meadjiunlem - Mathbox for Glauco Siliprandi

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Theorem meadjiunlem 46562

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 511	. . . . 5 ⊢ (𝜑 → (𝐺:𝑋⟶𝑆 ∧ 𝑋 ∈ 𝑉))
5		fex 7160	. . . . 5 ⊢ ((𝐺:𝑋⟶𝑆 ∧ 𝑋 ∈ 𝑉) → 𝐺 ∈ V)
6		rnexg 7832	. . . . 5 ⊢ (𝐺 ∈ V → ran 𝐺 ∈ V)
7	4, 5, 6	3syl 18	. . . 4 ⊢ (𝜑 → ran 𝐺 ∈ V)
8		difssd 4084	. . . 4 ⊢ (𝜑 → (ran 𝐺 ∖ {∅}) ⊆ ran 𝐺)
9		meadjiunlem.f	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ Meas)
10		meadjiunlem.3	. . . . . . 7 ⊢ 𝑆 = dom 𝑀
11	9, 10	meaf 46550	. . . . . 6 ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞))
12	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → 𝑀:𝑆⟶(0[,]+∞))
13	2	frnd 6659	. . . . . . 7 ⊢ (𝜑 → ran 𝐺 ⊆ 𝑆)
14	13	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → ran 𝐺 ⊆ 𝑆)
15	8	sselda 3929	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ ran 𝐺)
16	14, 15	sseldd 3930	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ 𝑆)
17	12, 16	ffvelcdmd 7018	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → (𝑀‘𝑘) ∈ (0[,]+∞))
18		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))) → 𝜑)
19		id 22	. . . . . . . 8 ⊢ (𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})))
20		dfin4 4225	. . . . . . . . 9 ⊢ (ran 𝐺 ∩ {∅}) = (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))
21	20	eqcomi 2740	. . . . . . . 8 ⊢ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})) = (ran 𝐺 ∩ {∅})
22	19, 21	eleqtrdi 2841	. . . . . . 7 ⊢ (𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ (ran 𝐺 ∩ {∅}))
23		elinel2 4149	. . . . . . . 8 ⊢ (𝑘 ∈ (ran 𝐺 ∩ {∅}) → 𝑘 ∈ {∅})
24		elsni 4590	. . . . . . . 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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))) → 𝑘 = ∅)
28		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = ∅) → 𝑘 = ∅)
29	28	fveq2d 6826	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = ∅) → (𝑀‘𝑘) = (𝑀‘∅))
30	9	mea0 46551	. . . . . . 7 ⊢ (𝜑 → (𝑀‘∅) = 0)
31	30	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = ∅) → (𝑀‘∅) = 0)
32	29, 31	eqtrd 2766	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = ∅) → (𝑀‘𝑘) = 0)
33	18, 27, 32	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))) → (𝑀‘𝑘) = 0)
34	1, 7, 8, 17, 33	sge0ss 46509	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))) = (Σ^{^}‘(𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘))))
35	34	eqcomd 2737	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘))) = (Σ^{^}‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))))
36	11, 13	feqresmpt 6891	. . 3 ⊢ (𝜑 → (𝑀 ↾ ran 𝐺) = (𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘)))
37	36	fveq2d 6826	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑀 ↾ ran 𝐺)) = (Σ^{^}‘(𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘))))
38	2	ffvelcdmda 7017	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐺‘𝑗) ∈ 𝑆)
39	2	feqmptd 6890	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑗 ∈ 𝑋 ↦ (𝐺‘𝑗)))
40	11	feqmptd 6890	. . . . 5 ⊢ (𝜑 → 𝑀 = (𝑘 ∈ 𝑆 ↦ (𝑀‘𝑘)))
41		fveq2 6822	. . . . 5 ⊢ (𝑘 = (𝐺‘𝑗) → (𝑀‘𝑘) = (𝑀‘(𝐺‘𝑗)))
42	38, 39, 40, 41	fmptco 7062	. . . 4 ⊢ (𝜑 → (𝑀 ∘ 𝐺) = (𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗))))
43	42	fveq2d 6826	. . 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 4027	. . . . . . 7 ⊢ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ⊆ 𝑋
47	46	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ⊆ 𝑋)
48	45, 47	eqsstrid 3968	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
49	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑀:𝑆⟶(0[,]+∞))
50	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐺:𝑋⟶𝑆)
51	48	sselda 3929	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 ∈ 𝑋)
52	50, 51	ffvelcdmd 7018	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗) ∈ 𝑆)
53	49, 52	ffvelcdmd 7018	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑀‘(𝐺‘𝑗)) ∈ (0[,]+∞))
54		eldifi 4078	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (𝑋 ∖ 𝑌) → 𝑗 ∈ 𝑋)
55	54	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → 𝑗 ∈ 𝑋)
56		fveq2 6822	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘𝑗) = ∅ → (𝑀‘(𝐺‘𝑗)) = (𝑀‘∅))
57	56	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → (𝑀‘(𝐺‘𝑗)) = (𝑀‘∅))
58	9	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → 𝑀 ∈ Meas)
59	58	mea0 46551	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → (𝑀‘∅) = 0)
60	57, 59	eqtrd 2766	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → (𝑀‘(𝐺‘𝑗)) = 0)
61	60	ad4ant14 752	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) ∧ (𝐺‘𝑗) = ∅) → (𝑀‘(𝐺‘𝑗)) = 0)
62		neneq 2934	. . . . . . . . . . . . 13 ⊢ ((𝑀‘(𝐺‘𝑗)) ≠ 0 → ¬ (𝑀‘(𝐺‘𝑗)) = 0)
63	62	ad2antlr 727	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) ∧ (𝐺‘𝑗) = ∅) → ¬ (𝑀‘(𝐺‘𝑗)) = 0)
64	61, 63	pm2.65da 816	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → ¬ (𝐺‘𝑗) = ∅)
65	64	neqned 2935	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → (𝐺‘𝑗) ≠ ∅)
66	55, 65	jca 511	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → (𝑗 ∈ 𝑋 ∧ (𝐺‘𝑗) ≠ ∅))
67		fveq2 6822	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (𝐺‘𝑖) = (𝐺‘𝑗))
68	67	neeq1d 2987	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((𝐺‘𝑖) ≠ ∅ ↔ (𝐺‘𝑗) ≠ ∅))
69	68	elrab 3642	. . . . . . . . 9 ⊢ (𝑗 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ↔ (𝑗 ∈ 𝑋 ∧ (𝐺‘𝑗) ≠ ∅))
70	66, 69	sylibr 234	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → 𝑗 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅})
71	70, 45	eleqtrrdi 2842	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → 𝑗 ∈ 𝑌)
72		eldifn 4079	. . . . . . . 8 ⊢ (𝑗 ∈ (𝑋 ∖ 𝑌) → ¬ 𝑗 ∈ 𝑌)
73	72	ad2antlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → ¬ 𝑗 ∈ 𝑌)
74	71, 73	pm2.65da 816	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) → ¬ (𝑀‘(𝐺‘𝑗)) ≠ 0)
75		nne 2932	. . . . . 6 ⊢ (¬ (𝑀‘(𝐺‘𝑗)) ≠ 0 ↔ (𝑀‘(𝐺‘𝑗)) = 0)
76	74, 75	sylib 218	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) → (𝑀‘(𝐺‘𝑗)) = 0)
77	44, 3, 48, 53, 76	sge0ss 46509	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗)))))
78	77	eqcomd 2737	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗)))))
79	3, 48	ssexd 5260	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ V)
80		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑖𝜑
81		eqid 2731	. . . . . . . . 9 ⊢ (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) = (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖))
82	2	ffnd 6652	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 Fn 𝑋)
83		dffn3 6663	. . . . . . . . . . . . 13 ⊢ (𝐺 Fn 𝑋 ↔ 𝐺:𝑋⟶ran 𝐺)
84	82, 83	sylib 218	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:𝑋⟶ran 𝐺)
85	84	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → 𝐺:𝑋⟶ran 𝐺)
86	48	sselda 3929	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → 𝑖 ∈ 𝑋)
87	85, 86	ffvelcdmd 7018	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → (𝐺‘𝑖) ∈ ran 𝐺)
88	45	eleq2i 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝑌 ↔ 𝑖 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅})
89		rabid 3416	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ↔ (𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅))
90	88, 89	bitri 275	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ 𝑌 ↔ (𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅))
91	90	biimpi 216	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ 𝑌 → (𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅))
92	91	simprd 495	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ 𝑌 → (𝐺‘𝑖) ≠ ∅)
93	92	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → (𝐺‘𝑖) ≠ ∅)
94		nelsn 4616	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑖) ≠ ∅ → ¬ (𝐺‘𝑖) ∈ {∅})
95	93, 94	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → ¬ (𝐺‘𝑖) ∈ {∅})
96	87, 95	eldifd 3908	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → (𝐺‘𝑖) ∈ (ran 𝐺 ∖ {∅}))
97		meadjiunlem.dj	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑖 ∈ 𝑋 (𝐺‘𝑖))
98		disjss1 5062	. . . . . . . . . 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 45279	. . . . . . . 8 ⊢ (𝜑 → (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)):𝑌–1-1→(ran 𝐺 ∖ {∅}))
101	2, 48	feqresmpt 6891	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ↾ 𝑌) = (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)))
102		f1eq1 6714	. . . . . . . . 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 257	. . . . . . 7 ⊢ (𝜑 → (𝐺 ↾ 𝑌):𝑌–1-1→(ran 𝐺 ∖ {∅}))
105	101	rneqd 5877	. . . . . . . . 9 ⊢ (𝜑 → ran (𝐺 ↾ 𝑌) = ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)))
106	96	ralrimiva 3124	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑖 ∈ 𝑌 (𝐺‘𝑖) ∈ (ran 𝐺 ∖ {∅}))
107	81	rnmptss 7056	. . . . . . . . . 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 3964	. . . . . . . 8 ⊢ (𝜑 → ran (𝐺 ↾ 𝑌) ⊆ (ran 𝐺 ∖ {∅}))
110		simpl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝜑)
111		eldifi 4078	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ran 𝐺 ∖ {∅}) → 𝑥 ∈ ran 𝐺)
112	111	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝑥 ∈ ran 𝐺)
113		eldifsni 4739	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ran 𝐺 ∖ {∅}) → 𝑥 ≠ ∅)
114	113	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝑥 ≠ ∅)
115		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺) → 𝑥 ∈ ran 𝐺)
116		fvelrnb 6882	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺) → (𝑥 ∈ ran 𝐺 ↔ ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥))
119	115, 118	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺) → ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥)
120	119	3adant3 1132	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺 ∧ 𝑥 ≠ ∅) → ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥)
121		id 22	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘𝑖) = 𝑥 → (𝐺‘𝑖) = 𝑥)
122	121	eqcomd 2737	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘𝑖) = 𝑥 → 𝑥 = (𝐺‘𝑖))
123	122	3ad2ant3 1135	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝑥 = (𝐺‘𝑖))
124		simp1l 1198	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝜑)
125		simp2 1137	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝑖 ∈ 𝑋)
126		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ≠ ∅ ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) = 𝑥)
127		simpl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ≠ ∅ ∧ (𝐺‘𝑖) = 𝑥) → 𝑥 ≠ ∅)
128	126, 127	eqnetrd 2995	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ≠ ∅ ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) ≠ ∅)
129	128	adantll 714	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) ≠ ∅)
130	129	3adant2 1131	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) ≠ ∅)
131	90	biimpri 228	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → 𝑖 ∈ 𝑌)
132		fvexd 6837	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → (𝐺‘𝑖) ∈ V)
133	81	elrnmpt1 5899	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ 𝑌 ∧ (𝐺‘𝑖) ∈ V) → (𝐺‘𝑖) ∈ ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)))
134	131, 132, 133	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → (𝐺‘𝑖) ∈ ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)))
135	134	3adant1 1130	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → (𝐺‘𝑖) ∈ ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)))
136	105	eqcomd 2737	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) = ran (𝐺 ↾ 𝑌))
137	136	3ad2ant1 1133	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) = ran (𝐺 ↾ 𝑌))
138	135, 137	eleqtrd 2833	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → (𝐺‘𝑖) ∈ ran (𝐺 ↾ 𝑌))
139	124, 125, 130, 138	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) ∈ ran (𝐺 ↾ 𝑌))
140	123, 139	eqeltrd 2831	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝑥 ∈ ran (𝐺 ↾ 𝑌))
141	140	3exp 1119	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ≠ ∅) → (𝑖 ∈ 𝑋 → ((𝐺‘𝑖) = 𝑥 → 𝑥 ∈ ran (𝐺 ↾ 𝑌))))
142	141	rexlimdv 3131	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ≠ ∅) → (∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥 → 𝑥 ∈ ran (𝐺 ↾ 𝑌)))
143	142	3adant2 1131	. . . . . . . . . 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 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝑥 ∈ ran (𝐺 ↾ 𝑌))
146	109, 145	eqelssd 3951	. . . . . . 7 ⊢ (𝜑 → ran (𝐺 ↾ 𝑌) = (ran 𝐺 ∖ {∅}))
147	104, 146	jca 511	. . . . . 6 ⊢ (𝜑 → ((𝐺 ↾ 𝑌):𝑌–1-1→(ran 𝐺 ∖ {∅}) ∧ ran (𝐺 ↾ 𝑌) = (ran 𝐺 ∖ {∅})))
148		dff1o5 6772	. . . . . 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 234	. . . . 5 ⊢ (𝜑 → (𝐺 ↾ 𝑌):𝑌–1-1-onto→(ran 𝐺 ∖ {∅}))
150		fvres 6841	. . . . . 6 ⊢ (𝑗 ∈ 𝑌 → ((𝐺 ↾ 𝑌)‘𝑗) = (𝐺‘𝑗))
151	150	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐺 ↾ 𝑌)‘𝑗) = (𝐺‘𝑗))
152	1, 44, 41, 79, 149, 151, 17	sge0f1o 46479	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))) = (Σ^{^}‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗)))))
153	152	eqcomd 2737	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗)))) = (Σ^{^}‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))))
154	43, 78, 153	3eqtrd 2770	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑀 ∘ 𝐺)) = (Σ^{^}‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))))
155	35, 37, 154	3eqtr4d 2776	1 ⊢ (𝜑 → (Σ^{^}‘(𝑀 ↾ ran 𝐺)) = (Σ^{^}‘(𝑀 ∘ 𝐺)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 {crab 3395 Vcvv 3436 ∖ cdif 3894 ∩ cin 3896 ⊆ wss 3897 ∅c0 4280 {csn 4573 Disj wdisj 5056 ↦ cmpt 5170 dom cdm 5614 ran crn 5615 ↾ cres 5616 ∘ ccom 5618 Fn wfn 6476 ⟶wf 6477 –1-1→wf1 6478 –1-1-onto→wf1o 6480 ‘cfv 6481 (class class class)co 7346 0cc0 11006 +∞cpnf 11143 [,]cicc 13248 Σ^{^}csumge0 46459 Meascmea 46546

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-disj 5057 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-n0 12382 df-z 12469 df-uz 12733 df-rp 12891 df-xadd 13012 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-sum 15594 df-sumge0 46460 df-mea 46547

This theorem is referenced by: meadjiun 46563

W3C validator