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Theorem sibfinima 34321
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 34183 . . . . . . . 8 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
31, 2sylib 218 . . . . . . 7 (𝜑𝑀 ∈ (measures‘dom 𝑀))
433ad2ant1 1132 . . . . . 6 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑀 ∈ (measures‘dom 𝑀))
5 dmmeas 34182 . . . . . . . . 9 (𝑀 ran measures → dom 𝑀 ran sigAlgebra)
61, 5syl 17 . . . . . . . 8 (𝜑 → dom 𝑀 ran sigAlgebra)
763ad2ant1 1132 . . . . . . 7 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → dom 𝑀 ran sigAlgebra)
8 sitgval.s . . . . . . . . . 10 𝑆 = (sigaGen‘𝐽)
9 sibfinima.j . . . . . . . . . . 11 (𝜑𝐽 ∈ Fre)
109sgsiga 34123 . . . . . . . . . 10 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
118, 10eqeltrid 2843 . . . . . . . . 9 (𝜑𝑆 ran sigAlgebra)
12113ad2ant1 1132 . . . . . . . 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 34317 . . . . . . . . 9 (𝜑𝐹 ∈ (dom 𝑀MblFnM𝑆))
21203ad2ant1 1132 . . . . . . . 8 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
22 sibfinima.w . . . . . . . . . . . 12 (𝜑𝑊 ∈ TopSp)
2314tpstop 22959 . . . . . . . . . . . 12 (𝑊 ∈ TopSp → 𝐽 ∈ Top)
24 cldssbrsiga 34168 . . . . . . . . . . . 12 (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
2522, 23, 243syl 18 . . . . . . . . . . 11 (𝜑 → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
2625, 8sseqtrrdi 4047 . . . . . . . . . 10 (𝜑 → (Clsd‘𝐽) ⊆ 𝑆)
27263ad2ant1 1132 . . . . . . . . 9 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → (Clsd‘𝐽) ⊆ 𝑆)
2893ad2ant1 1132 . . . . . . . . . 10 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝐽 ∈ Fre)
2913, 14, 8, 15, 16, 17, 18, 1, 19sibff 34318 . . . . . . . . . . . . 13 (𝜑𝐹: dom 𝑀 𝐽)
3029frnd 6745 . . . . . . . . . . . 12 (𝜑 → ran 𝐹 𝐽)
31303ad2ant1 1132 . . . . . . . . . . 11 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → ran 𝐹 𝐽)
32 simp2 1136 . . . . . . . . . . 11 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑋 ∈ ran 𝐹)
3331, 32sseldd 3996 . . . . . . . . . 10 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑋 𝐽)
34 eqid 2735 . . . . . . . . . . 11 𝐽 = 𝐽
3534t1sncld 23350 . . . . . . . . . 10 ((𝐽 ∈ Fre ∧ 𝑋 𝐽) → {𝑋} ∈ (Clsd‘𝐽))
3628, 33, 35syl2anc 584 . . . . . . . . 9 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → {𝑋} ∈ (Clsd‘𝐽))
3727, 36sseldd 3996 . . . . . . . 8 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → {𝑋} ∈ 𝑆)
387, 12, 21, 37mbfmcnvima 34237 . . . . . . 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 34317 . . . . . . . . 9 (𝜑𝐺 ∈ (dom 𝑀MblFnM𝑆))
41403ad2ant1 1132 . . . . . . . 8 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
4213, 14, 8, 15, 16, 17, 18, 1, 39sibff 34318 . . . . . . . . . . . . 13 (𝜑𝐺: dom 𝑀 𝐽)
4342frnd 6745 . . . . . . . . . . . 12 (𝜑 → ran 𝐺 𝐽)
44433ad2ant1 1132 . . . . . . . . . . 11 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → ran 𝐺 𝐽)
45 simp3 1137 . . . . . . . . . . 11 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑌 ∈ ran 𝐺)
4644, 45sseldd 3996 . . . . . . . . . 10 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑌 𝐽)
4734t1sncld 23350 . . . . . . . . . 10 ((𝐽 ∈ Fre ∧ 𝑌 𝐽) → {𝑌} ∈ (Clsd‘𝐽))
4828, 46, 47syl2anc 584 . . . . . . . . 9 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → {𝑌} ∈ (Clsd‘𝐽))
4927, 48sseldd 3996 . . . . . . . 8 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → {𝑌} ∈ 𝑆)
507, 12, 41, 49mbfmcnvima 34237 . . . . . . 7 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → (𝐺 “ {𝑌}) ∈ dom 𝑀)
51 inelsiga 34116 . . . . . . 7 ((dom 𝑀 ran sigAlgebra ∧ (𝐹 “ {𝑋}) ∈ dom 𝑀 ∧ (𝐺 “ {𝑌}) ∈ dom 𝑀) → ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ∈ dom 𝑀)
527, 38, 50, 51syl3anc 1370 . . . . . 6 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ∈ dom 𝑀)
53 measvxrge0 34186 . . . . . 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 13494 . . . . . 6 ((𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,]+∞) ↔ ((𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ* ∧ 0 ≤ (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})))))
5655simplbi 497 . . . . 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 480 . . 3 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ*)
59 0re 11261 . . . 4 0 ∈ ℝ
6059a1i 11 . . 3 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → 0 ∈ ℝ)
6155simprbi 496 . . . . 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 480 . . 3 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → 0 ≤ (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))))
6457adantr 480 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ*)
654adantr 480 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → 𝑀 ∈ (measures‘dom 𝑀))
6638adantr 480 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝐹 “ {𝑋}) ∈ dom 𝑀)
67 measvxrge0 34186 . . . . . . 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 13494 . . . . . . 7 ((𝑀‘(𝐹 “ {𝑋})) ∈ (0[,]+∞) ↔ ((𝑀‘(𝐹 “ {𝑋})) ∈ ℝ* ∧ 0 ≤ (𝑀‘(𝐹 “ {𝑋}))))
7069simplbi 497 . . . . . 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 11313 . . . . . 6 +∞ ∈ ℝ*
7372a1i 11 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → +∞ ∈ ℝ*)
7452adantr 480 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ∈ dom 𝑀)
75 inss1 4245 . . . . . . 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 34196 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ≤ (𝑀‘(𝐹 “ {𝑋})))
78 simpl1 1190 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → 𝜑)
7932anim1i 615 . . . . . . . 8 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑋 ∈ ran 𝐹𝑋0 ))
80 eldifsn 4791 . . . . . . . 8 (𝑋 ∈ (ran 𝐹 ∖ { 0 }) ↔ (𝑋 ∈ ran 𝐹𝑋0 ))
8179, 80sylibr 234 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → 𝑋 ∈ (ran 𝐹 ∖ { 0 }))
8213, 14, 8, 15, 16, 17, 18, 1, 19sibfima 34320 . . . . . . 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 13448 . . . . . . . 8 ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ((𝑀‘(𝐹 “ {𝑋})) ∈ (0[,)+∞) ↔ ((𝑀‘(𝐹 “ {𝑋})) ∈ ℝ ∧ 0 ≤ (𝑀‘(𝐹 “ {𝑋})) ∧ (𝑀‘(𝐹 “ {𝑋})) < +∞)))
8559, 72, 84mp2an 692 . . . . . . 7 ((𝑀‘(𝐹 “ {𝑋})) ∈ (0[,)+∞) ↔ ((𝑀‘(𝐹 “ {𝑋})) ∈ ℝ ∧ 0 ≤ (𝑀‘(𝐹 “ {𝑋})) ∧ (𝑀‘(𝐹 “ {𝑋})) < +∞))
8685simp3bi 1146 . . . . . 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 13199 . . . 4 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) < +∞)
8957adantr 480 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ*)
904adantr 480 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → 𝑀 ∈ (measures‘dom 𝑀))
9150adantr 480 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝐺 “ {𝑌}) ∈ dom 𝑀)
92 measvxrge0 34186 . . . . . . 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 13494 . . . . . . 7 ((𝑀‘(𝐺 “ {𝑌})) ∈ (0[,]+∞) ↔ ((𝑀‘(𝐺 “ {𝑌})) ∈ ℝ* ∧ 0 ≤ (𝑀‘(𝐺 “ {𝑌}))))
9594simplbi 497 . . . . . 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 480 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ∈ dom 𝑀)
99 inss2 4246 . . . . . . 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 34196 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ≤ (𝑀‘(𝐺 “ {𝑌})))
102 simpl1 1190 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → 𝜑)
10345anim1i 615 . . . . . . . 8 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑌 ∈ ran 𝐺𝑌0 ))
104 eldifsn 4791 . . . . . . . 8 (𝑌 ∈ (ran 𝐺 ∖ { 0 }) ↔ (𝑌 ∈ ran 𝐺𝑌0 ))
105103, 104sylibr 234 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → 𝑌 ∈ (ran 𝐺 ∖ { 0 }))
10613, 14, 8, 15, 16, 17, 18, 1, 39sibfima 34320 . . . . . . 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 13448 . . . . . . . 8 ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ((𝑀‘(𝐺 “ {𝑌})) ∈ (0[,)+∞) ↔ ((𝑀‘(𝐺 “ {𝑌})) ∈ ℝ ∧ 0 ≤ (𝑀‘(𝐺 “ {𝑌})) ∧ (𝑀‘(𝐺 “ {𝑌})) < +∞)))
10959, 72, 108mp2an 692 . . . . . . 7 ((𝑀‘(𝐺 “ {𝑌})) ∈ (0[,)+∞) ↔ ((𝑀‘(𝐺 “ {𝑌})) ∈ ℝ ∧ 0 ≤ (𝑀‘(𝐺 “ {𝑌})) ∧ (𝑀‘(𝐺 “ {𝑌})) < +∞))
110109simp3bi 1146 . . . . . 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 13199 . . . 4 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) < +∞)
11388, 112jaodan 959 . . 3 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) < +∞)
114 xrre3 13210 . . 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 839 . 2 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ)
116 elico2 13448 . . 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 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}))) < +∞))
118115, 63, 113, 117syl3anbrc 1342 1 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,)+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  cdif 3960  cin 3962  wss 3963  {csn 4631   cuni 4912   class class class wbr 5148  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692  cfv 6563  (class class class)co 7431  cr 11152  0cc0 11153  +∞cpnf 11290  *cxr 11292   < clt 11293  cle 11294  [,)cico 13386  [,]cicc 13387  Basecbs 17245  Scalarcsca 17301   ·𝑠 cvsca 17302  TopOpenctopn 17468  0gc0g 17486  Topctop 22915  TopSpctps 22954  Clsdccld 23040  Frect1 23331  ℝHomcrrh 33956  sigAlgebracsiga 34089  sigaGencsigagen 34119  measurescmeas 34176  MblFnMcmbfm 34230  sitgcsitg 34311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-ac2 10501  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232  ax-mulf 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-disj 5116  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-dju 9939  df-card 9977  df-acn 9980  df-ac 10154  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ioo 13388  df-ioc 13389  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-fl 13829  df-mod 13907  df-seq 14040  df-exp 14100  df-fac 14310  df-bc 14339  df-hash 14367  df-shft 15103  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-limsup 15504  df-clim 15521  df-rlim 15522  df-sum 15720  df-ef 16100  df-sin 16102  df-cos 16103  df-pi 16105  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-0g 17488  df-gsum 17489  df-topgen 17490  df-pt 17491  df-prds 17494  df-ordt 17548  df-xrs 17549  df-qtop 17554  df-imas 17555  df-xps 17557  df-mre 17631  df-mrc 17632  df-acs 17634  df-ps 18624  df-tsr 18625  df-plusf 18665  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-cntz 19348  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-cring 20254  df-subrng 20563  df-subrg 20587  df-abv 20827  df-lmod 20877  df-scaf 20878  df-sra 21190  df-rgmod 21191  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-fbas 21379  df-fg 21380  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-lp 23160  df-perf 23161  df-cn 23251  df-cnp 23252  df-t1 23338  df-haus 23339  df-tx 23586  df-hmeo 23779  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964  df-tmd 24096  df-tgp 24097  df-tsms 24151  df-trg 24184  df-xms 24346  df-ms 24347  df-tms 24348  df-nm 24611  df-ngp 24612  df-nrg 24614  df-nlm 24615  df-ii 24917  df-cncf 24918  df-limc 25916  df-dv 25917  df-log 26613  df-esum 34009  df-siga 34090  df-sigagen 34120  df-meas 34177  df-mbfm 34231  df-sitg 34312
This theorem is referenced by:  sibfof  34322  sitgaddlemb  34330
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