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Theorem sibfinima 32939
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 = (0g𝑊)
sitgval.x · = ( ·𝑠𝑊)
sitgval.h 𝐻 = (ℝHom‘(Scalar‘𝑊))
sitgval.1 (𝜑𝑊𝑉)
sitgval.2 (𝜑𝑀 ran measures)
sibfmbl.1 (𝜑𝐹 ∈ dom (𝑊sitg𝑀))
sibfinima.g (𝜑𝐺 ∈ dom (𝑊sitg𝑀))
sibfinima.w (𝜑𝑊 ∈ TopSp)
sibfinima.j (𝜑𝐽 ∈ Fre)
Assertion
Ref Expression
sibfinima (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,)+∞))

Proof of Theorem sibfinima
StepHypRef Expression
1 sitgval.2 . . . . . . . 8 (𝜑𝑀 ran measures)
2 measbasedom 32801 . . . . . . . 8 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
31, 2sylib 217 . . . . . . 7 (𝜑𝑀 ∈ (measures‘dom 𝑀))
433ad2ant1 1133 . . . . . 6 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑀 ∈ (measures‘dom 𝑀))
5 dmmeas 32800 . . . . . . . . 9 (𝑀 ran measures → dom 𝑀 ran sigAlgebra)
61, 5syl 17 . . . . . . . 8 (𝜑 → dom 𝑀 ran sigAlgebra)
763ad2ant1 1133 . . . . . . 7 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → dom 𝑀 ran sigAlgebra)
8 sitgval.s . . . . . . . . . 10 𝑆 = (sigaGen‘𝐽)
9 sibfinima.j . . . . . . . . . . 11 (𝜑𝐽 ∈ Fre)
109sgsiga 32741 . . . . . . . . . 10 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
118, 10eqeltrid 2842 . . . . . . . . 9 (𝜑𝑆 ran sigAlgebra)
12113ad2ant1 1133 . . . . . . . 8 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑆 ran sigAlgebra)
13 sitgval.b . . . . . . . . . 10 𝐵 = (Base‘𝑊)
14 sitgval.j . . . . . . . . . 10 𝐽 = (TopOpen‘𝑊)
15 sitgval.0 . . . . . . . . . 10 0 = (0g𝑊)
16 sitgval.x . . . . . . . . . 10 · = ( ·𝑠𝑊)
17 sitgval.h . . . . . . . . . 10 𝐻 = (ℝHom‘(Scalar‘𝑊))
18 sitgval.1 . . . . . . . . . 10 (𝜑𝑊𝑉)
19 sibfmbl.1 . . . . . . . . . 10 (𝜑𝐹 ∈ dom (𝑊sitg𝑀))
2013, 14, 8, 15, 16, 17, 18, 1, 19sibfmbl 32935 . . . . . . . . 9 (𝜑𝐹 ∈ (dom 𝑀MblFnM𝑆))
21203ad2ant1 1133 . . . . . . . 8 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
22 sibfinima.w . . . . . . . . . . . 12 (𝜑𝑊 ∈ TopSp)
2314tpstop 22286 . . . . . . . . . . . 12 (𝑊 ∈ TopSp → 𝐽 ∈ Top)
24 cldssbrsiga 32786 . . . . . . . . . . . 12 (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
2522, 23, 243syl 18 . . . . . . . . . . 11 (𝜑 → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
2625, 8sseqtrrdi 3995 . . . . . . . . . 10 (𝜑 → (Clsd‘𝐽) ⊆ 𝑆)
27263ad2ant1 1133 . . . . . . . . 9 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → (Clsd‘𝐽) ⊆ 𝑆)
2893ad2ant1 1133 . . . . . . . . . 10 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝐽 ∈ Fre)
2913, 14, 8, 15, 16, 17, 18, 1, 19sibff 32936 . . . . . . . . . . . . 13 (𝜑𝐹: dom 𝑀 𝐽)
3029frnd 6676 . . . . . . . . . . . 12 (𝜑 → ran 𝐹 𝐽)
31303ad2ant1 1133 . . . . . . . . . . 11 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → ran 𝐹 𝐽)
32 simp2 1137 . . . . . . . . . . 11 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑋 ∈ ran 𝐹)
3331, 32sseldd 3945 . . . . . . . . . 10 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑋 𝐽)
34 eqid 2736 . . . . . . . . . . 11 𝐽 = 𝐽
3534t1sncld 22677 . . . . . . . . . 10 ((𝐽 ∈ Fre ∧ 𝑋 𝐽) → {𝑋} ∈ (Clsd‘𝐽))
3628, 33, 35syl2anc 584 . . . . . . . . 9 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → {𝑋} ∈ (Clsd‘𝐽))
3727, 36sseldd 3945 . . . . . . . 8 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → {𝑋} ∈ 𝑆)
387, 12, 21, 37mbfmcnvima 32855 . . . . . . 7 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → (𝐹 “ {𝑋}) ∈ dom 𝑀)
39 sibfinima.g . . . . . . . . . 10 (𝜑𝐺 ∈ dom (𝑊sitg𝑀))
4013, 14, 8, 15, 16, 17, 18, 1, 39sibfmbl 32935 . . . . . . . . 9 (𝜑𝐺 ∈ (dom 𝑀MblFnM𝑆))
41403ad2ant1 1133 . . . . . . . 8 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
4213, 14, 8, 15, 16, 17, 18, 1, 39sibff 32936 . . . . . . . . . . . . 13 (𝜑𝐺: dom 𝑀 𝐽)
4342frnd 6676 . . . . . . . . . . . 12 (𝜑 → ran 𝐺 𝐽)
44433ad2ant1 1133 . . . . . . . . . . 11 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → ran 𝐺 𝐽)
45 simp3 1138 . . . . . . . . . . 11 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑌 ∈ ran 𝐺)
4644, 45sseldd 3945 . . . . . . . . . 10 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑌 𝐽)
4734t1sncld 22677 . . . . . . . . . 10 ((𝐽 ∈ Fre ∧ 𝑌 𝐽) → {𝑌} ∈ (Clsd‘𝐽))
4828, 46, 47syl2anc 584 . . . . . . . . 9 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → {𝑌} ∈ (Clsd‘𝐽))
4927, 48sseldd 3945 . . . . . . . 8 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → {𝑌} ∈ 𝑆)
507, 12, 41, 49mbfmcnvima 32855 . . . . . . 7 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → (𝐺 “ {𝑌}) ∈ dom 𝑀)
51 inelsiga 32734 . . . . . . 7 ((dom 𝑀 ran sigAlgebra ∧ (𝐹 “ {𝑋}) ∈ dom 𝑀 ∧ (𝐺 “ {𝑌}) ∈ dom 𝑀) → ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ∈ dom 𝑀)
527, 38, 50, 51syl3anc 1371 . . . . . 6 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ∈ dom 𝑀)
53 measvxrge0 32804 . . . . . 6 ((𝑀 ∈ (measures‘dom 𝑀) ∧ ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ∈ dom 𝑀) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,]+∞))
544, 52, 53syl2anc 584 . . . . 5 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,]+∞))
55 elxrge0 13374 . . . . . 6 ((𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,]+∞) ↔ ((𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ* ∧ 0 ≤ (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})))))
5655simplbi 498 . . . . 5 ((𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,]+∞) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ*)
5754, 56syl 17 . . . 4 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ*)
5857adantr 481 . . 3 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ*)
59 0re 11157 . . . 4 0 ∈ ℝ
6059a1i 11 . . 3 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → 0 ∈ ℝ)
6155simprbi 497 . . . . 5 ((𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,]+∞) → 0 ≤ (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))))
6254, 61syl 17 . . . 4 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 0 ≤ (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))))
6362adantr 481 . . 3 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → 0 ≤ (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))))
6457adantr 481 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ*)
654adantr 481 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → 𝑀 ∈ (measures‘dom 𝑀))
6638adantr 481 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝐹 “ {𝑋}) ∈ dom 𝑀)
67 measvxrge0 32804 . . . . . . 7 ((𝑀 ∈ (measures‘dom 𝑀) ∧ (𝐹 “ {𝑋}) ∈ dom 𝑀) → (𝑀‘(𝐹 “ {𝑋})) ∈ (0[,]+∞))
6865, 66, 67syl2anc 584 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘(𝐹 “ {𝑋})) ∈ (0[,]+∞))
69 elxrge0 13374 . . . . . . 7 ((𝑀‘(𝐹 “ {𝑋})) ∈ (0[,]+∞) ↔ ((𝑀‘(𝐹 “ {𝑋})) ∈ ℝ* ∧ 0 ≤ (𝑀‘(𝐹 “ {𝑋}))))
7069simplbi 498 . . . . . 6 ((𝑀‘(𝐹 “ {𝑋})) ∈ (0[,]+∞) → (𝑀‘(𝐹 “ {𝑋})) ∈ ℝ*)
7168, 70syl 17 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘(𝐹 “ {𝑋})) ∈ ℝ*)
72 pnfxr 11209 . . . . . 6 +∞ ∈ ℝ*
7372a1i 11 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → +∞ ∈ ℝ*)
7452adantr 481 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ∈ dom 𝑀)
75 inss1 4188 . . . . . . 7 ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ⊆ (𝐹 “ {𝑋})
7675a1i 11 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ⊆ (𝐹 “ {𝑋}))
7765, 74, 66, 76measssd 32814 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ≤ (𝑀‘(𝐹 “ {𝑋})))
78 simpl1 1191 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → 𝜑)
7932anim1i 615 . . . . . . . 8 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑋 ∈ ran 𝐹𝑋0 ))
80 eldifsn 4747 . . . . . . . 8 (𝑋 ∈ (ran 𝐹 ∖ { 0 }) ↔ (𝑋 ∈ ran 𝐹𝑋0 ))
8179, 80sylibr 233 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → 𝑋 ∈ (ran 𝐹 ∖ { 0 }))
8213, 14, 8, 15, 16, 17, 18, 1, 19sibfima 32938 . . . . . . 7 ((𝜑𝑋 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(𝐹 “ {𝑋})) ∈ (0[,)+∞))
8378, 81, 82syl2anc 584 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘(𝐹 “ {𝑋})) ∈ (0[,)+∞))
84 elico2 13328 . . . . . . . 8 ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ((𝑀‘(𝐹 “ {𝑋})) ∈ (0[,)+∞) ↔ ((𝑀‘(𝐹 “ {𝑋})) ∈ ℝ ∧ 0 ≤ (𝑀‘(𝐹 “ {𝑋})) ∧ (𝑀‘(𝐹 “ {𝑋})) < +∞)))
8559, 72, 84mp2an 690 . . . . . . 7 ((𝑀‘(𝐹 “ {𝑋})) ∈ (0[,)+∞) ↔ ((𝑀‘(𝐹 “ {𝑋})) ∈ ℝ ∧ 0 ≤ (𝑀‘(𝐹 “ {𝑋})) ∧ (𝑀‘(𝐹 “ {𝑋})) < +∞))
8685simp3bi 1147 . . . . . 6 ((𝑀‘(𝐹 “ {𝑋})) ∈ (0[,)+∞) → (𝑀‘(𝐹 “ {𝑋})) < +∞)
8783, 86syl 17 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘(𝐹 “ {𝑋})) < +∞)
8864, 71, 73, 77, 87xrlelttrd 13079 . . . 4 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) < +∞)
8957adantr 481 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ*)
904adantr 481 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → 𝑀 ∈ (measures‘dom 𝑀))
9150adantr 481 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝐺 “ {𝑌}) ∈ dom 𝑀)
92 measvxrge0 32804 . . . . . . 7 ((𝑀 ∈ (measures‘dom 𝑀) ∧ (𝐺 “ {𝑌}) ∈ dom 𝑀) → (𝑀‘(𝐺 “ {𝑌})) ∈ (0[,]+∞))
9390, 91, 92syl2anc 584 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘(𝐺 “ {𝑌})) ∈ (0[,]+∞))
94 elxrge0 13374 . . . . . . 7 ((𝑀‘(𝐺 “ {𝑌})) ∈ (0[,]+∞) ↔ ((𝑀‘(𝐺 “ {𝑌})) ∈ ℝ* ∧ 0 ≤ (𝑀‘(𝐺 “ {𝑌}))))
9594simplbi 498 . . . . . 6 ((𝑀‘(𝐺 “ {𝑌})) ∈ (0[,]+∞) → (𝑀‘(𝐺 “ {𝑌})) ∈ ℝ*)
9693, 95syl 17 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘(𝐺 “ {𝑌})) ∈ ℝ*)
9772a1i 11 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → +∞ ∈ ℝ*)
9852adantr 481 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ∈ dom 𝑀)
99 inss2 4189 . . . . . . 7 ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ⊆ (𝐺 “ {𝑌})
10099a1i 11 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ⊆ (𝐺 “ {𝑌}))
10190, 98, 91, 100measssd 32814 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ≤ (𝑀‘(𝐺 “ {𝑌})))
102 simpl1 1191 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → 𝜑)
10345anim1i 615 . . . . . . . 8 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑌 ∈ ran 𝐺𝑌0 ))
104 eldifsn 4747 . . . . . . . 8 (𝑌 ∈ (ran 𝐺 ∖ { 0 }) ↔ (𝑌 ∈ ran 𝐺𝑌0 ))
105103, 104sylibr 233 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → 𝑌 ∈ (ran 𝐺 ∖ { 0 }))
10613, 14, 8, 15, 16, 17, 18, 1, 39sibfima 32938 . . . . . . 7 ((𝜑𝑌 ∈ (ran 𝐺 ∖ { 0 })) → (𝑀‘(𝐺 “ {𝑌})) ∈ (0[,)+∞))
107102, 105, 106syl2anc 584 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘(𝐺 “ {𝑌})) ∈ (0[,)+∞))
108 elico2 13328 . . . . . . . 8 ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ((𝑀‘(𝐺 “ {𝑌})) ∈ (0[,)+∞) ↔ ((𝑀‘(𝐺 “ {𝑌})) ∈ ℝ ∧ 0 ≤ (𝑀‘(𝐺 “ {𝑌})) ∧ (𝑀‘(𝐺 “ {𝑌})) < +∞)))
10959, 72, 108mp2an 690 . . . . . . 7 ((𝑀‘(𝐺 “ {𝑌})) ∈ (0[,)+∞) ↔ ((𝑀‘(𝐺 “ {𝑌})) ∈ ℝ ∧ 0 ≤ (𝑀‘(𝐺 “ {𝑌})) ∧ (𝑀‘(𝐺 “ {𝑌})) < +∞))
110109simp3bi 1147 . . . . . 6 ((𝑀‘(𝐺 “ {𝑌})) ∈ (0[,)+∞) → (𝑀‘(𝐺 “ {𝑌})) < +∞)
111107, 110syl 17 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘(𝐺 “ {𝑌})) < +∞)
11289, 96, 97, 101, 111xrlelttrd 13079 . . . 4 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) < +∞)
11388, 112jaodan 956 . . 3 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) < +∞)
114 xrre3 13090 . . 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)
11558, 60, 63, 113, 114syl22anc 837 . 2 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ)
116 elico2 13328 . . 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}))) < +∞)))
11759, 72, 116mp2an 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}))) < +∞))
118115, 63, 113, 117syl3anbrc 1343 1 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → (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 2943  cdif 3907  cin 3909  wss 3910  {csn 4586   cuni 4865   class class class wbr 5105  ccnv 5632  dom cdm 5633  ran crn 5634  cima 5636  cfv 6496  (class class class)co 7357  cr 11050  0cc0 11051  +∞cpnf 11186  *cxr 11188   < clt 11189  cle 11190  [,)cico 13266  [,]cicc 13267  Basecbs 17083  Scalarcsca 17136   ·𝑠 cvsca 17137  TopOpenctopn 17303  0gc0g 17321  Topctop 22242  TopSpctps 22281  Clsdccld 22367  Frect1 22658  ℝHomcrrh 32574  sigAlgebracsiga 32707  sigaGencsigagen 32737  measurescmeas 32794  MblFnMcmbfm 32848  sitgcsitg 32929
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-ac2 10399  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-acn 9878  df-ac 10052  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-shft 14952  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-sum 15571  df-ef 15950  df-sin 15952  df-cos 15953  df-pi 15955  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-ordt 17383  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-ps 18455  df-tsr 18456  df-plusf 18496  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-subrg 20220  df-abv 20276  df-lmod 20324  df-scaf 20325  df-sra 20633  df-rgmod 20634  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lp 22487  df-perf 22488  df-cn 22578  df-cnp 22579  df-t1 22665  df-haus 22666  df-tx 22913  df-hmeo 23106  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-tmd 23423  df-tgp 23424  df-tsms 23478  df-trg 23511  df-xms 23673  df-ms 23674  df-tms 23675  df-nm 23938  df-ngp 23939  df-nrg 23941  df-nlm 23942  df-ii 24240  df-cncf 24241  df-limc 25230  df-dv 25231  df-log 25912  df-esum 32627  df-siga 32708  df-sigagen 32738  df-meas 32795  df-mbfm 32849  df-sitg 32930
This theorem is referenced by:  sibfof  32940  sitgaddlemb  32948
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