sibfinima - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > sibfinima		Structured version Visualization version GIF version

Theorem sibfinima 33407

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 33269	. . . . . . . 8 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
3	1, 2	sylib 217	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (measures‘dom 𝑀))
4	3	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑀 ∈ (measures‘dom 𝑀))
5		dmmeas 33268	. . . . . . . . 9 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)
6	1, 5	syl 17	. . . . . . . 8 ⊢ (𝜑 → dom 𝑀 ∈ ∪ ran sigAlgebra)
7	6	3ad2ant1 1133	. . . . . . 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 33209	. . . . . . . . . 10 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
11	8, 10	eqeltrid 2837	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
12	11	3ad2ant1 1133	. . . . . . . 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 33403	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
21	20	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
22		sibfinima.w	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ∈ TopSp)
23	14	tpstop 22446	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ TopSp → 𝐽 ∈ Top)
24		cldssbrsiga 33254	. . . . . . . . . . . 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 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (Clsd‘𝐽) ⊆ 𝑆)
28	9	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝐽 ∈ Fre)
29	13, 14, 8, 15, 16, 17, 18, 1, 19	sibff 33404	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝐽)
30	29	frnd 6725	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐹 ⊆ ∪ 𝐽)
31	30	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → ran 𝐹 ⊆ ∪ 𝐽)
32		simp2 1137	. . . . . . . . . . 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 2732	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
35	34	t1sncld 22837	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Fre ∧ 𝑋 ∈ ∪ 𝐽) → {𝑋} ∈ (Clsd‘𝐽))
36	28, 33, 35	syl2anc 584	. . . . . . . . 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 33323	. . . . . . 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 33403	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
41	40	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
42	13, 14, 8, 15, 16, 17, 18, 1, 39	sibff 33404	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:∪ dom 𝑀⟶∪ 𝐽)
43	42	frnd 6725	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐺 ⊆ ∪ 𝐽)
44	43	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → ran 𝐺 ⊆ ∪ 𝐽)
45		simp3 1138	. . . . . . . . . . 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 22837	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Fre ∧ 𝑌 ∈ ∪ 𝐽) → {𝑌} ∈ (Clsd‘𝐽))
48	28, 46, 47	syl2anc 584	. . . . . . . . 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 33323	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (^◡𝐺 “ {𝑌}) ∈ dom 𝑀)
51		inelsiga 33202	. . . . . . 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 1371	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → ((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌})) ∈ dom 𝑀)
53		measvxrge0 33272	. . . . . 6 ⊢ ((𝑀 ∈ (measures‘dom 𝑀) ∧ ((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌})) ∈ dom 𝑀) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ (0[,]+∞))
54	4, 52, 53	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ (0[,]+∞))
55		elxrge0 13436	. . . . . 6 ⊢ ((𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ (0[,]+∞) ↔ ((𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ ℝ^* ∧ 0 ≤ (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌})))))
56	55	simplbi 498	. . . . 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 481	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ ℝ^*)
59		0re 11218	. . . 4 ⊢ 0 ∈ ℝ
60	59	a1i 11	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → 0 ∈ ℝ)
61	55	simprbi 497	. . . . 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 481	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → 0 ≤ (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))))
64	57	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ ℝ^*)
65	4	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → 𝑀 ∈ (measures‘dom 𝑀))
66	38	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (^◡𝐹 “ {𝑋}) ∈ dom 𝑀)
67		measvxrge0 33272	. . . . . . 7 ⊢ ((𝑀 ∈ (measures‘dom 𝑀) ∧ (^◡𝐹 “ {𝑋}) ∈ dom 𝑀) → (𝑀‘(^◡𝐹 “ {𝑋})) ∈ (0[,]+∞))
68	65, 66, 67	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘(^◡𝐹 “ {𝑋})) ∈ (0[,]+∞))
69		elxrge0 13436	. . . . . . 7 ⊢ ((𝑀‘(^◡𝐹 “ {𝑋})) ∈ (0[,]+∞) ↔ ((𝑀‘(^◡𝐹 “ {𝑋})) ∈ ℝ^* ∧ 0 ≤ (𝑀‘(^◡𝐹 “ {𝑋}))))
70	69	simplbi 498	. . . . . 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 11270	. . . . . 6 ⊢ +∞ ∈ ℝ^*
73	72	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → +∞ ∈ ℝ^*)
74	52	adantr 481	. . . . . 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 33282	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ≤ (𝑀‘(^◡𝐹 “ {𝑋})))
78		simpl1 1191	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → 𝜑)
79	32	anim1i 615	. . . . . . . 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 33406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(^◡𝐹 “ {𝑋})) ∈ (0[,)+∞))
83	78, 81, 82	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘(^◡𝐹 “ {𝑋})) ∈ (0[,)+∞))
84		elico2 13390	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ^*) → ((𝑀‘(^◡𝐹 “ {𝑋})) ∈ (0[,)+∞) ↔ ((𝑀‘(^◡𝐹 “ {𝑋})) ∈ ℝ ∧ 0 ≤ (𝑀‘(^◡𝐹 “ {𝑋})) ∧ (𝑀‘(^◡𝐹 “ {𝑋})) < +∞)))
85	59, 72, 84	mp2an 690	. . . . . . 7 ⊢ ((𝑀‘(^◡𝐹 “ {𝑋})) ∈ (0[,)+∞) ↔ ((𝑀‘(^◡𝐹 “ {𝑋})) ∈ ℝ ∧ 0 ≤ (𝑀‘(^◡𝐹 “ {𝑋})) ∧ (𝑀‘(^◡𝐹 “ {𝑋})) < +∞))
86	85	simp3bi 1147	. . . . . 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 13141	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) < +∞)
89	57	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ ℝ^*)
90	4	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → 𝑀 ∈ (measures‘dom 𝑀))
91	50	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (^◡𝐺 “ {𝑌}) ∈ dom 𝑀)
92		measvxrge0 33272	. . . . . . 7 ⊢ ((𝑀 ∈ (measures‘dom 𝑀) ∧ (^◡𝐺 “ {𝑌}) ∈ dom 𝑀) → (𝑀‘(^◡𝐺 “ {𝑌})) ∈ (0[,]+∞))
93	90, 91, 92	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘(^◡𝐺 “ {𝑌})) ∈ (0[,]+∞))
94		elxrge0 13436	. . . . . . 7 ⊢ ((𝑀‘(^◡𝐺 “ {𝑌})) ∈ (0[,]+∞) ↔ ((𝑀‘(^◡𝐺 “ {𝑌})) ∈ ℝ^* ∧ 0 ≤ (𝑀‘(^◡𝐺 “ {𝑌}))))
95	94	simplbi 498	. . . . . 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 481	. . . . . 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 33282	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ≤ (𝑀‘(^◡𝐺 “ {𝑌})))
102		simpl1 1191	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → 𝜑)
103	45	anim1i 615	. . . . . . . 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 33406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ∈ (ran 𝐺 ∖ { 0 })) → (𝑀‘(^◡𝐺 “ {𝑌})) ∈ (0[,)+∞))
107	102, 105, 106	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘(^◡𝐺 “ {𝑌})) ∈ (0[,)+∞))
108		elico2 13390	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ^*) → ((𝑀‘(^◡𝐺 “ {𝑌})) ∈ (0[,)+∞) ↔ ((𝑀‘(^◡𝐺 “ {𝑌})) ∈ ℝ ∧ 0 ≤ (𝑀‘(^◡𝐺 “ {𝑌})) ∧ (𝑀‘(^◡𝐺 “ {𝑌})) < +∞)))
109	59, 72, 108	mp2an 690	. . . . . . 7 ⊢ ((𝑀‘(^◡𝐺 “ {𝑌})) ∈ (0[,)+∞) ↔ ((𝑀‘(^◡𝐺 “ {𝑌})) ∈ ℝ ∧ 0 ≤ (𝑀‘(^◡𝐺 “ {𝑌})) ∧ (𝑀‘(^◡𝐺 “ {𝑌})) < +∞))
110	109	simp3bi 1147	. . . . . 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 13141	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) < +∞)
113	88, 112	jaodan 956	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) < +∞)
114		xrre3 13152	. . 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 837	. 2 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((^◡𝐹 “ {𝑋}) ∩ (^◡𝐺 “ {𝑌}))) ∈ ℝ)
116		elico2 13390	. . 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 690	. 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 1343	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 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∖ 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 6543 (class class class)co 7411 ℝcr 11111 0cc0 11112 +∞cpnf 11247 ℝ^*cxr 11249 < clt 11250 ≤ cle 11251 [,)cico 13328 [,]cicc 13329 Basecbs 17146 Scalarcsca 17202 ·_𝑠 cvsca 17203 TopOpenctopn 17369 0_gc0g 17387 Topctop 22402 TopSpctps 22441 Clsdccld 22527 Frect1 22818 ℝHomcrrh 33042 sigAlgebracsiga 33175 sigaGencsigagen 33205 measurescmeas 33262 MblFnMcmbfm 33316 sitgcsitg 33397

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-ac2 10460 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 ax-addf 11191 ax-mulf 11192

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-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 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-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-ac 10113 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-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-ioo 13330 df-ioc 13331 df-ico 13332 df-icc 13333 df-fz 13487 df-fzo 13630 df-fl 13759 df-mod 13837 df-seq 13969 df-exp 14030 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15016 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-limsup 15417 df-clim 15434 df-rlim 15435 df-sum 15635 df-ef 16013 df-sin 16015 df-cos 16016 df-pi 16018 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-hom 17223 df-cco 17224 df-rest 17370 df-topn 17371 df-0g 17389 df-gsum 17390 df-topgen 17391 df-pt 17392 df-prds 17395 df-ordt 17449 df-xrs 17450 df-qtop 17455 df-imas 17456 df-xps 17458 df-mre 17532 df-mrc 17533 df-acs 17535 df-ps 18521 df-tsr 18522 df-plusf 18562 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-mhm 18673 df-submnd 18674 df-grp 18824 df-minusg 18825 df-sbg 18826 df-mulg 18953 df-subg 19005 df-cntz 19183 df-cmn 19652 df-abl 19653 df-mgp 19990 df-ur 20007 df-ring 20060 df-cring 20061 df-subrg 20321 df-abv 20429 df-lmod 20477 df-scaf 20478 df-sra 20791 df-rgmod 20792 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-fbas 20947 df-fg 20948 df-cnfld 20951 df-top 22403 df-topon 22420 df-topsp 22442 df-bases 22456 df-cld 22530 df-ntr 22531 df-cls 22532 df-nei 22609 df-lp 22647 df-perf 22648 df-cn 22738 df-cnp 22739 df-t1 22825 df-haus 22826 df-tx 23073 df-hmeo 23266 df-fil 23357 df-fm 23449 df-flim 23450 df-flf 23451 df-tmd 23583 df-tgp 23584 df-tsms 23638 df-trg 23671 df-xms 23833 df-ms 23834 df-tms 23835 df-nm 24098 df-ngp 24099 df-nrg 24101 df-nlm 24102 df-ii 24400 df-cncf 24401 df-limc 25390 df-dv 25391 df-log 26072 df-esum 33095 df-siga 33176 df-sigagen 33206 df-meas 33263 df-mbfm 33317 df-sitg 33398

This theorem is referenced by: sibfof 33408 sitgaddlemb 33416

W3C validator