Theorem sibfinima 30401

Description: The measure of the intersection of any two preimages by simple functions is a real number. (Contributed by Thierry Arnoux, 21-Mar-2018.)

Hypotheses

Ref	Expression
sitgval.b	⊢ 𝐵 = (Base‘𝑊)
sitgval.j	⊢ 𝐽 = (TopOpen‘𝑊)
sitgval.s	⊢ 𝑆 = (sigaGen‘𝐽)
sitgval.0	⊢ 0 = (0g‘𝑊)
sitgval.x	⊢ · = ( ·𝑠 ‘𝑊)
sitgval.h	⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
sitgval.1	⊢ (𝜑 → 𝑊 ∈ 𝑉)
sitgval.2	⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
sibfmbl.1	⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))
sibfinima.g	⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀))
sibfinima.w	⊢ (𝜑 → 𝑊 ∈ TopSp)
sibfinima.j	⊢ (𝜑 → 𝐽 ∈ Fre)

Assertion

Ref	Expression
sibfinima	⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,)+∞))

Proof of Theorem sibfinima

Step	Hyp	Ref	Expression
1		sitgval.2	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
2		measbasedom 30265	. . . . . . . 8 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
3	1, 2	sylib 208	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (measures‘dom 𝑀))
4	3	3ad2ant1 1082	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑀 ∈ (measures‘dom 𝑀))
5		dmmeas 30264	. . . . . . . . 9 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)
6	1, 5	syl 17	. . . . . . . 8 ⊢ (𝜑 → dom 𝑀 ∈ ∪ ran sigAlgebra)
7	6	3ad2ant1 1082	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → dom 𝑀 ∈ ∪ ran sigAlgebra)
8		sitgval.s	. . . . . . . . . 10 ⊢ 𝑆 = (sigaGen‘𝐽)
9		sibfinima.j	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ Fre)
10	9	sgsiga 30205	. . . . . . . . . 10 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
11	8, 10	syl5eqel 2705	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
12	11	3ad2ant1 1082	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑆 ∈ ∪ ran sigAlgebra)
13		sitgval.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑊)
14		sitgval.j	. . . . . . . . . 10 ⊢ 𝐽 = (TopOpen‘𝑊)
15		sitgval.0	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑊)
16		sitgval.x	. . . . . . . . . 10 ⊢ · = ( ·𝑠 ‘𝑊)
17		sitgval.h	. . . . . . . . . 10 ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
18		sitgval.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ 𝑉)
19		sibfmbl.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))
20	13, 14, 8, 15, 16, 17, 18, 1, 19	sibfmbl 30397	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
21	20	3ad2ant1 1082	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
22		sibfinima.w	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ∈ TopSp)
23	14	tpstop 20741	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ TopSp → 𝐽 ∈ Top)
24		cldssbrsiga 30250	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
25	22, 23, 24	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
26	25, 8	syl6sseqr 3652	. . . . . . . . . 10 ⊢ (𝜑 → (Clsd‘𝐽) ⊆ 𝑆)
27	26	3ad2ant1 1082	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (Clsd‘𝐽) ⊆ 𝑆)
28	9	3ad2ant1 1082	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝐽 ∈ Fre)
29	13, 14, 8, 15, 16, 17, 18, 1, 19	sibff 30398	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝐽)
30		frn 6053	. . . . . . . . . . . . 13 ⊢ (𝐹:∪ dom 𝑀⟶∪ 𝐽 → ran 𝐹 ⊆ ∪ 𝐽)
31	29, 30	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐹 ⊆ ∪ 𝐽)
32	31	3ad2ant1 1082	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → ran 𝐹 ⊆ ∪ 𝐽)
33		simp2 1062	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑋 ∈ ran 𝐹)
34	32, 33	sseldd 3604	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑋 ∈ ∪ 𝐽)
35		eqid 2622	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
36	35	t1sncld 21130	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Fre ∧ 𝑋 ∈ ∪ 𝐽) → {𝑋} ∈ (Clsd‘𝐽))
37	28, 34, 36	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → {𝑋} ∈ (Clsd‘𝐽))
38	27, 37	sseldd 3604	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → {𝑋} ∈ 𝑆)
39	7, 12, 21, 38	mbfmcnvima 30319	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (◡𝐹 “ {𝑋}) ∈ dom 𝑀)
40		sibfinima.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀))
41	13, 14, 8, 15, 16, 17, 18, 1, 40	sibfmbl 30397	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
42	41	3ad2ant1 1082	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
43	13, 14, 8, 15, 16, 17, 18, 1, 40	sibff 30398	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:∪ dom 𝑀⟶∪ 𝐽)
44		frn 6053	. . . . . . . . . . . . 13 ⊢ (𝐺:∪ dom 𝑀⟶∪ 𝐽 → ran 𝐺 ⊆ ∪ 𝐽)
45	43, 44	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐺 ⊆ ∪ 𝐽)
46	45	3ad2ant1 1082	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → ran 𝐺 ⊆ ∪ 𝐽)
47		simp3 1063	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑌 ∈ ran 𝐺)
48	46, 47	sseldd 3604	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑌 ∈ ∪ 𝐽)
49	35	t1sncld 21130	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Fre ∧ 𝑌 ∈ ∪ 𝐽) → {𝑌} ∈ (Clsd‘𝐽))
50	28, 48, 49	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → {𝑌} ∈ (Clsd‘𝐽))
51	27, 50	sseldd 3604	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → {𝑌} ∈ 𝑆)
52	7, 12, 42, 51	mbfmcnvima 30319	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (◡𝐺 “ {𝑌}) ∈ dom 𝑀)
53		inelsiga 30198	. . . . . . 7 ⊢ ((dom 𝑀 ∈ ∪ ran sigAlgebra ∧ (◡𝐹 “ {𝑋}) ∈ dom 𝑀 ∧ (◡𝐺 “ {𝑌}) ∈ dom 𝑀) → ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ∈ dom 𝑀)
54	7, 39, 52, 53	syl3anc 1326	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ∈ dom 𝑀)
55		measvxrge0 30268	. . . . . 6 ⊢ ((𝑀 ∈ (measures‘dom 𝑀) ∧ ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ∈ dom 𝑀) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,]+∞))
56	4, 54, 55	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,]+∞))
57		elxrge0 12281	. . . . . 6 ⊢ ((𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,]+∞) ↔ ((𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ* ∧ 0 ≤ (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})))))
58	57	simplbi 476	. . . . 5 ⊢ ((𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,]+∞) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ*)
59	56, 58	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ*)
60	59	adantr 481	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ*)
61		0re 10040	. . . 4 ⊢ 0 ∈ ℝ
62	61	a1i 11	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → 0 ∈ ℝ)
63	57	simprbi 480	. . . . 5 ⊢ ((𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,]+∞) → 0 ≤ (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))))
64	56, 63	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 0 ≤ (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))))
65	64	adantr 481	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → 0 ≤ (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))))
66	59	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ*)
67	4	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → 𝑀 ∈ (measures‘dom 𝑀))
68	39	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (◡𝐹 “ {𝑋}) ∈ dom 𝑀)
69		measvxrge0 30268	. . . . . . 7 ⊢ ((𝑀 ∈ (measures‘dom 𝑀) ∧ (◡𝐹 “ {𝑋}) ∈ dom 𝑀) → (𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,]+∞))
70	67, 68, 69	syl2anc 693	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,]+∞))
71		elxrge0 12281	. . . . . . 7 ⊢ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,]+∞) ↔ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ ℝ* ∧ 0 ≤ (𝑀‘(◡𝐹 “ {𝑋}))))
72	71	simplbi 476	. . . . . 6 ⊢ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,]+∞) → (𝑀‘(◡𝐹 “ {𝑋})) ∈ ℝ*)
73	70, 72	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘(◡𝐹 “ {𝑋})) ∈ ℝ*)
74		pnfxr 10092	. . . . . 6 ⊢ +∞ ∈ ℝ*
75	74	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → +∞ ∈ ℝ*)
76	54	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ∈ dom 𝑀)
77		inss1 3833	. . . . . . 7 ⊢ ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ⊆ (◡𝐹 “ {𝑋})
78	77	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ⊆ (◡𝐹 “ {𝑋}))
79	67, 76, 68, 78	measssd 30278	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ≤ (𝑀‘(◡𝐹 “ {𝑋})))
80		simpl1 1064	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → 𝜑)
81	33	anim1i 592	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑋 ∈ ran 𝐹 ∧ 𝑋 ≠ 0 ))
82		eldifsn 4317	. . . . . . . 8 ⊢ (𝑋 ∈ (ran 𝐹 ∖ { 0 }) ↔ (𝑋 ∈ ran 𝐹 ∧ 𝑋 ≠ 0 ))
83	81, 82	sylibr 224	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ (ran 𝐹 ∖ { 0 }))
84	13, 14, 8, 15, 16, 17, 18, 1, 19	sibfima 30400	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,)+∞))
85	80, 83, 84	syl2anc 693	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,)+∞))
86		elico2 12237	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ((𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,)+∞) ↔ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ ℝ ∧ 0 ≤ (𝑀‘(◡𝐹 “ {𝑋})) ∧ (𝑀‘(◡𝐹 “ {𝑋})) < +∞)))
87	61, 74, 86	mp2an 708	. . . . . . 7 ⊢ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,)+∞) ↔ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ ℝ ∧ 0 ≤ (𝑀‘(◡𝐹 “ {𝑋})) ∧ (𝑀‘(◡𝐹 “ {𝑋})) < +∞))
88	87	simp3bi 1078	. . . . . 6 ⊢ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,)+∞) → (𝑀‘(◡𝐹 “ {𝑋})) < +∞)
89	85, 88	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘(◡𝐹 “ {𝑋})) < +∞)
90	66, 73, 75, 79, 89	xrlelttrd 11991	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) < +∞)
91	59	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ*)
92	4	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → 𝑀 ∈ (measures‘dom 𝑀))
93	52	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (◡𝐺 “ {𝑌}) ∈ dom 𝑀)
94		measvxrge0 30268	. . . . . . 7 ⊢ ((𝑀 ∈ (measures‘dom 𝑀) ∧ (◡𝐺 “ {𝑌}) ∈ dom 𝑀) → (𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,]+∞))
95	92, 93, 94	syl2anc 693	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,]+∞))
96		elxrge0 12281	. . . . . . 7 ⊢ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,]+∞) ↔ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ ℝ* ∧ 0 ≤ (𝑀‘(◡𝐺 “ {𝑌}))))
97	96	simplbi 476	. . . . . 6 ⊢ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,]+∞) → (𝑀‘(◡𝐺 “ {𝑌})) ∈ ℝ*)
98	95, 97	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘(◡𝐺 “ {𝑌})) ∈ ℝ*)
99	74	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → +∞ ∈ ℝ*)
100	54	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ∈ dom 𝑀)
101		inss2 3834	. . . . . . 7 ⊢ ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ⊆ (◡𝐺 “ {𝑌})
102	101	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ⊆ (◡𝐺 “ {𝑌}))
103	92, 100, 93, 102	measssd 30278	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ≤ (𝑀‘(◡𝐺 “ {𝑌})))
104		simpl1 1064	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → 𝜑)
105	47	anim1i 592	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑌 ∈ ran 𝐺 ∧ 𝑌 ≠ 0 ))
106		eldifsn 4317	. . . . . . . 8 ⊢ (𝑌 ∈ (ran 𝐺 ∖ { 0 }) ↔ (𝑌 ∈ ran 𝐺 ∧ 𝑌 ≠ 0 ))
107	105, 106	sylibr 224	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → 𝑌 ∈ (ran 𝐺 ∖ { 0 }))
108	13, 14, 8, 15, 16, 17, 18, 1, 40	sibfima 30400	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ∈ (ran 𝐺 ∖ { 0 })) → (𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,)+∞))
109	104, 107, 108	syl2anc 693	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,)+∞))
110		elico2 12237	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ((𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,)+∞) ↔ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ ℝ ∧ 0 ≤ (𝑀‘(◡𝐺 “ {𝑌})) ∧ (𝑀‘(◡𝐺 “ {𝑌})) < +∞)))
111	61, 74, 110	mp2an 708	. . . . . . 7 ⊢ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,)+∞) ↔ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ ℝ ∧ 0 ≤ (𝑀‘(◡𝐺 “ {𝑌})) ∧ (𝑀‘(◡𝐺 “ {𝑌})) < +∞))
112	111	simp3bi 1078	. . . . . 6 ⊢ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,)+∞) → (𝑀‘(◡𝐺 “ {𝑌})) < +∞)
113	109, 112	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘(◡𝐺 “ {𝑌})) < +∞)
114	91, 98, 99, 103, 113	xrlelttrd 11991	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) < +∞)
115	90, 114	jaodan 826	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) < +∞)
116		xrre3 12002	. . 3 ⊢ ((((𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ* ∧ 0 ∈ ℝ) ∧ (0 ≤ (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∧ (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) < +∞)) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ)
117	60, 62, 65, 115, 116	syl22anc 1327	. 2 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ)
118		elico2 12237	. . 3 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ((𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,)+∞) ↔ ((𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ ∧ 0 ≤ (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∧ (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) < +∞)))
119	61, 74, 118	mp2an 708	. 2 ⊢ ((𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,)+∞) ↔ ((𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ ∧ 0 ≤ (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∧ (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) < +∞))
120	117, 65, 115, 119	syl3anbrc 1246	1 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,)+∞))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  {csn 4177  ∪ cuni 4436   class class class wbr 4653  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   “ cima 5117  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ℝcr 9935  0cc0 9936  +∞cpnf 10071  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075  [,)cico 12177  [,]cicc 12178  Basecbs 15857  Scalarcsca 15944   ·_𝑠 cvsca 15945  TopOpenctopn 16082  0_gc0g 16100  Topctop 20698  TopSpctps 20736  Clsdccld 20820  Frect1 21111  ℝHomcrrh 30037  sigAlgebracsiga 30170  sigaGencsigagen 30201  measurescmeas 30258  MblFnMcmbfm 30312  sitgcsitg 30391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-ordt 16161  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-ps 17200  df-tsr 17201  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-subrg 18778  df-abv 18817  df-lmod 18865  df-scaf 18866  df-sra 19172  df-rgmod 19173  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-t1 21118  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-tmd 21876  df-tgp 21877  df-tsms 21930  df-trg 21963  df-xms 22125  df-ms 22126  df-tms 22127  df-nm 22387  df-ngp 22388  df-nrg 22390  df-nlm 22391  df-ii 22680  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-esum 30090  df-siga 30171  df-sigagen 30202  df-meas 30259  df-mbfm 30313  df-sitg 30392

This theorem is referenced by:  sibfof  30402  sitgaddlemb  30410