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Theorem sibfinima 29728
 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 29592 . . . . . . . 8 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
31, 2sylib 207 . . . . . . 7 (𝜑𝑀 ∈ (measures‘dom 𝑀))
433ad2ant1 1075 . . . . . 6 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑀 ∈ (measures‘dom 𝑀))
5 dmmeas 29591 . . . . . . . . 9 (𝑀 ran measures → dom 𝑀 ran sigAlgebra)
61, 5syl 17 . . . . . . . 8 (𝜑 → dom 𝑀 ran sigAlgebra)
763ad2ant1 1075 . . . . . . 7 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → dom 𝑀 ran sigAlgebra)
8 sitgval.s . . . . . . . . . 10 𝑆 = (sigaGen‘𝐽)
9 sibfinima.j . . . . . . . . . . 11 (𝜑𝐽 ∈ Fre)
109sgsiga 29532 . . . . . . . . . 10 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
118, 10syl5eqel 2692 . . . . . . . . 9 (𝜑𝑆 ran sigAlgebra)
12113ad2ant1 1075 . . . . . . . 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 29724 . . . . . . . . 9 (𝜑𝐹 ∈ (dom 𝑀MblFnM𝑆))
21203ad2ant1 1075 . . . . . . . 8 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
22 sibfinima.w . . . . . . . . . . . 12 (𝜑𝑊 ∈ TopSp)
2314tpstop 20554 . . . . . . . . . . . 12 (𝑊 ∈ TopSp → 𝐽 ∈ Top)
24 cldssbrsiga 29577 . . . . . . . . . . . 12 (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
2522, 23, 243syl 18 . . . . . . . . . . 11 (𝜑 → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
2625, 8syl6sseqr 3615 . . . . . . . . . 10 (𝜑 → (Clsd‘𝐽) ⊆ 𝑆)
27263ad2ant1 1075 . . . . . . . . 9 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → (Clsd‘𝐽) ⊆ 𝑆)
2893ad2ant1 1075 . . . . . . . . . 10 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝐽 ∈ Fre)
2913, 14, 8, 15, 16, 17, 18, 1, 19sibff 29725 . . . . . . . . . . . . 13 (𝜑𝐹: dom 𝑀 𝐽)
30 frn 5966 . . . . . . . . . . . . 13 (𝐹: dom 𝑀 𝐽 → ran 𝐹 𝐽)
3129, 30syl 17 . . . . . . . . . . . 12 (𝜑 → ran 𝐹 𝐽)
32313ad2ant1 1075 . . . . . . . . . . 11 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → ran 𝐹 𝐽)
33 simp2 1055 . . . . . . . . . . 11 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑋 ∈ ran 𝐹)
3432, 33sseldd 3569 . . . . . . . . . 10 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑋 𝐽)
35 eqid 2610 . . . . . . . . . . 11 𝐽 = 𝐽
3635t1sncld 20940 . . . . . . . . . 10 ((𝐽 ∈ Fre ∧ 𝑋 𝐽) → {𝑋} ∈ (Clsd‘𝐽))
3728, 34, 36syl2anc 691 . . . . . . . . 9 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → {𝑋} ∈ (Clsd‘𝐽))
3827, 37sseldd 3569 . . . . . . . 8 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → {𝑋} ∈ 𝑆)
397, 12, 21, 38mbfmcnvima 29646 . . . . . . 7 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → (𝐹 “ {𝑋}) ∈ dom 𝑀)
40 sibfinima.g . . . . . . . . . 10 (𝜑𝐺 ∈ dom (𝑊sitg𝑀))
4113, 14, 8, 15, 16, 17, 18, 1, 40sibfmbl 29724 . . . . . . . . 9 (𝜑𝐺 ∈ (dom 𝑀MblFnM𝑆))
42413ad2ant1 1075 . . . . . . . 8 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
4313, 14, 8, 15, 16, 17, 18, 1, 40sibff 29725 . . . . . . . . . . . . 13 (𝜑𝐺: dom 𝑀 𝐽)
44 frn 5966 . . . . . . . . . . . . 13 (𝐺: dom 𝑀 𝐽 → ran 𝐺 𝐽)
4543, 44syl 17 . . . . . . . . . . . 12 (𝜑 → ran 𝐺 𝐽)
46453ad2ant1 1075 . . . . . . . . . . 11 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → ran 𝐺 𝐽)
47 simp3 1056 . . . . . . . . . . 11 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑌 ∈ ran 𝐺)
4846, 47sseldd 3569 . . . . . . . . . 10 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑌 𝐽)
4935t1sncld 20940 . . . . . . . . . 10 ((𝐽 ∈ Fre ∧ 𝑌 𝐽) → {𝑌} ∈ (Clsd‘𝐽))
5028, 48, 49syl2anc 691 . . . . . . . . 9 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → {𝑌} ∈ (Clsd‘𝐽))
5127, 50sseldd 3569 . . . . . . . 8 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → {𝑌} ∈ 𝑆)
527, 12, 42, 51mbfmcnvima 29646 . . . . . . 7 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → (𝐺 “ {𝑌}) ∈ dom 𝑀)
53 inelsiga 29525 . . . . . . 7 ((dom 𝑀 ran sigAlgebra ∧ (𝐹 “ {𝑋}) ∈ dom 𝑀 ∧ (𝐺 “ {𝑌}) ∈ dom 𝑀) → ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ∈ dom 𝑀)
547, 39, 52, 53syl3anc 1318 . . . . . 6 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ∈ dom 𝑀)
55 measvxrge0 29595 . . . . . 6 ((𝑀 ∈ (measures‘dom 𝑀) ∧ ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ∈ dom 𝑀) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,]+∞))
564, 54, 55syl2anc 691 . . . . 5 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,]+∞))
57 elxrge0 12152 . . . . . 6 ((𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,]+∞) ↔ ((𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ* ∧ 0 ≤ (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})))))
5857simplbi 475 . . . . 5 ((𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,]+∞) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ*)
5956, 58syl 17 . . . 4 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ*)
6059adantr 480 . . 3 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ*)
61 0re 9919 . . . 4 0 ∈ ℝ
6261a1i 11 . . 3 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → 0 ∈ ℝ)
6357simprbi 479 . . . . 5 ((𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,]+∞) → 0 ≤ (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))))
6456, 63syl 17 . . . 4 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 0 ≤ (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))))
6564adantr 480 . . 3 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → 0 ≤ (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))))
6659adantr 480 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ*)
674adantr 480 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → 𝑀 ∈ (measures‘dom 𝑀))
6839adantr 480 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝐹 “ {𝑋}) ∈ dom 𝑀)
69 measvxrge0 29595 . . . . . . 7 ((𝑀 ∈ (measures‘dom 𝑀) ∧ (𝐹 “ {𝑋}) ∈ dom 𝑀) → (𝑀‘(𝐹 “ {𝑋})) ∈ (0[,]+∞))
7067, 68, 69syl2anc 691 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘(𝐹 “ {𝑋})) ∈ (0[,]+∞))
71 elxrge0 12152 . . . . . . 7 ((𝑀‘(𝐹 “ {𝑋})) ∈ (0[,]+∞) ↔ ((𝑀‘(𝐹 “ {𝑋})) ∈ ℝ* ∧ 0 ≤ (𝑀‘(𝐹 “ {𝑋}))))
7271simplbi 475 . . . . . 6 ((𝑀‘(𝐹 “ {𝑋})) ∈ (0[,]+∞) → (𝑀‘(𝐹 “ {𝑋})) ∈ ℝ*)
7370, 72syl 17 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘(𝐹 “ {𝑋})) ∈ ℝ*)
74 pnfxr 9971 . . . . . 6 +∞ ∈ ℝ*
7574a1i 11 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → +∞ ∈ ℝ*)
7654adantr 480 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ∈ dom 𝑀)
77 inss1 3795 . . . . . . 7 ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ⊆ (𝐹 “ {𝑋})
7877a1i 11 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ⊆ (𝐹 “ {𝑋}))
7967, 76, 68, 78measssd 29605 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ≤ (𝑀‘(𝐹 “ {𝑋})))
80 simpl1 1057 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → 𝜑)
8133anim1i 590 . . . . . . . 8 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑋 ∈ ran 𝐹𝑋0 ))
82 eldifsn 4260 . . . . . . . 8 (𝑋 ∈ (ran 𝐹 ∖ { 0 }) ↔ (𝑋 ∈ ran 𝐹𝑋0 ))
8381, 82sylibr 223 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → 𝑋 ∈ (ran 𝐹 ∖ { 0 }))
8413, 14, 8, 15, 16, 17, 18, 1, 19sibfima 29727 . . . . . . 7 ((𝜑𝑋 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(𝐹 “ {𝑋})) ∈ (0[,)+∞))
8580, 83, 84syl2anc 691 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘(𝐹 “ {𝑋})) ∈ (0[,)+∞))
86 elico2 12108 . . . . . . . 8 ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ((𝑀‘(𝐹 “ {𝑋})) ∈ (0[,)+∞) ↔ ((𝑀‘(𝐹 “ {𝑋})) ∈ ℝ ∧ 0 ≤ (𝑀‘(𝐹 “ {𝑋})) ∧ (𝑀‘(𝐹 “ {𝑋})) < +∞)))
8761, 74, 86mp2an 704 . . . . . . 7 ((𝑀‘(𝐹 “ {𝑋})) ∈ (0[,)+∞) ↔ ((𝑀‘(𝐹 “ {𝑋})) ∈ ℝ ∧ 0 ≤ (𝑀‘(𝐹 “ {𝑋})) ∧ (𝑀‘(𝐹 “ {𝑋})) < +∞))
8887simp3bi 1071 . . . . . 6 ((𝑀‘(𝐹 “ {𝑋})) ∈ (0[,)+∞) → (𝑀‘(𝐹 “ {𝑋})) < +∞)
8985, 88syl 17 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘(𝐹 “ {𝑋})) < +∞)
9066, 73, 75, 79, 89xrlelttrd 11867 . . . 4 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) < +∞)
9159adantr 480 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ*)
924adantr 480 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → 𝑀 ∈ (measures‘dom 𝑀))
9352adantr 480 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝐺 “ {𝑌}) ∈ dom 𝑀)
94 measvxrge0 29595 . . . . . . 7 ((𝑀 ∈ (measures‘dom 𝑀) ∧ (𝐺 “ {𝑌}) ∈ dom 𝑀) → (𝑀‘(𝐺 “ {𝑌})) ∈ (0[,]+∞))
9592, 93, 94syl2anc 691 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘(𝐺 “ {𝑌})) ∈ (0[,]+∞))
96 elxrge0 12152 . . . . . . 7 ((𝑀‘(𝐺 “ {𝑌})) ∈ (0[,]+∞) ↔ ((𝑀‘(𝐺 “ {𝑌})) ∈ ℝ* ∧ 0 ≤ (𝑀‘(𝐺 “ {𝑌}))))
9796simplbi 475 . . . . . 6 ((𝑀‘(𝐺 “ {𝑌})) ∈ (0[,]+∞) → (𝑀‘(𝐺 “ {𝑌})) ∈ ℝ*)
9895, 97syl 17 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘(𝐺 “ {𝑌})) ∈ ℝ*)
9974a1i 11 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → +∞ ∈ ℝ*)
10054adantr 480 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ∈ dom 𝑀)
101 inss2 3796 . . . . . . 7 ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ⊆ (𝐺 “ {𝑌})
102101a1i 11 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ⊆ (𝐺 “ {𝑌}))
10392, 100, 93, 102measssd 29605 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ≤ (𝑀‘(𝐺 “ {𝑌})))
104 simpl1 1057 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → 𝜑)
10547anim1i 590 . . . . . . . 8 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑌 ∈ ran 𝐺𝑌0 ))
106 eldifsn 4260 . . . . . . . 8 (𝑌 ∈ (ran 𝐺 ∖ { 0 }) ↔ (𝑌 ∈ ran 𝐺𝑌0 ))
107105, 106sylibr 223 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → 𝑌 ∈ (ran 𝐺 ∖ { 0 }))
10813, 14, 8, 15, 16, 17, 18, 1, 40sibfima 29727 . . . . . . 7 ((𝜑𝑌 ∈ (ran 𝐺 ∖ { 0 })) → (𝑀‘(𝐺 “ {𝑌})) ∈ (0[,)+∞))
109104, 107, 108syl2anc 691 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘(𝐺 “ {𝑌})) ∈ (0[,)+∞))
110 elico2 12108 . . . . . . . 8 ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ((𝑀‘(𝐺 “ {𝑌})) ∈ (0[,)+∞) ↔ ((𝑀‘(𝐺 “ {𝑌})) ∈ ℝ ∧ 0 ≤ (𝑀‘(𝐺 “ {𝑌})) ∧ (𝑀‘(𝐺 “ {𝑌})) < +∞)))
11161, 74, 110mp2an 704 . . . . . . 7 ((𝑀‘(𝐺 “ {𝑌})) ∈ (0[,)+∞) ↔ ((𝑀‘(𝐺 “ {𝑌})) ∈ ℝ ∧ 0 ≤ (𝑀‘(𝐺 “ {𝑌})) ∧ (𝑀‘(𝐺 “ {𝑌})) < +∞))
112111simp3bi 1071 . . . . . 6 ((𝑀‘(𝐺 “ {𝑌})) ∈ (0[,)+∞) → (𝑀‘(𝐺 “ {𝑌})) < +∞)
113109, 112syl 17 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘(𝐺 “ {𝑌})) < +∞)
11491, 98, 99, 103, 113xrlelttrd 11867 . . . 4 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) < +∞)
11590, 114jaodan 822 . . 3 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) < +∞)
116 xrre3 11876 . . 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)
11760, 62, 65, 115, 116syl22anc 1319 . 2 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ)
118 elico2 12108 . . 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}))) < +∞)))
11961, 74, 118mp2an 704 . 2 ((𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,)+∞) ↔ ((𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ ∧ 0 ≤ (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∧ (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) < +∞))
120117, 65, 115, 119syl3anbrc 1239 1 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,)+∞))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  {csn 4125  ∪ cuni 4372   class class class wbr 4583  ◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  +∞cpnf 9950  ℝ*cxr 9952   < clt 9953   ≤ cle 9954  [,)cico 12048  [,]cicc 12049  Basecbs 15695  Scalarcsca 15771   ·𝑠 cvsca 15772  TopOpenctopn 15905  0gc0g 15923  Topctop 20517  TopSpctps 20519  Clsdccld 20630  Frect1 20921  ℝHomcrrh 29365  sigAlgebracsiga 29497  sigaGencsigagen 29528  measurescmeas 29585  MblFnMcmbfm 29639  sitgcsitg 29718 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-ac2 9168  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-ac 8822  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-shft 13655  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265  df-ef 14637  df-sin 14639  df-cos 14640  df-pi 14642  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-ordt 15984  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-ps 17023  df-tsr 17024  df-plusf 17064  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-subrg 18601  df-abv 18640  df-lmod 18688  df-scaf 18689  df-sra 18993  df-rgmod 18994  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cn 20841  df-cnp 20842  df-t1 20928  df-haus 20929  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-tmd 21686  df-tgp 21687  df-tsms 21740  df-trg 21773  df-xms 21935  df-ms 21936  df-tms 21937  df-nm 22197  df-ngp 22198  df-nrg 22200  df-nlm 22201  df-ii 22488  df-cncf 22489  df-limc 23436  df-dv 23437  df-log 24107  df-esum 29417  df-siga 29498  df-sigagen 29529  df-meas 29586  df-mbfm 29640  df-sitg 29719 This theorem is referenced by:  sibfof  29729  sitgaddlemb  29737
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