meadjiunlem - Mathbox for Glauco Siliprandi

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Theorem meadjiunlem 45266

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 1917	. . . 4 ⊢ Ⅎ𝑘𝜑
2		meadjiunlem.g	. . . . . 6 ⊢ (𝜑 → 𝐺:𝑋⟶𝑆)
3		meadjiunlem.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
4	2, 3	jca 512	. . . . 5 ⊢ (𝜑 → (𝐺:𝑋⟶𝑆 ∧ 𝑋 ∈ 𝑉))
5		fex 7230	. . . . 5 ⊢ ((𝐺:𝑋⟶𝑆 ∧ 𝑋 ∈ 𝑉) → 𝐺 ∈ V)
6		rnexg 7897	. . . . 5 ⊢ (𝐺 ∈ V → ran 𝐺 ∈ V)
7	4, 5, 6	3syl 18	. . . 4 ⊢ (𝜑 → ran 𝐺 ∈ V)
8		difssd 4132	. . . 4 ⊢ (𝜑 → (ran 𝐺 ∖ {∅}) ⊆ ran 𝐺)
9		meadjiunlem.f	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ Meas)
10		meadjiunlem.3	. . . . . . 7 ⊢ 𝑆 = dom 𝑀
11	9, 10	meaf 45254	. . . . . 6 ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞))
12	11	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → 𝑀:𝑆⟶(0[,]+∞))
13	2	frnd 6725	. . . . . . 7 ⊢ (𝜑 → ran 𝐺 ⊆ 𝑆)
14	13	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → ran 𝐺 ⊆ 𝑆)
15	8	sselda 3982	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ ran 𝐺)
16	14, 15	sseldd 3983	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ 𝑆)
17	12, 16	ffvelcdmd 7087	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → (𝑀‘𝑘) ∈ (0[,]+∞))
18		simpl 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))) → 𝜑)
19		id 22	. . . . . . . 8 ⊢ (𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})))
20		dfin4 4267	. . . . . . . . 9 ⊢ (ran 𝐺 ∩ {∅}) = (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))
21	20	eqcomi 2741	. . . . . . . 8 ⊢ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})) = (ran 𝐺 ∩ {∅})
22	19, 21	eleqtrdi 2843	. . . . . . 7 ⊢ (𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ (ran 𝐺 ∩ {∅}))
23		elinel2 4196	. . . . . . . 8 ⊢ (𝑘 ∈ (ran 𝐺 ∩ {∅}) → 𝑘 ∈ {∅})
24		elsni 4645	. . . . . . . 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 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))) → 𝑘 = ∅)
28		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = ∅) → 𝑘 = ∅)
29	28	fveq2d 6895	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = ∅) → (𝑀‘𝑘) = (𝑀‘∅))
30	9	mea0 45255	. . . . . . 7 ⊢ (𝜑 → (𝑀‘∅) = 0)
31	30	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = ∅) → (𝑀‘∅) = 0)
32	29, 31	eqtrd 2772	. . . . 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 45213	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))) = (Σ^{^}‘(𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘))))
35	34	eqcomd 2738	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘))) = (Σ^{^}‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))))
36	11, 13	feqresmpt 6961	. . 3 ⊢ (𝜑 → (𝑀 ↾ ran 𝐺) = (𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘)))
37	36	fveq2d 6895	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑀 ↾ ran 𝐺)) = (Σ^{^}‘(𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘))))
38	2	ffvelcdmda 7086	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐺‘𝑗) ∈ 𝑆)
39	2	feqmptd 6960	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑗 ∈ 𝑋 ↦ (𝐺‘𝑗)))
40	11	feqmptd 6960	. . . . 5 ⊢ (𝜑 → 𝑀 = (𝑘 ∈ 𝑆 ↦ (𝑀‘𝑘)))
41		fveq2 6891	. . . . 5 ⊢ (𝑘 = (𝐺‘𝑗) → (𝑀‘𝑘) = (𝑀‘(𝐺‘𝑗)))
42	38, 39, 40, 41	fmptco 7129	. . . 4 ⊢ (𝜑 → (𝑀 ∘ 𝐺) = (𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗))))
43	42	fveq2d 6895	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑀 ∘ 𝐺)) = (Σ^{^}‘(𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗)))))
44		nfv 1917	. . . . 5 ⊢ Ⅎ𝑗𝜑
45		meadjiunlem.y	. . . . . 6 ⊢ 𝑌 = {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅}
46		ssrab2 4077	. . . . . . 7 ⊢ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ⊆ 𝑋
47	46	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ⊆ 𝑋)
48	45, 47	eqsstrid 4030	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
49	11	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑀:𝑆⟶(0[,]+∞))
50	2	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐺:𝑋⟶𝑆)
51	48	sselda 3982	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 ∈ 𝑋)
52	50, 51	ffvelcdmd 7087	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗) ∈ 𝑆)
53	49, 52	ffvelcdmd 7087	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑀‘(𝐺‘𝑗)) ∈ (0[,]+∞))
54		eldifi 4126	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (𝑋 ∖ 𝑌) → 𝑗 ∈ 𝑋)
55	54	ad2antlr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → 𝑗 ∈ 𝑋)
56		fveq2 6891	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘𝑗) = ∅ → (𝑀‘(𝐺‘𝑗)) = (𝑀‘∅))
57	56	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → (𝑀‘(𝐺‘𝑗)) = (𝑀‘∅))
58	9	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → 𝑀 ∈ Meas)
59	58	mea0 45255	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → (𝑀‘∅) = 0)
60	57, 59	eqtrd 2772	. . . . . . . . . . . . 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 2946	. . . . . . . . . . . . 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 2947	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → (𝐺‘𝑗) ≠ ∅)
66	55, 65	jca 512	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → (𝑗 ∈ 𝑋 ∧ (𝐺‘𝑗) ≠ ∅))
67		fveq2 6891	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (𝐺‘𝑖) = (𝐺‘𝑗))
68	67	neeq1d 3000	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → ((𝐺‘𝑖) ≠ ∅ ↔ (𝐺‘𝑗) ≠ ∅))
69	68	elrab 3683	. . . . . . . . 9 ⊢ (𝑗 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ↔ (𝑗 ∈ 𝑋 ∧ (𝐺‘𝑗) ≠ ∅))
70	66, 69	sylibr 233	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → 𝑗 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅})
71	70, 45	eleqtrrdi 2844	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → 𝑗 ∈ 𝑌)
72		eldifn 4127	. . . . . . . 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 2944	. . . . . 6 ⊢ (¬ (𝑀‘(𝐺‘𝑗)) ≠ 0 ↔ (𝑀‘(𝐺‘𝑗)) = 0)
76	74, 75	sylib 217	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) → (𝑀‘(𝐺‘𝑗)) = 0)
77	44, 3, 48, 53, 76	sge0ss 45213	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗)))))
78	77	eqcomd 2738	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗)))) = (Σ^{^}‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗)))))
79	3, 48	ssexd 5324	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ V)
80		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑖𝜑
81		eqid 2732	. . . . . . . . 9 ⊢ (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) = (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖))
82	2	ffnd 6718	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 Fn 𝑋)
83		dffn3 6730	. . . . . . . . . . . . 13 ⊢ (𝐺 Fn 𝑋 ↔ 𝐺:𝑋⟶ran 𝐺)
84	82, 83	sylib 217	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:𝑋⟶ran 𝐺)
85	84	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → 𝐺:𝑋⟶ran 𝐺)
86	48	sselda 3982	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → 𝑖 ∈ 𝑋)
87	85, 86	ffvelcdmd 7087	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → (𝐺‘𝑖) ∈ ran 𝐺)
88	45	eleq2i 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝑌 ↔ 𝑖 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅})
89		rabid 3452	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ↔ (𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅))
90	88, 89	bitri 274	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ 𝑌 ↔ (𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅))
91	90	biimpi 215	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ 𝑌 → (𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅))
92	91	simprd 496	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ 𝑌 → (𝐺‘𝑖) ≠ ∅)
93	92	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → (𝐺‘𝑖) ≠ ∅)
94		nelsn 4668	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑖) ≠ ∅ → ¬ (𝐺‘𝑖) ∈ {∅})
95	93, 94	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → ¬ (𝐺‘𝑖) ∈ {∅})
96	87, 95	eldifd 3959	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → (𝐺‘𝑖) ∈ (ran 𝐺 ∖ {∅}))
97		meadjiunlem.dj	. . . . . . . . . 10 ⊢ (𝜑 → Disj 𝑖 ∈ 𝑋 (𝐺‘𝑖))
98		disjss1 5119	. . . . . . . . . 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 43967	. . . . . . . 8 ⊢ (𝜑 → (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)):𝑌–1-1→(ran 𝐺 ∖ {∅}))
101	2, 48	feqresmpt 6961	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ↾ 𝑌) = (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)))
102		f1eq1 6782	. . . . . . . . 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 256	. . . . . . 7 ⊢ (𝜑 → (𝐺 ↾ 𝑌):𝑌–1-1→(ran 𝐺 ∖ {∅}))
105	101	rneqd 5937	. . . . . . . . 9 ⊢ (𝜑 → ran (𝐺 ↾ 𝑌) = ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)))
106	96	ralrimiva 3146	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑖 ∈ 𝑌 (𝐺‘𝑖) ∈ (ran 𝐺 ∖ {∅}))
107	81	rnmptss 7124	. . . . . . . . . 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 4020	. . . . . . . 8 ⊢ (𝜑 → ran (𝐺 ↾ 𝑌) ⊆ (ran 𝐺 ∖ {∅}))
110		simpl 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝜑)
111		eldifi 4126	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ran 𝐺 ∖ {∅}) → 𝑥 ∈ ran 𝐺)
112	111	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝑥 ∈ ran 𝐺)
113		eldifsni 4793	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ran 𝐺 ∖ {∅}) → 𝑥 ≠ ∅)
114	113	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝑥 ≠ ∅)
115		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺) → 𝑥 ∈ ran 𝐺)
116		fvelrnb 6952	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺) → (𝑥 ∈ ran 𝐺 ↔ ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥))
119	115, 118	mpbid 231	. . . . . . . . . . 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 2738	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘𝑖) = 𝑥 → 𝑥 = (𝐺‘𝑖))
123	122	3ad2ant3 1135	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝑥 = (𝐺‘𝑖))
124		simp1l 1197	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝜑)
125		simp2 1137	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝑖 ∈ 𝑋)
126		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ≠ ∅ ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) = 𝑥)
127		simpl 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ≠ ∅ ∧ (𝐺‘𝑖) = 𝑥) → 𝑥 ≠ ∅)
128	126, 127	eqnetrd 3008	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ≠ ∅ ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) ≠ ∅)
129	128	adantll 712	. . . . . . . . . . . . . . . 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 227	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → 𝑖 ∈ 𝑌)
132		fvexd 6906	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → (𝐺‘𝑖) ∈ V)
133	81	elrnmpt1 5957	. . . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . 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 2835	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → (𝐺‘𝑖) ∈ ran (𝐺 ↾ 𝑌))
139	124, 125, 130, 138	syl3anc 1371	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) ∈ ran (𝐺 ↾ 𝑌))
140	123, 139	eqeltrd 2833	. . . . . . . . . . . . 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 3153	. . . . . . . . . . 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 1371	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝑥 ∈ ran (𝐺 ↾ 𝑌))
146	109, 145	eqelssd 4003	. . . . . . 7 ⊢ (𝜑 → ran (𝐺 ↾ 𝑌) = (ran 𝐺 ∖ {∅}))
147	104, 146	jca 512	. . . . . 6 ⊢ (𝜑 → ((𝐺 ↾ 𝑌):𝑌–1-1→(ran 𝐺 ∖ {∅}) ∧ ran (𝐺 ↾ 𝑌) = (ran 𝐺 ∖ {∅})))
148		dff1o5 6842	. . . . . 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 233	. . . . 5 ⊢ (𝜑 → (𝐺 ↾ 𝑌):𝑌–1-1-onto→(ran 𝐺 ∖ {∅}))
150		fvres 6910	. . . . . 6 ⊢ (𝑗 ∈ 𝑌 → ((𝐺 ↾ 𝑌)‘𝑗) = (𝐺‘𝑗))
151	150	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐺 ↾ 𝑌)‘𝑗) = (𝐺‘𝑗))
152	1, 44, 41, 79, 149, 151, 17	sge0f1o 45183	. . . 4 ⊢ (𝜑 → (Σ^{^}‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))) = (Σ^{^}‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗)))))
153	152	eqcomd 2738	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗)))) = (Σ^{^}‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))))
154	43, 78, 153	3eqtrd 2776	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑀 ∘ 𝐺)) = (Σ^{^}‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))))
155	35, 37, 154	3eqtr4d 2782	1 ⊢ (𝜑 → (Σ^{^}‘(𝑀 ↾ ran 𝐺)) = (Σ^{^}‘(𝑀 ∘ 𝐺)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3432 Vcvv 3474 ∖ cdif 3945 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 {csn 4628 Disj wdisj 5113 ↦ cmpt 5231 dom cdm 5676 ran crn 5677 ↾ cres 5678 ∘ ccom 5680 Fn wfn 6538 ⟶wf 6539 –1-1→wf1 6540 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7411 0cc0 11112 +∞cpnf 11247 [,]cicc 13329 Σ^{^}csumge0 45163 Meascmea 45250

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-n0 12475 df-z 12561 df-uz 12825 df-rp 12977 df-xadd 13095 df-ico 13332 df-icc 13333 df-fz 13487 df-fzo 13630 df-seq 13969 df-exp 14030 df-hash 14293 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-sum 15635 df-sumge0 45164 df-mea 45251

This theorem is referenced by: meadjiun 45267

W3C validator