meadjiunlem - Mathbox for Glauco Siliprandi

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Theorem meadjiunlem 42767

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 1915	. . . 4 ⊢ Ⅎ𝑘𝜑
2		meadjiunlem.g	. . . . . 6 ⊢ (𝜑 → 𝐺:𝑋⟶𝑆)
3		meadjiunlem.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
4	2, 3	jca 514	. . . . 5 ⊢ (𝜑 → (𝐺:𝑋⟶𝑆 ∧ 𝑋 ∈ 𝑉))
5		fex 6989	. . . . 5 ⊢ ((𝐺:𝑋⟶𝑆 ∧ 𝑋 ∈ 𝑉) → 𝐺 ∈ V)
6		rnexg 7614	. . . . 5 ⊢ (𝐺 ∈ V → ran 𝐺 ∈ V)
7	4, 5, 6	3syl 18	. . . 4 ⊢ (𝜑 → ran 𝐺 ∈ V)
8		difssd 4109	. . . 4 ⊢ (𝜑 → (ran 𝐺 ∖ {∅}) ⊆ ran 𝐺)
9		meadjiunlem.f	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ Meas)
10		meadjiunlem.3	. . . . . . 7 ⊢ 𝑆 = dom 𝑀
11	9, 10	meaf 42755	. . . . . 6 ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞))
12	11	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → 𝑀:𝑆⟶(0[,]+∞))
13	2	frnd 6521	. . . . . . 7 ⊢ (𝜑 → ran 𝐺 ⊆ 𝑆)
14	13	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → ran 𝐺 ⊆ 𝑆)
15	8	sselda 3967	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ ran 𝐺)
16	14, 15	sseldd 3968	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ 𝑆)
17	12, 16	ffvelrnd 6852	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → (𝑀‘𝑘) ∈ (0[,]+∞))
18		simpl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))) → 𝜑)
19		id 22	. . . . . . . 8 ⊢ (𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})))
20		dfin4 4244	. . . . . . . . 9 ⊢ (ran 𝐺 ∩ {∅}) = (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))
21	20	eqcomi 2830	. . . . . . . 8 ⊢ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})) = (ran 𝐺 ∩ {∅})
22	19, 21	eleqtrdi 2923	. . . . . . 7 ⊢ (𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ (ran 𝐺 ∩ {∅}))
23		elinel2 4173	. . . . . . . 8 ⊢ (𝑘 ∈ (ran 𝐺 ∩ {∅}) → 𝑘 ∈ {∅})
24		elsni 4584	. . . . . . . 8 ⊢ (𝑘 ∈ {∅} → 𝑘 = ∅)
25	23, 24	syl 17	. . . . . . 7 ⊢ (𝑘 ∈ (ran 𝐺 ∩ {∅}) → 𝑘 = ∅)
26	22, 25	syl 17	. . . . . 6 ⊢ (𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})) → 𝑘 = ∅)
27	26	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))) → 𝑘 = ∅)
28		simpr 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = ∅) → 𝑘 = ∅)
29	28	fveq2d 6674	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = ∅) → (𝑀‘𝑘) = (𝑀‘∅))
30	9	mea0 42756	. . . . . . 7 ⊢ (𝜑 → (𝑀‘∅) = 0)
31	30	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = ∅) → (𝑀‘∅) = 0)
32	29, 31	eqtrd 2856	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = ∅) → (𝑀‘𝑘) = 0)
33	18, 27, 32	syl2anc 586	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))) → (𝑀‘𝑘) = 0)
34	1, 7, 8, 17, 33	sge0ss 42714	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))) = (Σ^{^}‘(𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘))))
35	34	eqcomd 2827	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘))) = (Σ^{^}‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))))
36	11, 13	feqresmpt 6734	. . 3 ⊢ (𝜑 → (𝑀 ↾ ran 𝐺) = (𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘)))
37	36	fveq2d 6674	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑀 ↾ ran 𝐺)) = (Σ^{^}‘(𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘))))
38	2	ffvelrnda 6851	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐺‘𝑗) ∈ 𝑆)
39	2	feqmptd 6733	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑗 ∈ 𝑋 ↦ (𝐺‘𝑗)))
40	11	feqmptd 6733	. . . . 5 ⊢ (𝜑 → 𝑀 = (𝑘 ∈ 𝑆 ↦ (𝑀‘𝑘)))
41		fveq2 6670	. . . . 5 ⊢ (𝑘 = (𝐺‘𝑗) → (𝑀‘𝑘) = (𝑀‘(𝐺‘𝑗)))
42	38, 39, 40, 41	fmptco 6891	. . . 4 ⊢ (𝜑 → (𝑀 ∘ 𝐺) = (𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗))))
43	42	fveq2d 6674	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑀 ∘ 𝐺)) = (Σ^{^}‘(𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗)))))
44		nfv 1915	. . . . 5 ⊢ Ⅎ𝑗𝜑
45		meadjiunlem.y	. . . . . 6 ⊢ 𝑌 = {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅}
46		ssrab2 4056	. . . . . . 7 ⊢ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ⊆ 𝑋
47	46	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ⊆ 𝑋)
48	45, 47	eqsstrid 4015	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
49	11	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑀:𝑆⟶(0[,]+∞))
50	2	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐺:𝑋⟶𝑆)
51	48	sselda 3967	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 ∈ 𝑋)
52	50, 51	ffvelrnd 6852	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗) ∈ 𝑆)
53	49, 52	ffvelrnd 6852	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑀‘(𝐺‘𝑗)) ∈ (0[,]+∞))
54		eldifi 4103	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (𝑋 ∖ 𝑌) → 𝑗 ∈ 𝑋)
55	54	ad2antlr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → 𝑗 ∈ 𝑋)
56		fveq2 6670	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘𝑗) = ∅ → (𝑀‘(𝐺‘𝑗)) = (𝑀‘∅))
57	56	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → (𝑀‘(𝐺‘𝑗)) = (𝑀‘∅))
58	9	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → 𝑀 ∈ Meas)
59	58	mea0 42756	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → (𝑀‘∅) = 0)
60	57, 59	eqtrd 2856	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → (𝑀‘(𝐺‘𝑗)) = 0)
61	60	ad4ant14 750	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) ∧ (𝐺‘𝑗) = ∅) → (𝑀‘(𝐺‘𝑗)) = 0)
62		neneq 3022	. . . . . . . . . . . . 13 ⊢ ((𝑀‘(𝐺‘𝑗)) ≠ 0 → ¬ (𝑀‘(𝐺‘𝑗)) = 0)
63	62	ad2antlr 725	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) ∧ (𝐺‘𝑗) = ∅) → ¬ (𝑀‘(𝐺‘𝑗)) = 0)
64	61, 63	pm2.65da 815	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → ¬ (𝐺‘𝑗) = ∅)
65	64	neqned 3023	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → (𝐺‘𝑗) ≠ ∅)
66	55, 65	jca 514	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → (𝑗 ∈ 𝑋 ∧ (𝐺‘𝑗) ≠ ∅))
67		fveq2 6670	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (𝐺‘𝑖) = (𝐺‘𝑗))
68	67	neeq1d 3075	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((𝐺‘𝑖) ≠ ∅ ↔ (𝐺‘𝑗) ≠ ∅))
69	68	elrab 3680	. . . . . . . . 9 ⊢ (𝑗 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ↔ (𝑗 ∈ 𝑋 ∧ (𝐺‘𝑗) ≠ ∅))
70	66, 69	sylibr 236	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → 𝑗 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅})
71	70, 45	eleqtrrdi 2924	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → 𝑗 ∈ 𝑌)
72		eldifn 4104	. . . . . . . 8 ⊢ (𝑗 ∈ (𝑋 ∖ 𝑌) → ¬ 𝑗 ∈ 𝑌)
73	72	ad2antlr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → ¬ 𝑗 ∈ 𝑌)
74	71, 73	pm2.65da 815	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) → ¬ (𝑀‘(𝐺‘𝑗)) ≠ 0)
75		nne 3020	. . . . . 6 ⊢ (¬ (𝑀‘(𝐺‘𝑗)) ≠ 0 ↔ (𝑀‘(𝐺‘𝑗)) = 0)
76	74, 75	sylib 220	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) → (𝑀‘(𝐺‘𝑗)) = 0)
77	44, 3, 48, 53, 76	sge0ss 42714	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗)))))
78	77	eqcomd 2827	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗)))))
79	3, 48	ssexd 5228	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ V)
80		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑖𝜑
81		eqid 2821	. . . . . . . . 9 ⊢ (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) = (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖))
82	2	ffnd 6515	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 Fn 𝑋)
83		dffn3 6525	. . . . . . . . . . . . 13 ⊢ (𝐺 Fn 𝑋 ↔ 𝐺:𝑋⟶ran 𝐺)
84	82, 83	sylib 220	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:𝑋⟶ran 𝐺)
85	84	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → 𝐺:𝑋⟶ran 𝐺)
86	48	sselda 3967	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → 𝑖 ∈ 𝑋)
87	85, 86	ffvelrnd 6852	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → (𝐺‘𝑖) ∈ ran 𝐺)
88	45	eleq2i 2904	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝑌 ↔ 𝑖 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅})
89		rabid 3378	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ↔ (𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅))
90	88, 89	bitri 277	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ 𝑌 ↔ (𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅))
91	90	biimpi 218	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ 𝑌 → (𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅))
92	91	simprd 498	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ 𝑌 → (𝐺‘𝑖) ≠ ∅)
93	92	adantl 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → (𝐺‘𝑖) ≠ ∅)
94		nelsn 4605	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑖) ≠ ∅ → ¬ (𝐺‘𝑖) ∈ {∅})
95	93, 94	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → ¬ (𝐺‘𝑖) ∈ {∅})
96	87, 95	eldifd 3947	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → (𝐺‘𝑖) ∈ (ran 𝐺 ∖ {∅}))
97		meadjiunlem.dj	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑖 ∈ 𝑋 (𝐺‘𝑖))
98		disjss1 5037	. . . . . . . . . 10 ⊢ (𝑌 ⊆ 𝑋 → (Disj 𝑖 ∈ 𝑋 (𝐺‘𝑖) → Disj 𝑖 ∈ 𝑌 (𝐺‘𝑖)))
99	48, 97, 98	sylc 65	. . . . . . . . 9 ⊢ (𝜑 → Disj 𝑖 ∈ 𝑌 (𝐺‘𝑖))
100	80, 81, 96, 93, 99	disjf1 41463	. . . . . . . 8 ⊢ (𝜑 → (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)):𝑌–1-1→(ran 𝐺 ∖ {∅}))
101	2, 48	feqresmpt 6734	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ↾ 𝑌) = (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)))
102		f1eq1 6570	. . . . . . . . 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 17	. . . . . . . 8 ⊢ (𝜑 → ((𝐺 ↾ 𝑌):𝑌–1-1→(ran 𝐺 ∖ {∅}) ↔ (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)):𝑌–1-1→(ran 𝐺 ∖ {∅})))
104	100, 103	mpbird 259	. . . . . . 7 ⊢ (𝜑 → (𝐺 ↾ 𝑌):𝑌–1-1→(ran 𝐺 ∖ {∅}))
105	101	rneqd 5808	. . . . . . . . 9 ⊢ (𝜑 → ran (𝐺 ↾ 𝑌) = ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)))
106	96	ralrimiva 3182	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑖 ∈ 𝑌 (𝐺‘𝑖) ∈ (ran 𝐺 ∖ {∅}))
107	81	rnmptss 6886	. . . . . . . . . 10 ⊢ (∀𝑖 ∈ 𝑌 (𝐺‘𝑖) ∈ (ran 𝐺 ∖ {∅}) → ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) ⊆ (ran 𝐺 ∖ {∅}))
108	106, 107	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) ⊆ (ran 𝐺 ∖ {∅}))
109	105, 108	eqsstrd 4005	. . . . . . . 8 ⊢ (𝜑 → ran (𝐺 ↾ 𝑌) ⊆ (ran 𝐺 ∖ {∅}))
110		simpl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝜑)
111		eldifi 4103	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ran 𝐺 ∖ {∅}) → 𝑥 ∈ ran 𝐺)
112	111	adantl 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝑥 ∈ ran 𝐺)
113		eldifsni 4722	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ran 𝐺 ∖ {∅}) → 𝑥 ≠ ∅)
114	113	adantl 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝑥 ≠ ∅)
115		simpr 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺) → 𝑥 ∈ ran 𝐺)
116		fvelrnb 6726	. . . . . . . . . . . . . 14 ⊢ (𝐺 Fn 𝑋 → (𝑥 ∈ ran 𝐺 ↔ ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥))
117	82, 116	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ ran 𝐺 ↔ ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥))
118	117	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺) → (𝑥 ∈ ran 𝐺 ↔ ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥))
119	115, 118	mpbid 234	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺) → ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥)
120	119	3adant3 1128	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺 ∧ 𝑥 ≠ ∅) → ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥)
121		id 22	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘𝑖) = 𝑥 → (𝐺‘𝑖) = 𝑥)
122	121	eqcomd 2827	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘𝑖) = 𝑥 → 𝑥 = (𝐺‘𝑖))
123	122	3ad2ant3 1131	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝑥 = (𝐺‘𝑖))
124		simp1l 1193	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝜑)
125		simp2 1133	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝑖 ∈ 𝑋)
126		simpr 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ≠ ∅ ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) = 𝑥)
127		simpl 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ≠ ∅ ∧ (𝐺‘𝑖) = 𝑥) → 𝑥 ≠ ∅)
128	126, 127	eqnetrd 3083	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ≠ ∅ ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) ≠ ∅)
129	128	adantll 712	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) ≠ ∅)
130	129	3adant2 1127	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) ≠ ∅)
131	90	biimpri 230	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → 𝑖 ∈ 𝑌)
132		fvexd 6685	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → (𝐺‘𝑖) ∈ V)
133	81	elrnmpt1 5830	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ 𝑌 ∧ (𝐺‘𝑖) ∈ V) → (𝐺‘𝑖) ∈ ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)))
134	131, 132, 133	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → (𝐺‘𝑖) ∈ ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)))
135	134	3adant1 1126	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → (𝐺‘𝑖) ∈ ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)))
136	105	eqcomd 2827	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) = ran (𝐺 ↾ 𝑌))
137	136	3ad2ant1 1129	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) = ran (𝐺 ↾ 𝑌))
138	135, 137	eleqtrd 2915	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → (𝐺‘𝑖) ∈ ran (𝐺 ↾ 𝑌))
139	124, 125, 130, 138	syl3anc 1367	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) ∈ ran (𝐺 ↾ 𝑌))
140	123, 139	eqeltrd 2913	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝑥 ∈ ran (𝐺 ↾ 𝑌))
141	140	3exp 1115	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ≠ ∅) → (𝑖 ∈ 𝑋 → ((𝐺‘𝑖) = 𝑥 → 𝑥 ∈ ran (𝐺 ↾ 𝑌))))
142	141	rexlimdv 3283	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ≠ ∅) → (∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥 → 𝑥 ∈ ran (𝐺 ↾ 𝑌)))
143	142	3adant2 1127	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺 ∧ 𝑥 ≠ ∅) → (∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥 → 𝑥 ∈ ran (𝐺 ↾ 𝑌)))
144	120, 143	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺 ∧ 𝑥 ≠ ∅) → 𝑥 ∈ ran (𝐺 ↾ 𝑌))
145	110, 112, 114, 144	syl3anc 1367	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝑥 ∈ ran (𝐺 ↾ 𝑌))
146	109, 145	eqelssd 3988	. . . . . . 7 ⊢ (𝜑 → ran (𝐺 ↾ 𝑌) = (ran 𝐺 ∖ {∅}))
147	104, 146	jca 514	. . . . . 6 ⊢ (𝜑 → ((𝐺 ↾ 𝑌):𝑌–1-1→(ran 𝐺 ∖ {∅}) ∧ ran (𝐺 ↾ 𝑌) = (ran 𝐺 ∖ {∅})))
148		dff1o5 6624	. . . . . 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 236	. . . . 5 ⊢ (𝜑 → (𝐺 ↾ 𝑌):𝑌–1-1-onto→(ran 𝐺 ∖ {∅}))
150		fvres 6689	. . . . . 6 ⊢ (𝑗 ∈ 𝑌 → ((𝐺 ↾ 𝑌)‘𝑗) = (𝐺‘𝑗))
151	150	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐺 ↾ 𝑌)‘𝑗) = (𝐺‘𝑗))
152	1, 44, 41, 79, 149, 151, 17	sge0f1o 42684	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))) = (Σ^{^}‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗)))))
153	152	eqcomd 2827	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗)))) = (Σ^{^}‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))))
154	43, 78, 153	3eqtrd 2860	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑀 ∘ 𝐺)) = (Σ^{^}‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))))
155	35, 37, 154	3eqtr4d 2866	1 ⊢ (𝜑 → (Σ^{^}‘(𝑀 ↾ ran 𝐺)) = (Σ^{^}‘(𝑀 ∘ 𝐺)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 {crab 3142 Vcvv 3494 ∖ cdif 3933 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 {csn 4567 Disj wdisj 5031 ↦ cmpt 5146 dom cdm 5555 ran crn 5556 ↾ cres 5557 ∘ ccom 5559 Fn wfn 6350 ⟶wf 6351 –1-1→wf1 6352 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 0cc0 10537 +∞cpnf 10672 [,]cicc 12742 Σ^{^}csumge0 42664 Meascmea 42751

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-disj 5032 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-xadd 12509 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 df-sumge0 42665 df-mea 42752

This theorem is referenced by: meadjiun 42768

W3C validator