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Theorem sibfinima 31618
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 31482 . . . . . . . 8 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
31, 2sylib 220 . . . . . . 7 (𝜑𝑀 ∈ (measures‘dom 𝑀))
433ad2ant1 1128 . . . . . 6 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑀 ∈ (measures‘dom 𝑀))
5 dmmeas 31481 . . . . . . . . 9 (𝑀 ran measures → dom 𝑀 ran sigAlgebra)
61, 5syl 17 . . . . . . . 8 (𝜑 → dom 𝑀 ran sigAlgebra)
763ad2ant1 1128 . . . . . . 7 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → dom 𝑀 ran sigAlgebra)
8 sitgval.s . . . . . . . . . 10 𝑆 = (sigaGen‘𝐽)
9 sibfinima.j . . . . . . . . . . 11 (𝜑𝐽 ∈ Fre)
109sgsiga 31422 . . . . . . . . . 10 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
118, 10eqeltrid 2916 . . . . . . . . 9 (𝜑𝑆 ran sigAlgebra)
12113ad2ant1 1128 . . . . . . . 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 31614 . . . . . . . . 9 (𝜑𝐹 ∈ (dom 𝑀MblFnM𝑆))
21203ad2ant1 1128 . . . . . . . 8 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
22 sibfinima.w . . . . . . . . . . . 12 (𝜑𝑊 ∈ TopSp)
2314tpstop 21540 . . . . . . . . . . . 12 (𝑊 ∈ TopSp → 𝐽 ∈ Top)
24 cldssbrsiga 31467 . . . . . . . . . . . 12 (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
2522, 23, 243syl 18 . . . . . . . . . . 11 (𝜑 → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
2625, 8sseqtrrdi 4011 . . . . . . . . . 10 (𝜑 → (Clsd‘𝐽) ⊆ 𝑆)
27263ad2ant1 1128 . . . . . . . . 9 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → (Clsd‘𝐽) ⊆ 𝑆)
2893ad2ant1 1128 . . . . . . . . . 10 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝐽 ∈ Fre)
2913, 14, 8, 15, 16, 17, 18, 1, 19sibff 31615 . . . . . . . . . . . . 13 (𝜑𝐹: dom 𝑀 𝐽)
3029frnd 6514 . . . . . . . . . . . 12 (𝜑 → ran 𝐹 𝐽)
31303ad2ant1 1128 . . . . . . . . . . 11 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → ran 𝐹 𝐽)
32 simp2 1132 . . . . . . . . . . 11 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑋 ∈ ran 𝐹)
3331, 32sseldd 3961 . . . . . . . . . 10 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑋 𝐽)
34 eqid 2820 . . . . . . . . . . 11 𝐽 = 𝐽
3534t1sncld 21929 . . . . . . . . . 10 ((𝐽 ∈ Fre ∧ 𝑋 𝐽) → {𝑋} ∈ (Clsd‘𝐽))
3628, 33, 35syl2anc 586 . . . . . . . . 9 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → {𝑋} ∈ (Clsd‘𝐽))
3727, 36sseldd 3961 . . . . . . . 8 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → {𝑋} ∈ 𝑆)
387, 12, 21, 37mbfmcnvima 31536 . . . . . . 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 31614 . . . . . . . . 9 (𝜑𝐺 ∈ (dom 𝑀MblFnM𝑆))
41403ad2ant1 1128 . . . . . . . 8 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
4213, 14, 8, 15, 16, 17, 18, 1, 39sibff 31615 . . . . . . . . . . . . 13 (𝜑𝐺: dom 𝑀 𝐽)
4342frnd 6514 . . . . . . . . . . . 12 (𝜑 → ran 𝐺 𝐽)
44433ad2ant1 1128 . . . . . . . . . . 11 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → ran 𝐺 𝐽)
45 simp3 1133 . . . . . . . . . . 11 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑌 ∈ ran 𝐺)
4644, 45sseldd 3961 . . . . . . . . . 10 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑌 𝐽)
4734t1sncld 21929 . . . . . . . . . 10 ((𝐽 ∈ Fre ∧ 𝑌 𝐽) → {𝑌} ∈ (Clsd‘𝐽))
4828, 46, 47syl2anc 586 . . . . . . . . 9 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → {𝑌} ∈ (Clsd‘𝐽))
4927, 48sseldd 3961 . . . . . . . 8 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → {𝑌} ∈ 𝑆)
507, 12, 41, 49mbfmcnvima 31536 . . . . . . 7 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → (𝐺 “ {𝑌}) ∈ dom 𝑀)
51 inelsiga 31415 . . . . . . 7 ((dom 𝑀 ran sigAlgebra ∧ (𝐹 “ {𝑋}) ∈ dom 𝑀 ∧ (𝐺 “ {𝑌}) ∈ dom 𝑀) → ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ∈ dom 𝑀)
527, 38, 50, 51syl3anc 1366 . . . . . 6 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ∈ dom 𝑀)
53 measvxrge0 31485 . . . . . 6 ((𝑀 ∈ (measures‘dom 𝑀) ∧ ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ∈ dom 𝑀) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,]+∞))
544, 52, 53syl2anc 586 . . . . 5 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,]+∞))
55 elxrge0 12839 . . . . . 6 ((𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,]+∞) ↔ ((𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ* ∧ 0 ≤ (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})))))
5655simplbi 500 . . . . 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 483 . . 3 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ*)
59 0re 10636 . . . 4 0 ∈ ℝ
6059a1i 11 . . 3 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → 0 ∈ ℝ)
6155simprbi 499 . . . . 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 483 . . 3 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → 0 ≤ (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))))
6457adantr 483 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ*)
654adantr 483 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → 𝑀 ∈ (measures‘dom 𝑀))
6638adantr 483 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝐹 “ {𝑋}) ∈ dom 𝑀)
67 measvxrge0 31485 . . . . . . 7 ((𝑀 ∈ (measures‘dom 𝑀) ∧ (𝐹 “ {𝑋}) ∈ dom 𝑀) → (𝑀‘(𝐹 “ {𝑋})) ∈ (0[,]+∞))
6865, 66, 67syl2anc 586 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘(𝐹 “ {𝑋})) ∈ (0[,]+∞))
69 elxrge0 12839 . . . . . . 7 ((𝑀‘(𝐹 “ {𝑋})) ∈ (0[,]+∞) ↔ ((𝑀‘(𝐹 “ {𝑋})) ∈ ℝ* ∧ 0 ≤ (𝑀‘(𝐹 “ {𝑋}))))
7069simplbi 500 . . . . . 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 10688 . . . . . 6 +∞ ∈ ℝ*
7372a1i 11 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → +∞ ∈ ℝ*)
7452adantr 483 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ∈ dom 𝑀)
75 inss1 4198 . . . . . . 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 31495 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ≤ (𝑀‘(𝐹 “ {𝑋})))
78 simpl1 1186 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → 𝜑)
7932anim1i 616 . . . . . . . 8 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑋 ∈ ran 𝐹𝑋0 ))
80 eldifsn 4712 . . . . . . . 8 (𝑋 ∈ (ran 𝐹 ∖ { 0 }) ↔ (𝑋 ∈ ran 𝐹𝑋0 ))
8179, 80sylibr 236 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → 𝑋 ∈ (ran 𝐹 ∖ { 0 }))
8213, 14, 8, 15, 16, 17, 18, 1, 19sibfima 31617 . . . . . . 7 ((𝜑𝑋 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(𝐹 “ {𝑋})) ∈ (0[,)+∞))
8378, 81, 82syl2anc 586 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘(𝐹 “ {𝑋})) ∈ (0[,)+∞))
84 elico2 12794 . . . . . . . 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 1142 . . . . . 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 12547 . . . 4 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) < +∞)
8957adantr 483 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ*)
904adantr 483 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → 𝑀 ∈ (measures‘dom 𝑀))
9150adantr 483 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝐺 “ {𝑌}) ∈ dom 𝑀)
92 measvxrge0 31485 . . . . . . 7 ((𝑀 ∈ (measures‘dom 𝑀) ∧ (𝐺 “ {𝑌}) ∈ dom 𝑀) → (𝑀‘(𝐺 “ {𝑌})) ∈ (0[,]+∞))
9390, 91, 92syl2anc 586 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘(𝐺 “ {𝑌})) ∈ (0[,]+∞))
94 elxrge0 12839 . . . . . . 7 ((𝑀‘(𝐺 “ {𝑌})) ∈ (0[,]+∞) ↔ ((𝑀‘(𝐺 “ {𝑌})) ∈ ℝ* ∧ 0 ≤ (𝑀‘(𝐺 “ {𝑌}))))
9594simplbi 500 . . . . . 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 483 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ∈ dom 𝑀)
99 inss2 4199 . . . . . . 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 31495 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ≤ (𝑀‘(𝐺 “ {𝑌})))
102 simpl1 1186 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → 𝜑)
10345anim1i 616 . . . . . . . 8 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑌 ∈ ran 𝐺𝑌0 ))
104 eldifsn 4712 . . . . . . . 8 (𝑌 ∈ (ran 𝐺 ∖ { 0 }) ↔ (𝑌 ∈ ran 𝐺𝑌0 ))
105103, 104sylibr 236 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → 𝑌 ∈ (ran 𝐺 ∖ { 0 }))
10613, 14, 8, 15, 16, 17, 18, 1, 39sibfima 31617 . . . . . . 7 ((𝜑𝑌 ∈ (ran 𝐺 ∖ { 0 })) → (𝑀‘(𝐺 “ {𝑌})) ∈ (0[,)+∞))
107102, 105, 106syl2anc 586 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘(𝐺 “ {𝑌})) ∈ (0[,)+∞))
108 elico2 12794 . . . . . . . 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 1142 . . . . . 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 12547 . . . 4 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) < +∞)
11388, 112jaodan 954 . . 3 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) < +∞)
114 xrre3 12558 . . 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 836 . 2 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ)
116 elico2 12794 . . 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 1338 1 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,)+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wo 843  w3a 1082   = wceq 1536  wcel 2113  wne 3015  cdif 3926  cin 3928  wss 3929  {csn 4560   cuni 4831   class class class wbr 5059  ccnv 5547  dom cdm 5548  ran crn 5549  cima 5551  cfv 6348  (class class class)co 7149  cr 10529  0cc0 10530  +∞cpnf 10665  *cxr 10667   < clt 10668  cle 10669  [,)cico 12734  [,]cicc 12735  Basecbs 16478  Scalarcsca 16563   ·𝑠 cvsca 16564  TopOpenctopn 16690  0gc0g 16708  Topctop 21496  TopSpctps 21535  Clsdccld 21619  Frect1 21910  ℝHomcrrh 31255  sigAlgebracsiga 31388  sigaGencsigagen 31418  measurescmeas 31475  MblFnMcmbfm 31529  sitgcsitg 31608
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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454  ax-inf2 9097  ax-ac2 9878  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608  ax-addf 10609  ax-mulf 10610
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-nel 3123  df-ral 3142  df-rex 3143  df-reu 3144  df-rmo 3145  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-pss 3947  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4870  df-iun 4914  df-iin 4915  df-disj 5025  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7402  df-om 7574  df-1st 7682  df-2nd 7683  df-supp 7824  df-wrecs 7940  df-recs 8001  df-rdg 8039  df-1o 8095  df-2o 8096  df-oadd 8099  df-er 8282  df-map 8401  df-pm 8402  df-ixp 8455  df-en 8503  df-dom 8504  df-sdom 8505  df-fin 8506  df-fsupp 8827  df-fi 8868  df-sup 8899  df-inf 8900  df-oi 8967  df-dju 9323  df-card 9361  df-acn 9364  df-ac 9535  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11632  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-q 12343  df-rp 12384  df-xneg 12501  df-xadd 12502  df-xmul 12503  df-ioo 12736  df-ioc 12737  df-ico 12738  df-icc 12739  df-fz 12890  df-fzo 13031  df-fl 13159  df-mod 13235  df-seq 13367  df-exp 13427  df-fac 13631  df-bc 13660  df-hash 13688  df-shft 14421  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-limsup 14823  df-clim 14840  df-rlim 14841  df-sum 15038  df-ef 15416  df-sin 15418  df-cos 15419  df-pi 15421  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-starv 16575  df-sca 16576  df-vsca 16577  df-ip 16578  df-tset 16579  df-ple 16580  df-ds 16582  df-unif 16583  df-hom 16584  df-cco 16585  df-rest 16691  df-topn 16692  df-0g 16710  df-gsum 16711  df-topgen 16712  df-pt 16713  df-prds 16716  df-ordt 16769  df-xrs 16770  df-qtop 16775  df-imas 16776  df-xps 16778  df-mre 16852  df-mrc 16853  df-acs 16855  df-ps 17805  df-tsr 17806  df-plusf 17846  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-mhm 17951  df-submnd 17952  df-grp 18101  df-minusg 18102  df-sbg 18103  df-mulg 18220  df-subg 18271  df-cntz 18442  df-cmn 18903  df-abl 18904  df-mgp 19235  df-ur 19247  df-ring 19294  df-cring 19295  df-subrg 19528  df-abv 19583  df-lmod 19631  df-scaf 19632  df-sra 19939  df-rgmod 19940  df-psmet 20532  df-xmet 20533  df-met 20534  df-bl 20535  df-mopn 20536  df-fbas 20537  df-fg 20538  df-cnfld 20541  df-top 21497  df-topon 21514  df-topsp 21536  df-bases 21549  df-cld 21622  df-ntr 21623  df-cls 21624  df-nei 21701  df-lp 21739  df-perf 21740  df-cn 21830  df-cnp 21831  df-t1 21917  df-haus 21918  df-tx 22165  df-hmeo 22358  df-fil 22449  df-fm 22541  df-flim 22542  df-flf 22543  df-tmd 22675  df-tgp 22676  df-tsms 22730  df-trg 22763  df-xms 22925  df-ms 22926  df-tms 22927  df-nm 23187  df-ngp 23188  df-nrg 23190  df-nlm 23191  df-ii 23480  df-cncf 23481  df-limc 24461  df-dv 24462  df-log 25138  df-esum 31308  df-siga 31389  df-sigagen 31419  df-meas 31476  df-mbfm 31530  df-sitg 31609
This theorem is referenced by:  sibfof  31619  sitgaddlemb  31627
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