Theorem sibfinima 35038

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 34902	. . . . . . . 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 34901	. . . . . . . . 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 34842	. . . . . . . . . 10 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
11	8, 10	eqeltrid 2868	. . . . . . . . 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 35034	. . . . . . . . 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 23282	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ TopSp → 𝐽 ∈ Top)
24		cldssbrsiga 34887	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
25	22, 23, 24	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
26	25, 8	sseqtrrdi 3954	. . . . . . . . . 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 35035	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝐽)
30	29	frnd 6737	. . . . . . . . . . . 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 3914	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑋 ∈ ∪ 𝐽)
34		eqid 2763	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
35	34	t1sncld 23671	. . . . . . . . . 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 3914	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → {𝑋} ∈ 𝑆)
38	7, 12, 21, 37	mbfmcnvima 34955	. . . . . . 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 35034	. . . . . . . . 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 35035	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:∪ dom 𝑀⟶∪ 𝐽)
43	42	frnd 6737	. . . . . . . . . . . 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 3914	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑌 ∈ ∪ 𝐽)
47	34	t1sncld 23671	. . . . . . . . . 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 3914	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → {𝑌} ∈ 𝑆)
50	7, 12, 41, 49	mbfmcnvima 34955	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (◡𝐺 “ {𝑌}) ∈ dom 𝑀)
51		inelsiga 34835	. . . . . . 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 1399	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ∈ dom 𝑀)
53		measvxrge0 34905	. . . . . 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 13612	. . . . . 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 11334	. . . 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 34905	. . . . . . 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 13612	. . . . . . 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 11387	. . . . . 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 4136	. . . . . . 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 34915	. . . . 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 35037	. . . . . . 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 13565	. . . . . . . 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 13313	. . . 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 34905	. . . . . . 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 13612	. . . . . . 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 4137	. . . . . . 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 34915	. . . . 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 35037	. . . . . . 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 13565	. . . . . . . 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 13313	. . . 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 13325	. . 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 13565	. . 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 2147   ≠ wne 2959   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092  {csn 4588  ∪ cuni 4870   class class class wbr 5106  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   “ cima 5658  ‘cfv 6555  (class class class)co 7440  ℝcr 11223  0cc0 11224  +∞cpnf 11364  ℝ^*cxr 11366   < clt 11367   ≤ cle 11368  [,)cico 13502  [,]cicc 13503  Basecbs 17413  Scalarcsca 17457   ·_𝑠 cvsca 17458  TopOpenctopn 17618  0_gc0g 17636  Topctop 23238  TopSpctps 23277  Clsdccld 23361  Frect1 23652  ℝHomcrrh 34692  sigAlgebracsiga 34807  sigaGencsigagen 34838  measurescmeas 34895  MblFnMcmbfm 34949  sitgcsitg 35028

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-inf2 9666  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301  ax-pre-sup 11302  ax-addf 11303  ax-mulf 11304

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-disj 5074  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  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 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-isom 6564  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-of 7714  df-om 7899  df-1st 8022  df-2nd 8023  df-supp 8202  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-er 8742  df-map 8874  df-pm 8875  df-ixp 8951  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-fsupp 9379  df-fi 9427  df-sup 9458  df-inf 9459  df-oi 9528  df-dju 10006  df-card 10044  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-div 11998  df-nn 12360  df-2 12429  df-3 12430  df-4 12431  df-5 12432  df-6 12433  df-7 12434  df-8 12435  df-9 12436  df-n0 12631  df-z 12718  df-dec 12839  df-uz 12990  df-q 13100  df-rp 13145  df-xneg 13265  df-xadd 13266  df-xmul 13267  df-ioo 13504  df-ioc 13505  df-ico 13506  df-icc 13507  df-fz 13664  df-fzo 13813  df-fl 13956  df-mod 14034  df-seq 14169  df-exp 14229  df-fac 14442  df-bc 14471  df-hash 14499  df-shft 15246  df-cj 15292  df-re 15293  df-im 15294  df-sqrt 15428  df-abs 15429  df-limsup 15664  df-clim 15681  df-rlim 15682  df-sum 15880  df-ef 16259  df-sin 16261  df-cos 16262  df-pi 16264  df-struct 17351  df-sets 17368  df-slot 17386  df-ndx 17398  df-base 17414  df-ress 17435  df-plusg 17467  df-mulr 17468  df-starv 17469  df-sca 17470  df-vsca 17471  df-ip 17472  df-tset 17473  df-ple 17474  df-ds 17476  df-unif 17477  df-hom 17478  df-cco 17479  df-rest 17619  df-topn 17620  df-0g 17638  df-gsum 17639  df-topgen 17640  df-pt 17641  df-prds 17644  df-ordt 17699  df-xrs 17700  df-qtop 17705  df-imas 17706  df-xps 17708  df-mre 17782  df-mrc 17783  df-acs 17785  df-ps 18766  df-tsr 18767  df-plusf 18841  df-mgm 18842  df-sgrp 18934  df-mnd 18950  df-mhm 19004  df-submnd 19005  df-grp 19173  df-minusg 19174  df-sbg 19175  df-mulg 19304  df-subg 19359  df-cntz 19558  df-cmn 20023  df-abl 20024  df-mgp 20388  df-rng 20402  df-ur 20435  df-ring 20488  df-cring 20489  df-subrng 20825  df-subrg 20849  df-abv 21093  df-lmod 21164  df-scaf 21165  df-sra 21475  df-rgmod 21476  df-psmet 21697  df-xmet 21698  df-met 21699  df-bl 21700  df-mopn 21701  df-fbas 21702  df-fg 21703  df-cnfld 21706  df-top 23239  df-topon 23256  df-topsp 23278  df-bases 23291  df-cld 23364  df-ntr 23365  df-cls 23366  df-nei 23443  df-lp 23481  df-perf 23482  df-cn 23572  df-cnp 23573  df-t1 23659  df-haus 23660  df-tx 23908  df-hmeo 24101  df-fil 24192  df-fm 24284  df-flim 24285  df-flf 24286  df-tmd 24418  df-tgp 24419  df-tsms 24473  df-trg 24506  df-xms 24666  df-ms 24667  df-tms 24668  df-nm 24928  df-ngp 24929  df-nrg 24931  df-nlm 24932  df-ii 25225  df-cncf 25226  df-limc 26213  df-dv 26214  df-log 26911  df-esum 34727  df-siga 34808  df-sigagen 34839  df-meas 34896  df-mbfm 34950  df-sitg 35029

This theorem is used by:  sibfof  35039  sitgaddlemb  35047