sibfinima - Mathbox for Thierry Arnoux

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Theorem sibfinima 33327

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 = (0_g‘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 33189	. . . . . . . 8 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
3	1, 2	sylib 217	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (measures‘dom 𝑀))
4	3	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑀 ∈ (measures‘dom 𝑀))
5		dmmeas 33188	. . . . . . . . 9 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)
6	1, 5	syl 17	. . . . . . . 8 ⊢ (𝜑 → dom 𝑀 ∈ ∪ ran sigAlgebra)
7	6	3ad2ant1 1134	. . . . . . 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 33129	. . . . . . . . . 10 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
11	8, 10	eqeltrid 2838	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
12	11	3ad2ant1 1134	. . . . . . . 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 = (0_g‘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 33323	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
21	20	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
22		sibfinima.w	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ∈ TopSp)
23	14	tpstop 22431	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ TopSp → 𝐽 ∈ Top)
24		cldssbrsiga 33174	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
25	22, 23, 24	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
26	25, 8	sseqtrrdi 4033	. . . . . . . . . 10 ⊢ (𝜑 → (Clsd‘𝐽) ⊆ 𝑆)
27	26	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (Clsd‘𝐽) ⊆ 𝑆)
28	9	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝐽 ∈ Fre)
29	13, 14, 8, 15, 16, 17, 18, 1, 19	sibff 33324	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝐽)
30	29	frnd 6723	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐹 ⊆ ∪ 𝐽)
31	30	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → ran 𝐹 ⊆ ∪ 𝐽)
32		simp2 1138	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑋 ∈ ran 𝐹)
33	31, 32	sseldd 3983	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑋 ∈ ∪ 𝐽)
34		eqid 2733	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
35	34	t1sncld 22822	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Fre ∧ 𝑋 ∈ ∪ 𝐽) → {𝑋} ∈ (Clsd‘𝐽))
36	28, 33, 35	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → {𝑋} ∈ (Clsd‘𝐽))
37	27, 36	sseldd 3983	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → {𝑋} ∈ 𝑆)
38	7, 12, 21, 37	mbfmcnvima 33243	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (^◡𝐹 “ {𝑋}) ∈ dom 𝑀)
39		sibfinima.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀))
40	13, 14, 8, 15, 16, 17, 18, 1, 39	sibfmbl 33323	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
41	40	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
42	13, 14, 8, 15, 16, 17, 18, 1, 39	sibff 33324	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:∪ dom 𝑀⟶∪ 𝐽)
43	42	frnd 6723	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐺 ⊆ ∪ 𝐽)
44	43	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → ran 𝐺 ⊆ ∪ 𝐽)
45		simp3 1139	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑌 ∈ ran 𝐺)
46	44, 45	sseldd 3983	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑌 ∈ ∪ 𝐽)
47	34	t1sncld 22822	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Fre ∧ 𝑌 ∈ ∪ 𝐽) → {𝑌} ∈ (Clsd‘𝐽))
48	28, 46, 47	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → {𝑌} ∈ (Clsd‘𝐽))
49	27, 48	sseldd 3983	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → {𝑌} ∈ 𝑆)
50	7, 12, 41, 49	mbfmcnvima 33243	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (^◡𝐺 “ {𝑌}) ∈ dom 𝑀)
51		inelsiga 33122	. . . . . . 7 ⊢ ((dom 𝑀 ∈ ∪ ran sigAlgebra ∧ (^◡𝐹 “ {𝑋}) ∈ dom 𝑀 ∧ (^◡𝐺 “ {𝑌}) ∈ dom 𝑀) → ((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌})) ∈ dom 𝑀)
52	7, 38, 50, 51	syl3anc 1372	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → ((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌})) ∈ dom 𝑀)
53		measvxrge0 33192	. . . . . 6 ⊢ ((𝑀 ∈ (measures‘dom 𝑀) ∧ ((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌})) ∈ dom 𝑀) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ (0[,]+∞))
54	4, 52, 53	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ (0[,]+∞))
55		elxrge0 13431	. . . . . 6 ⊢ ((𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ (0[,]+∞) ↔ ((𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ ℝ^* ∧ 0 ≤ (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌})))))
56	55	simplbi 499	. . . . 5 ⊢ ((𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ (0[,]+∞) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ ℝ^*)
57	54, 56	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ ℝ^*)
58	57	adantr 482	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ ℝ^*)
59		0re 11213	. . . 4 ⊢ 0 ∈ ℝ
60	59	a1i 11	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → 0 ∈ ℝ)
61	55	simprbi 498	. . . . 5 ⊢ ((𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ (0[,]+∞) → 0 ≤ (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))))
62	54, 61	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 0 ≤ (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))))
63	62	adantr 482	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → 0 ≤ (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))))
64	57	adantr 482	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ ℝ^*)
65	4	adantr 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → 𝑀 ∈ (measures‘dom 𝑀))
66	38	adantr 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (^◡𝐹 “ {𝑋}) ∈ dom 𝑀)
67		measvxrge0 33192	. . . . . . 7 ⊢ ((𝑀 ∈ (measures‘dom 𝑀) ∧ (^◡𝐹 “ {𝑋}) ∈ dom 𝑀) → (𝑀‘(^◡𝐹 “ {𝑋})) ∈ (0[,]+∞))
68	65, 66, 67	syl2anc 585	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘(^◡𝐹 “ {𝑋})) ∈ (0[,]+∞))
69		elxrge0 13431	. . . . . . 7 ⊢ ((𝑀‘(^◡𝐹 “ {𝑋})) ∈ (0[,]+∞) ↔ ((𝑀‘(^◡𝐹 “ {𝑋})) ∈ ℝ^* ∧ 0 ≤ (𝑀‘(^◡𝐹 “ {𝑋}))))
70	69	simplbi 499	. . . . . 6 ⊢ ((𝑀‘(^◡𝐹 “ {𝑋})) ∈ (0[,]+∞) → (𝑀‘(^◡𝐹 “ {𝑋})) ∈ ℝ^*)
71	68, 70	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘(^◡𝐹 “ {𝑋})) ∈ ℝ^*)
72		pnfxr 11265	. . . . . 6 ⊢ +∞ ∈ ℝ^*
73	72	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → +∞ ∈ ℝ^*)
74	52	adantr 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → ((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌})) ∈ dom 𝑀)
75		inss1 4228	. . . . . . 7 ⊢ ((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌})) ⊆ (^◡𝐹 “ {𝑋})
76	75	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → ((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌})) ⊆ (^◡𝐹 “ {𝑋}))
77	65, 74, 66, 76	measssd 33202	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ≤ (𝑀‘(^◡𝐹 “ {𝑋})))
78		simpl1 1192	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → 𝜑)
79	32	anim1i 616	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑋 ∈ ran 𝐹 ∧ 𝑋 ≠ 0 ))
80		eldifsn 4790	. . . . . . . 8 ⊢ (𝑋 ∈ (ran 𝐹 ∖ { 0 }) ↔ (𝑋 ∈ ran 𝐹 ∧ 𝑋 ≠ 0 ))
81	79, 80	sylibr 233	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ (ran 𝐹 ∖ { 0 }))
82	13, 14, 8, 15, 16, 17, 18, 1, 19	sibfima 33326	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(^◡𝐹 “ {𝑋})) ∈ (0[,)+∞))
83	78, 81, 82	syl2anc 585	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘(^◡𝐹 “ {𝑋})) ∈ (0[,)+∞))
84		elico2 13385	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ^*) → ((𝑀‘(^◡𝐹 “ {𝑋})) ∈ (0[,)+∞) ↔ ((𝑀‘(^◡𝐹 “ {𝑋})) ∈ ℝ ∧ 0 ≤ (𝑀‘(^◡𝐹 “ {𝑋})) ∧ (𝑀‘(^◡𝐹 “ {𝑋})) < +∞)))
85	59, 72, 84	mp2an 691	. . . . . . 7 ⊢ ((𝑀‘(^◡𝐹 “ {𝑋})) ∈ (0[,)+∞) ↔ ((𝑀‘(^◡𝐹 “ {𝑋})) ∈ ℝ ∧ 0 ≤ (𝑀‘(^◡𝐹 “ {𝑋})) ∧ (𝑀‘(^◡𝐹 “ {𝑋})) < +∞))
86	85	simp3bi 1148	. . . . . 6 ⊢ ((𝑀‘(^◡𝐹 “ {𝑋})) ∈ (0[,)+∞) → (𝑀‘(^◡𝐹 “ {𝑋})) < +∞)
87	83, 86	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘(^◡𝐹 “ {𝑋})) < +∞)
88	64, 71, 73, 77, 87	xrlelttrd 13136	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) < +∞)
89	57	adantr 482	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ ℝ^*)
90	4	adantr 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → 𝑀 ∈ (measures‘dom 𝑀))
91	50	adantr 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (^◡𝐺 “ {𝑌}) ∈ dom 𝑀)
92		measvxrge0 33192	. . . . . . 7 ⊢ ((𝑀 ∈ (measures‘dom 𝑀) ∧ (^◡𝐺 “ {𝑌}) ∈ dom 𝑀) → (𝑀‘(^◡𝐺 “ {𝑌})) ∈ (0[,]+∞))
93	90, 91, 92	syl2anc 585	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘(^◡𝐺 “ {𝑌})) ∈ (0[,]+∞))
94		elxrge0 13431	. . . . . . 7 ⊢ ((𝑀‘(^◡𝐺 “ {𝑌})) ∈ (0[,]+∞) ↔ ((𝑀‘(^◡𝐺 “ {𝑌})) ∈ ℝ^* ∧ 0 ≤ (𝑀‘(^◡𝐺 “ {𝑌}))))
95	94	simplbi 499	. . . . . 6 ⊢ ((𝑀‘(^◡𝐺 “ {𝑌})) ∈ (0[,]+∞) → (𝑀‘(^◡𝐺 “ {𝑌})) ∈ ℝ^*)
96	93, 95	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘(^◡𝐺 “ {𝑌})) ∈ ℝ^*)
97	72	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → +∞ ∈ ℝ^*)
98	52	adantr 482	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → ((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌})) ∈ dom 𝑀)
99		inss2 4229	. . . . . . 7 ⊢ ((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌})) ⊆ (^◡𝐺 “ {𝑌})
100	99	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → ((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌})) ⊆ (^◡𝐺 “ {𝑌}))
101	90, 98, 91, 100	measssd 33202	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ≤ (𝑀‘(^◡𝐺 “ {𝑌})))
102		simpl1 1192	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → 𝜑)
103	45	anim1i 616	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑌 ∈ ran 𝐺 ∧ 𝑌 ≠ 0 ))
104		eldifsn 4790	. . . . . . . 8 ⊢ (𝑌 ∈ (ran 𝐺 ∖ { 0 }) ↔ (𝑌 ∈ ran 𝐺 ∧ 𝑌 ≠ 0 ))
105	103, 104	sylibr 233	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → 𝑌 ∈ (ran 𝐺 ∖ { 0 }))
106	13, 14, 8, 15, 16, 17, 18, 1, 39	sibfima 33326	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ∈ (ran 𝐺 ∖ { 0 })) → (𝑀‘(^◡𝐺 “ {𝑌})) ∈ (0[,)+∞))
107	102, 105, 106	syl2anc 585	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘(^◡𝐺 “ {𝑌})) ∈ (0[,)+∞))
108		elico2 13385	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ^*) → ((𝑀‘(^◡𝐺 “ {𝑌})) ∈ (0[,)+∞) ↔ ((𝑀‘(^◡𝐺 “ {𝑌})) ∈ ℝ ∧ 0 ≤ (𝑀‘(^◡𝐺 “ {𝑌})) ∧ (𝑀‘(^◡𝐺 “ {𝑌})) < +∞)))
109	59, 72, 108	mp2an 691	. . . . . . 7 ⊢ ((𝑀‘(^◡𝐺 “ {𝑌})) ∈ (0[,)+∞) ↔ ((𝑀‘(^◡𝐺 “ {𝑌})) ∈ ℝ ∧ 0 ≤ (𝑀‘(^◡𝐺 “ {𝑌})) ∧ (𝑀‘(^◡𝐺 “ {𝑌})) < +∞))
110	109	simp3bi 1148	. . . . . 6 ⊢ ((𝑀‘(^◡𝐺 “ {𝑌})) ∈ (0[,)+∞) → (𝑀‘(^◡𝐺 “ {𝑌})) < +∞)
111	107, 110	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘(^◡𝐺 “ {𝑌})) < +∞)
112	89, 96, 97, 101, 111	xrlelttrd 13136	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) < +∞)
113	88, 112	jaodan 957	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) < +∞)
114		xrre3 13147	. . 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)
115	58, 60, 63, 113, 114	syl22anc 838	. 2 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ ℝ)
116		elico2 13385	. . 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}))) < +∞)))
117	59, 72, 116	mp2an 691	. 2 ⊢ ((𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ (0[,)+∞) ↔ ((𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ ℝ ∧ 0 ≤ (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∧ (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) < +∞))
118	115, 63, 113, 117	syl3anbrc 1344	1 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ (0[,)+∞))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∖ cdif 3945 ∩ cin 3947 ⊆ wss 3948 {csn 4628 ∪ cuni 4908 class class class wbr 5148 ^◡ccnv 5675 dom cdm 5676 ran crn 5677 “ cima 5679 ‘cfv 6541 (class class class)co 7406 ℝcr 11106 0cc0 11107 +∞cpnf 11242 ℝ^*cxr 11244 < clt 11245 ≤ cle 11246 [,)cico 13323 [,]cicc 13324 Basecbs 17141 Scalarcsca 17197 ·_𝑠 cvsca 17198 TopOpenctopn 17364 0_gc0g 17382 Topctop 22387 TopSpctps 22426 Clsdccld 22512 Frect1 22803 ℝHomcrrh 32962 sigAlgebracsiga 33095 sigaGencsigagen 33125 measurescmeas 33182 MblFnMcmbfm 33236 sitgcsitg 33317

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-ac2 10455 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 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-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 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 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-supp 8144 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-2o 8464 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-dju 9893 df-card 9931 df-acn 9934 df-ac 10108 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ioc 13326 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15011 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-limsup 15412 df-clim 15429 df-rlim 15430 df-sum 15630 df-ef 16008 df-sin 16010 df-cos 16011 df-pi 16013 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-starv 17209 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-hom 17218 df-cco 17219 df-rest 17365 df-topn 17366 df-0g 17384 df-gsum 17385 df-topgen 17386 df-pt 17387 df-prds 17390 df-ordt 17444 df-xrs 17445 df-qtop 17450 df-imas 17451 df-xps 17453 df-mre 17527 df-mrc 17528 df-acs 17530 df-ps 18516 df-tsr 18517 df-plusf 18557 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-mhm 18668 df-submnd 18669 df-grp 18819 df-minusg 18820 df-sbg 18821 df-mulg 18946 df-subg 18998 df-cntz 19176 df-cmn 19645 df-abl 19646 df-mgp 19983 df-ur 20000 df-ring 20052 df-cring 20053 df-subrg 20354 df-abv 20418 df-lmod 20466 df-scaf 20467 df-sra 20778 df-rgmod 20779 df-psmet 20929 df-xmet 20930 df-met 20931 df-bl 20932 df-mopn 20933 df-fbas 20934 df-fg 20935 df-cnfld 20938 df-top 22388 df-topon 22405 df-topsp 22427 df-bases 22441 df-cld 22515 df-ntr 22516 df-cls 22517 df-nei 22594 df-lp 22632 df-perf 22633 df-cn 22723 df-cnp 22724 df-t1 22810 df-haus 22811 df-tx 23058 df-hmeo 23251 df-fil 23342 df-fm 23434 df-flim 23435 df-flf 23436 df-tmd 23568 df-tgp 23569 df-tsms 23623 df-trg 23656 df-xms 23818 df-ms 23819 df-tms 23820 df-nm 24083 df-ngp 24084 df-nrg 24086 df-nlm 24087 df-ii 24385 df-cncf 24386 df-limc 25375 df-dv 25376 df-log 26057 df-esum 33015 df-siga 33096 df-sigagen 33126 df-meas 33183 df-mbfm 33237 df-sitg 33318

This theorem is referenced by: sibfof 33328 sitgaddlemb 33336

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