Theorem sibfinima 34883

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 34746	. . . . . . . 8 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
3	1, 2	sylib 221	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (measures‘dom 𝑀))
4	3	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑀 ∈ (measures‘dom 𝑀))
5		dmmeas 34745	. . . . . . . . 9 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra)
6	1, 5	syl 18	. . . . . . . 8 ⊢ (𝜑 → dom 𝑀 ∈ ∪ ran sigAlgebra)
7	6	3ad2ant1 1151	. . . . . . 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 34686	. . . . . . . . . 10 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
11	8, 10	eqeltrid 2864	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
12	11	3ad2ant1 1151	. . . . . . . 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 34879	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
21	20	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
22		sibfinima.w	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ∈ TopSp)
23	14	tpstop 23194	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ TopSp → 𝐽 ∈ Top)
24		cldssbrsiga 34731	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
25	22, 23, 24	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
26	25, 8	sseqtrrdi 3972	. . . . . . . . . 10 ⊢ (𝜑 → (Clsd‘𝐽) ⊆ 𝑆)
27	26	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (Clsd‘𝐽) ⊆ 𝑆)
28	9	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝐽 ∈ Fre)
29	13, 14, 8, 15, 16, 17, 18, 1, 19	sibff 34880	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝐽)
30	29	frnd 6714	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐹 ⊆ ∪ 𝐽)
31	30	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → ran 𝐹 ⊆ ∪ 𝐽)
32		simp2 1155	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑋 ∈ ran 𝐹)
33	31, 32	sseldd 3932	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑋 ∈ ∪ 𝐽)
34		eqid 2760	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
35	34	t1sncld 23583	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Fre ∧ 𝑋 ∈ ∪ 𝐽) → {𝑋} ∈ (Clsd‘𝐽))
36	28, 33, 35	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → {𝑋} ∈ (Clsd‘𝐽))
37	27, 36	sseldd 3932	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → {𝑋} ∈ 𝑆)
38	7, 12, 21, 37	mbfmcnvima 34799	. . . . . . 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 34879	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
41	40	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
42	13, 14, 8, 15, 16, 17, 18, 1, 39	sibff 34880	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:∪ dom 𝑀⟶∪ 𝐽)
43	42	frnd 6714	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐺 ⊆ ∪ 𝐽)
44	43	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → ran 𝐺 ⊆ ∪ 𝐽)
45		simp3 1156	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑌 ∈ ran 𝐺)
46	44, 45	sseldd 3932	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑌 ∈ ∪ 𝐽)
47	34	t1sncld 23583	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Fre ∧ 𝑌 ∈ ∪ 𝐽) → {𝑌} ∈ (Clsd‘𝐽))
48	28, 46, 47	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → {𝑌} ∈ (Clsd‘𝐽))
49	27, 48	sseldd 3932	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → {𝑌} ∈ 𝑆)
50	7, 12, 41, 49	mbfmcnvima 34799	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (◡𝐺 “ {𝑌}) ∈ dom 𝑀)
51		inelsiga 34679	. . . . . . 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 1398	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ∈ dom 𝑀)
53		measvxrge0 34749	. . . . . 6 ⊢ ((𝑀 ∈ (measures‘dom 𝑀) ∧ ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ∈ dom 𝑀) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,]+∞))
54	4, 52, 53	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,]+∞))
55		elxrge0 13535	. . . . . 6 ⊢ ((𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,]+∞) ↔ ((𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ* ∧ 0 ≤ (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})))))
56	55	simplbi 502	. . . . 5 ⊢ ((𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,]+∞) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ*)
57	54, 56	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ*)
58	57	adantr 486	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ*)
59		0re 11259	. . . 4 ⊢ 0 ∈ ℝ
60	59	a1i 11	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → 0 ∈ ℝ)
61	55	simprbi 503	. . . . 5 ⊢ ((𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,]+∞) → 0 ≤ (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))))
62	54, 61	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 0 ≤ (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))))
63	62	adantr 486	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → 0 ≤ (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))))
64	57	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ*)
65	4	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → 𝑀 ∈ (measures‘dom 𝑀))
66	38	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (◡𝐹 “ {𝑋}) ∈ dom 𝑀)
67		measvxrge0 34749	. . . . . . 7 ⊢ ((𝑀 ∈ (measures‘dom 𝑀) ∧ (◡𝐹 “ {𝑋}) ∈ dom 𝑀) → (𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,]+∞))
68	65, 66, 67	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,]+∞))
69		elxrge0 13535	. . . . . . 7 ⊢ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,]+∞) ↔ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ ℝ* ∧ 0 ≤ (𝑀‘(◡𝐹 “ {𝑋}))))
70	69	simplbi 502	. . . . . 6 ⊢ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,]+∞) → (𝑀‘(◡𝐹 “ {𝑋})) ∈ ℝ*)
71	68, 70	syl 18	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘(◡𝐹 “ {𝑋})) ∈ ℝ*)
72		pnfxr 11312	. . . . . 6 ⊢ +∞ ∈ ℝ*
73	72	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → +∞ ∈ ℝ*)
74	52	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ∈ dom 𝑀)
75		inss1 4182	. . . . . . 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 34759	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ≤ (𝑀‘(◡𝐹 “ {𝑋})))
78		simpl1 1210	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → 𝜑)
79	32	anim1i 627	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑋 ∈ ran 𝐹 ∧ 𝑋 ≠ 0 ))
80		eldifsn 4748	. . . . . . . 8 ⊢ (𝑋 ∈ (ran 𝐹 ∖ { 0 }) ↔ (𝑋 ∈ ran 𝐹 ∧ 𝑋 ≠ 0 ))
81	79, 80	sylibr 237	. . . . . . 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 34882	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,)+∞))
83	78, 81, 82	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,)+∞))
84		elico2 13488	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ((𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,)+∞) ↔ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ ℝ ∧ 0 ≤ (𝑀‘(◡𝐹 “ {𝑋})) ∧ (𝑀‘(◡𝐹 “ {𝑋})) < +∞)))
85	59, 72, 84	mp2an 705	. . . . . . 7 ⊢ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,)+∞) ↔ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ ℝ ∧ 0 ≤ (𝑀‘(◡𝐹 “ {𝑋})) ∧ (𝑀‘(◡𝐹 “ {𝑋})) < +∞))
86	85	simp3bi 1165	. . . . . 6 ⊢ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,)+∞) → (𝑀‘(◡𝐹 “ {𝑋})) < +∞)
87	83, 86	syl 18	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘(◡𝐹 “ {𝑋})) < +∞)
88	64, 71, 73, 77, 87	xrlelttrd 13236	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) < +∞)
89	57	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ*)
90	4	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → 𝑀 ∈ (measures‘dom 𝑀))
91	50	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (◡𝐺 “ {𝑌}) ∈ dom 𝑀)
92		measvxrge0 34749	. . . . . . 7 ⊢ ((𝑀 ∈ (measures‘dom 𝑀) ∧ (◡𝐺 “ {𝑌}) ∈ dom 𝑀) → (𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,]+∞))
93	90, 91, 92	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,]+∞))
94		elxrge0 13535	. . . . . . 7 ⊢ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,]+∞) ↔ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ ℝ* ∧ 0 ≤ (𝑀‘(◡𝐺 “ {𝑌}))))
95	94	simplbi 502	. . . . . 6 ⊢ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,]+∞) → (𝑀‘(◡𝐺 “ {𝑌})) ∈ ℝ*)
96	93, 95	syl 18	. . . . 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 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ∈ dom 𝑀)
99		inss2 4183	. . . . . . 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 34759	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ≤ (𝑀‘(◡𝐺 “ {𝑌})))
102		simpl1 1210	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → 𝜑)
103	45	anim1i 627	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑌 ∈ ran 𝐺 ∧ 𝑌 ≠ 0 ))
104		eldifsn 4748	. . . . . . . 8 ⊢ (𝑌 ∈ (ran 𝐺 ∖ { 0 }) ↔ (𝑌 ∈ ran 𝐺 ∧ 𝑌 ≠ 0 ))
105	103, 104	sylibr 237	. . . . . . 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 34882	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ∈ (ran 𝐺 ∖ { 0 })) → (𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,)+∞))
107	102, 105, 106	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,)+∞))
108		elico2 13488	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ((𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,)+∞) ↔ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ ℝ ∧ 0 ≤ (𝑀‘(◡𝐺 “ {𝑌})) ∧ (𝑀‘(◡𝐺 “ {𝑌})) < +∞)))
109	59, 72, 108	mp2an 705	. . . . . . 7 ⊢ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,)+∞) ↔ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ ℝ ∧ 0 ≤ (𝑀‘(◡𝐺 “ {𝑌})) ∧ (𝑀‘(◡𝐺 “ {𝑌})) < +∞))
110	109	simp3bi 1165	. . . . . 6 ⊢ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,)+∞) → (𝑀‘(◡𝐺 “ {𝑌})) < +∞)
111	107, 110	syl 18	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘(◡𝐺 “ {𝑌})) < +∞)
112	89, 96, 97, 101, 111	xrlelttrd 13236	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) < +∞)
113	88, 112	jaodan 972	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) < +∞)
114		xrre3 13248	. . 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 852	. 2 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ)
116		elico2 13488	. . 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 705	. 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 1362	1 ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,)+∞))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  {csn 4584  ∪ cuni 4867   class class class wbr 5103  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   “ cima 5658  ‘cfv 6535  (class class class)co 7416  ℝcr 11148  0cc0 11149  +∞cpnf 11289  ℝ^*cxr 11291   < clt 11292   ≤ cle 11293  [,)cico 13425  [,]cicc 13426  Basecbs 17326  Scalarcsca 17370   ·_𝑠 cvsca 17371  TopOpenctopn 17531  0_gc0g 17549  Topctop 23150  TopSpctps 23189  Clsdccld 23273  Frect1 23564  ℝHomcrrh 34536  sigAlgebracsiga 34651  sigaGencsigagen 34682  measurescmeas 34739  MblFnMcmbfm 34793  sitgcsitg 34873

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742  ax-inf2 9627  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226  ax-pre-sup 11227  ax-addf 11228  ax-mulf 11229

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-disj 5071  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6301  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-isom 6544  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7684  df-om 7869  df-1st 7992  df-2nd 7993  df-supp 8164  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-1o 8462  df-2o 8463  df-er 8703  df-map 8835  df-pm 8836  df-ixp 8912  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-fsupp 9339  df-fi 9388  df-sup 9419  df-inf 9420  df-oi 9489  df-dju 9931  df-card 9969  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11492  df-neg 11493  df-div 11921  df-nn 12283  df-2 12352  df-3 12353  df-4 12354  df-5 12355  df-6 12356  df-7 12357  df-8 12358  df-9 12359  df-n0 12554  df-z 12641  df-dec 12762  df-uz 12913  df-q 13023  df-rp 13068  df-xneg 13188  df-xadd 13189  df-xmul 13190  df-ioo 13427  df-ioc 13428  df-ico 13429  df-icc 13430  df-fz 13587  df-fzo 13735  df-fl 13878  df-mod 13956  df-seq 14091  df-exp 14151  df-fac 14363  df-bc 14392  df-hash 14420  df-shft 15165  df-cj 15211  df-re 15212  df-im 15213  df-sqrt 15347  df-abs 15348  df-limsup 15583  df-clim 15600  df-rlim 15601  df-sum 15799  df-ef 16178  df-sin 16180  df-cos 16181  df-pi 16183  df-struct 17264  df-sets 17281  df-slot 17299  df-ndx 17311  df-base 17327  df-ress 17348  df-plusg 17380  df-mulr 17381  df-starv 17382  df-sca 17383  df-vsca 17384  df-ip 17385  df-tset 17386  df-ple 17387  df-ds 17389  df-unif 17390  df-hom 17391  df-cco 17392  df-rest 17532  df-topn 17533  df-0g 17551  df-gsum 17552  df-topgen 17553  df-pt 17554  df-prds 17557  df-ordt 17612  df-xrs 17613  df-qtop 17618  df-imas 17619  df-xps 17621  df-mre 17695  df-mrc 17696  df-acs 17698  df-ps 18679  df-tsr 18680  df-plusf 18754  df-mgm 18755  df-sgrp 18847  df-mnd 18863  df-mhm 18917  df-submnd 18918  df-grp 19086  df-minusg 19087  df-sbg 19088  df-mulg 19217  df-subg 19272  df-cntz 19470  df-cmn 19935  df-abl 19936  df-mgp 20300  df-rng 20314  df-ur 20347  df-ring 20400  df-cring 20401  df-subrng 20737  df-subrg 20761  df-abv 21005  df-lmod 21076  df-scaf 21077  df-sra 21387  df-rgmod 21388  df-psmet 21609  df-xmet 21610  df-met 21611  df-bl 21612  df-mopn 21613  df-fbas 21614  df-fg 21615  df-cnfld 21618  df-top 23151  df-topon 23168  df-topsp 23190  df-bases 23203  df-cld 23276  df-ntr 23277  df-cls 23278  df-nei 23355  df-lp 23393  df-perf 23394  df-cn 23484  df-cnp 23485  df-t1 23571  df-haus 23572  df-tx 23820  df-hmeo 24013  df-fil 24104  df-fm 24196  df-flim 24197  df-flf 24198  df-tmd 24330  df-tgp 24331  df-tsms 24385  df-trg 24418  df-xms 24578  df-ms 24579  df-tms 24580  df-nm 24840  df-ngp 24841  df-nrg 24843  df-nlm 24844  df-ii 25137  df-cncf 25138  df-limc 26125  df-dv 26126  df-log 26825  df-esum 34571  df-siga 34652  df-sigagen 34683  df-meas 34740  df-mbfm 34794  df-sitg 34874

This theorem is used by:  sibfof  34884  sitgaddlemb  34892