Theorem sibfinima 34376

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 34238	. . . . . . . 8 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
3	1, 2	sylib 218	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (measures‘dom 𝑀))
4	3	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑀 ∈ (measures‘dom 𝑀))
5		dmmeas 34237	. . . . . . . . 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 34178	. . . . . . . . . 10 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
11	8, 10	eqeltrid 2839	. . . . . . . . 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 = (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 34372	. . . . . . . . 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 22880	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ TopSp → 𝐽 ∈ Top)
24		cldssbrsiga 34223	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
25	22, 23, 24	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
26	25, 8	sseqtrrdi 4005	. . . . . . . . . 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 34373	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝐽)
30	29	frnd 6719	. . . . . . . . . . . 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 3964	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑋 ∈ ∪ 𝐽)
34		eqid 2736	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
35	34	t1sncld 23269	. . . . . . . . . 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 3964	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → {𝑋} ∈ 𝑆)
38	7, 12, 21, 37	mbfmcnvima 34292	. . . . . . 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 34372	. . . . . . . . 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 34373	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:∪ dom 𝑀⟶∪ 𝐽)
43	42	frnd 6719	. . . . . . . . . . . 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 3964	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑌 ∈ ∪ 𝐽)
47	34	t1sncld 23269	. . . . . . . . . 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 3964	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → {𝑌} ∈ 𝑆)
50	7, 12, 41, 49	mbfmcnvima 34292	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (◡𝐺 “ {𝑌}) ∈ dom 𝑀)
51		inelsiga 34171	. . . . . . 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 1373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ∈ dom 𝑀)
53		measvxrge0 34241	. . . . . 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 13479	. . . . . 6 ⊢ ((𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,]+∞) ↔ ((𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ* ∧ 0 ≤ (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})))))
56	55	simplbi 497	. . . . 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 480	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ*)
59		0re 11242	. . . 4 ⊢ 0 ∈ ℝ
60	59	a1i 11	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → 0 ∈ ℝ)
61	55	simprbi 496	. . . . 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 480	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → 0 ≤ (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))))
64	57	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ*)
65	4	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → 𝑀 ∈ (measures‘dom 𝑀))
66	38	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (◡𝐹 “ {𝑋}) ∈ dom 𝑀)
67		measvxrge0 34241	. . . . . . 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 13479	. . . . . . 7 ⊢ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,]+∞) ↔ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ ℝ* ∧ 0 ≤ (𝑀‘(◡𝐹 “ {𝑋}))))
70	69	simplbi 497	. . . . . 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 11294	. . . . . 6 ⊢ +∞ ∈ ℝ*
73	72	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → +∞ ∈ ℝ*)
74	52	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ∈ dom 𝑀)
75		inss1 4217	. . . . . . 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 34251	. . . . 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 615	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑋 ∈ ran 𝐹 ∧ 𝑋 ≠ 0 ))
80		eldifsn 4767	. . . . . . . 8 ⊢ (𝑋 ∈ (ran 𝐹 ∖ { 0 }) ↔ (𝑋 ∈ ran 𝐹 ∧ 𝑋 ≠ 0 ))
81	79, 80	sylibr 234	. . . . . . 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 34375	. . . . . . 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 13432	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ((𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,)+∞) ↔ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ ℝ ∧ 0 ≤ (𝑀‘(◡𝐹 “ {𝑋})) ∧ (𝑀‘(◡𝐹 “ {𝑋})) < +∞)))
85	59, 72, 84	mp2an 692	. . . . . . 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 13181	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) < +∞)
89	57	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ*)
90	4	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → 𝑀 ∈ (measures‘dom 𝑀))
91	50	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (◡𝐺 “ {𝑌}) ∈ dom 𝑀)
92		measvxrge0 34241	. . . . . . 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 13479	. . . . . . 7 ⊢ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,]+∞) ↔ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ ℝ* ∧ 0 ≤ (𝑀‘(◡𝐺 “ {𝑌}))))
95	94	simplbi 497	. . . . . 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 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ∈ dom 𝑀)
99		inss2 4218	. . . . . . 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 34251	. . . . 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 615	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑌 ∈ ran 𝐺 ∧ 𝑌 ≠ 0 ))
104		eldifsn 4767	. . . . . . . 8 ⊢ (𝑌 ∈ (ran 𝐺 ∖ { 0 }) ↔ (𝑌 ∈ ran 𝐺 ∧ 𝑌 ≠ 0 ))
105	103, 104	sylibr 234	. . . . . . 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 34375	. . . . . . 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 13432	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ((𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,)+∞) ↔ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ ℝ ∧ 0 ≤ (𝑀‘(◡𝐺 “ {𝑌})) ∧ (𝑀‘(◡𝐺 “ {𝑌})) < +∞)))
109	59, 72, 108	mp2an 692	. . . . . . 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 13181	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) < +∞)
113	88, 112	jaodan 959	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) < +∞)
114		xrre3 13192	. . 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 13432	. . 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 692	. 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[,)+∞))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2933   ∖ cdif 3928   ∩ cin 3930   ⊆ wss 3931  {csn 4606  ∪ cuni 4888   class class class wbr 5124  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   “ cima 5662  ‘cfv 6536  (class class class)co 7410  ℝcr 11133  0cc0 11134  +∞cpnf 11271  ℝ^*cxr 11273   < clt 11274   ≤ cle 11275  [,)cico 13369  [,]cicc 13370  Basecbs 17233  Scalarcsca 17279   ·_𝑠 cvsca 17280  TopOpenctopn 17440  0_gc0g 17458  Topctop 22836  TopSpctps 22875  Clsdccld 22959  Frect1 23250  ℝHomcrrh 34029  sigAlgebracsiga 34144  sigaGencsigagen 34174  measurescmeas 34231  MblFnMcmbfm 34285  sitgcsitg 34366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660  ax-ac2 10482  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211  ax-pre-sup 11212  ax-addf 11213  ax-mulf 11214

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-iin 4975  df-disj 5092  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7676  df-om 7867  df-1st 7993  df-2nd 7994  df-supp 8165  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-er 8724  df-map 8847  df-pm 8848  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9379  df-fi 9428  df-sup 9459  df-inf 9460  df-oi 9529  df-dju 9920  df-card 9958  df-acn 9961  df-ac 10135  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-div 11900  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-z 12594  df-dec 12714  df-uz 12858  df-q 12970  df-rp 13014  df-xneg 13133  df-xadd 13134  df-xmul 13135  df-ioo 13371  df-ioc 13372  df-ico 13373  df-icc 13374  df-fz 13530  df-fzo 13677  df-fl 13814  df-mod 13892  df-seq 14025  df-exp 14085  df-fac 14297  df-bc 14326  df-hash 14354  df-shft 15091  df-cj 15123  df-re 15124  df-im 15125  df-sqrt 15259  df-abs 15260  df-limsup 15492  df-clim 15509  df-rlim 15510  df-sum 15708  df-ef 16088  df-sin 16090  df-cos 16091  df-pi 16093  df-struct 17171  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-ress 17257  df-plusg 17289  df-mulr 17290  df-starv 17291  df-sca 17292  df-vsca 17293  df-ip 17294  df-tset 17295  df-ple 17296  df-ds 17298  df-unif 17299  df-hom 17300  df-cco 17301  df-rest 17441  df-topn 17442  df-0g 17460  df-gsum 17461  df-topgen 17462  df-pt 17463  df-prds 17466  df-ordt 17520  df-xrs 17521  df-qtop 17526  df-imas 17527  df-xps 17529  df-mre 17603  df-mrc 17604  df-acs 17606  df-ps 18581  df-tsr 18582  df-plusf 18622  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-mhm 18766  df-submnd 18767  df-grp 18924  df-minusg 18925  df-sbg 18926  df-mulg 19056  df-subg 19111  df-cntz 19305  df-cmn 19768  df-abl 19769  df-mgp 20106  df-rng 20118  df-ur 20147  df-ring 20200  df-cring 20201  df-subrng 20511  df-subrg 20535  df-abv 20774  df-lmod 20824  df-scaf 20825  df-sra 21136  df-rgmod 21137  df-psmet 21312  df-xmet 21313  df-met 21314  df-bl 21315  df-mopn 21316  df-fbas 21317  df-fg 21318  df-cnfld 21321  df-top 22837  df-topon 22854  df-topsp 22876  df-bases 22889  df-cld 22962  df-ntr 22963  df-cls 22964  df-nei 23041  df-lp 23079  df-perf 23080  df-cn 23170  df-cnp 23171  df-t1 23257  df-haus 23258  df-tx 23505  df-hmeo 23698  df-fil 23789  df-fm 23881  df-flim 23882  df-flf 23883  df-tmd 24015  df-tgp 24016  df-tsms 24070  df-trg 24103  df-xms 24264  df-ms 24265  df-tms 24266  df-nm 24526  df-ngp 24527  df-nrg 24529  df-nlm 24530  df-ii 24826  df-cncf 24827  df-limc 25824  df-dv 25825  df-log 26522  df-esum 34064  df-siga 34145  df-sigagen 34175  df-meas 34232  df-mbfm 34286  df-sitg 34367

This theorem is referenced by:  sibfof  34377  sitgaddlemb  34385