅}) ↔ (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)):𝑌–1-1→(ran 𝐺 ∖ {∅})))
104	100, 103	mpbird 260	. . . . . . 7 ⊢ (𝜑 → (𝐺 ↾ 𝑌):𝑌–1-1→(ran 𝐺 ∖ {∅}))
105	101	rneqd 5931	. . . . . . . . 9 ⊢ (𝜑 → ran (𝐺 ↾ 𝑌) = ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)))
106	96	ralrimiva 3163	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑖 ∈ 𝑌 (𝐺‘𝑖) ∈ (ran 𝐺 ∖ {∅}))
107	81	rnmptss 7121	. . . . . . . . . 10 ⊢ (∀𝑖 ∈ 𝑌 (𝐺‘𝑖) ∈ (ran 𝐺 ∖ {∅}) → ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) ⊆ (ran 𝐺 ∖ {∅}))
108	106, 107	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) ⊆ (ran 𝐺 ∖ {∅}))
109	105, 108	eqsstrd 3979	. . . . . . . 8 ⊢ (𝜑 → ran (𝐺 ↾ 𝑌) ⊆ (ran 𝐺 ∖ {∅}))
110		simpl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝜑)
111		eldifi 4093	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ran 𝐺 ∖ {∅}) → 𝑥 ∈ ran 𝐺)
112	111	adantl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝑥 ∈ ran 𝐺)
113		eldifsni 4762	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ran 𝐺 ∖ {∅}) → 𝑥 ≠ ∅)
114	113	adantl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝑥 ≠ ∅)
115		simpr 489	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺) → 𝑥 ∈ ran 𝐺)
116		fvelrnb 6944	. . . . . . . . . . . . . 14 ⊢ (𝐺 Fn 𝑋 → (𝑥 ∈ ran 𝐺 ↔ ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥))
117	82, 116	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ ran 𝐺 ↔ ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥))
118	117	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺) → (𝑥 ∈ ran 𝐺 ↔ ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥))
119	115, 118	mpbid 235	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺) → ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥)
120	119	3adant3 1148	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺 ∧ 𝑥 ≠ ∅) → ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥)
121		id 23	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘𝑖) = 𝑥 → (𝐺‘𝑖) = 𝑥)
122	121	eqcomd 2775	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘𝑖) = 𝑥 → 𝑥 = (𝐺‘𝑖))
123	122	3ad2ant3 1151	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝑥 = (𝐺‘𝑖))
124		simp1l 1214	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝜑)
125		simp2 1153	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝑖 ∈ 𝑋)
126		simpr 489	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ≠ ∅ ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) = 𝑥)
127		simpl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ≠ ∅ ∧ (𝐺‘𝑖) = 𝑥) → 𝑥 ≠ ∅)
128	126, 127	eqnetrd 3031	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ≠ ∅ ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) ≠ ∅)
129	128	adantll 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) ≠ ∅)
130	129	3adant2 1147	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) ≠ ∅)
131	90	biimpri 231	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → 𝑖 ∈ 𝑌)
132		fvexd 6899	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → (𝐺‘𝑖) ∈ V)
133	81	elrnmpt1 5953	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ 𝑌 ∧ (𝐺‘𝑖) ∈ V) → (𝐺‘𝑖) ∈ ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)))
134	131, 132, 133	syl2anc 595	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → (𝐺‘𝑖) ∈ ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)))
135	134	3adant1 1146	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → (𝐺‘𝑖) ∈ ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)))
136	105	eqcomd 2775	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) = ran (𝐺 ↾ 𝑌))
137	136	3ad2ant1 1149	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) = ran (𝐺 ↾ 𝑌))
138	135, 137	eleqtrd 2871	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → (𝐺‘𝑖) ∈ ran (𝐺 ↾ 𝑌))
139	124, 125, 130, 138	syl3anc 1396	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) ∈ ran (𝐺 ↾ 𝑌))
140	123, 139	eqeltrd 2869	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝑥 ∈ ran (𝐺 ↾ 𝑌))
141	140	3exp 1135	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ≠ ∅) → (𝑖 ∈ 𝑋 → ((𝐺‘𝑖) = 𝑥 → 𝑥 ∈ ran (𝐺 ↾ 𝑌))))
142	141	rexlimdv 3170	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ≠ ∅) → (∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥 → 𝑥 ∈ ran (𝐺 ↾ 𝑌)))
143	142	3adant2 1147	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺 ∧ 𝑥 ≠ ∅) → (∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥 → 𝑥 ∈ ran (𝐺 ↾ 𝑌)))
144	120, 143	mpd 16	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺 ∧ 𝑥 ≠ ∅) → 𝑥 ∈ ran (𝐺 ↾ 𝑌))
145	110, 112, 114, 144	syl3anc 1396	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝑥 ∈ ran (𝐺 ↾ 𝑌))
146	109, 145	eqelssd 3966	. . . . . . 7 ⊢ (𝜑 → ran (𝐺 ↾ 𝑌) = (ran 𝐺 ∖ {∅}))
147	104, 146	jca 520	. . . . . 6 ⊢ (𝜑 → ((𝐺 ↾ 𝑌):𝑌–1-1→(ran 𝐺 ∖ {∅}) ∧ ran (𝐺 ↾ 𝑌) = (ran 𝐺 ∖ {∅})))
148		dff1o5 6833	. . . . . 6 ⊢ ((𝐺 ↾ 𝑌):𝑌–1-1-onto→(ran 𝐺 ∖ {∅}) ↔ ((𝐺 ↾ 𝑌):𝑌–1-1→(ran 𝐺 ∖ {∅}) ∧ ran (𝐺 ↾ 𝑌) = (ran 𝐺 ∖ {∅})))
149	147, 148	sylibr 237	. . . . 5 ⊢ (𝜑 → (𝐺 ↾ 𝑌):𝑌–1-1-onto→(ran 𝐺 ∖ {∅}))
150		fvres 6903	. . . . . 6 ⊢ (𝑗 ∈ 𝑌 → ((𝐺 ↾ 𝑌)‘𝑗) = (𝐺‘𝑗))
151	150	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐺 ↾ 𝑌)‘𝑗) = (𝐺‘𝑗))
152	1, 44, 41, 79, 149, 151, 17	sge0f1o 47025	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))) = (Σ^{^}‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗)))))
153	152	eqcomd 2775	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗)))) = (Σ^{^}‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))))
154	43, 78, 153	3eqtrd 2808	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑀 ∘ 𝐺)) = (Σ^{^}‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))))
155	35, 37, 154	3eqtr4d 2814	1 ⊢ (𝜑 → (Σ^{^}‘(𝑀 ↾ ran 𝐺)) = (Σ^{^}‘(𝑀 ∘ 𝐺)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∀wral 3085 ∃wrex 3095 {crab 3423 Vcvv 3463 ∖ cdif 3910 ∩ cin 3912 ⊆ wss 3913 ∅c0 4294 {csn 4594 Disj wdisj 5080 ↦ cmpt 5196 dom cdm 5664 ran crn 5665 ↾ cres 5666 ∘ ccom 5668 Fn wfn 6534 ⟶wf 6535 –1-1→wf1 6536 –1-1-onto→wf1o 6538 ‘cfv 6539 (class class class)co 7413 0cc0 11102 +∞cpnf 11242 [,]cicc 13377 Σ^{^}csumge0 47005 Meascmea 47092

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 5273 ax-pow 5339 ax-pr 5407 ax-un 7735 ax-inf2 9612 ax-cnex 11158 ax-resscn 11159 ax-1cn 11160 ax-icn 11161 ax-addcl 11162 ax-addrcl 11163 ax-mulcl 11164 ax-mulrcl 11165 ax-mulcom 11166 ax-addass 11167 ax-mulass 11168 ax-distr 11169 ax-i2m1 11170 ax-1ne0 11171 ax-1rid 11172 ax-rnegex 11173 ax-rrecex 11174 ax-cnre 11175 ax-pre-lttri 11176 ax-pre-lttrn 11177 ax-pre-ltadd 11178 ax-pre-mulgt0 11179 ax-pre-sup 11180

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 5559 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-se 5618 df-we 5619 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-pred 6305 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6495 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7865 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9404 df-oi 9474 df-card 9927 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11445 df-neg 11446 df-div 11874 df-nn 12236 df-2 12305 df-3 12306 df-n0 12507 df-z 12594 df-uz 12865 df-rp 13019 df-xadd 13140 df-ico 13380 df-icc 13381 df-fz 13538 df-fzo 13685 df-seq 14040 df-exp 14100 df-hash 14369 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-clim 15541 df-sum 15740 df-sumge0 47006 df-mea 47093

This theorem is referenced by: meadjiun 47109

W3C validator