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Theorem sibfinima 30529
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 30393 . . . . . . . 8 (𝑀 ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀))
31, 2sylib 208 . . . . . . 7 (𝜑𝑀 ∈ (measures‘dom 𝑀))
433ad2ant1 1102 . . . . . 6 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑀 ∈ (measures‘dom 𝑀))
5 dmmeas 30392 . . . . . . . . 9 (𝑀 ran measures → dom 𝑀 ran sigAlgebra)
61, 5syl 17 . . . . . . . 8 (𝜑 → dom 𝑀 ran sigAlgebra)
763ad2ant1 1102 . . . . . . 7 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → dom 𝑀 ran sigAlgebra)
8 sitgval.s . . . . . . . . . 10 𝑆 = (sigaGen‘𝐽)
9 sibfinima.j . . . . . . . . . . 11 (𝜑𝐽 ∈ Fre)
109sgsiga 30333 . . . . . . . . . 10 (𝜑 → (sigaGen‘𝐽) ∈ ran sigAlgebra)
118, 10syl5eqel 2734 . . . . . . . . 9 (𝜑𝑆 ran sigAlgebra)
12113ad2ant1 1102 . . . . . . . 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 30525 . . . . . . . . 9 (𝜑𝐹 ∈ (dom 𝑀MblFnM𝑆))
21203ad2ant1 1102 . . . . . . . 8 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
22 sibfinima.w . . . . . . . . . . . 12 (𝜑𝑊 ∈ TopSp)
2314tpstop 20789 . . . . . . . . . . . 12 (𝑊 ∈ TopSp → 𝐽 ∈ Top)
24 cldssbrsiga 30378 . . . . . . . . . . . 12 (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
2522, 23, 243syl 18 . . . . . . . . . . 11 (𝜑 → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽))
2625, 8syl6sseqr 3685 . . . . . . . . . 10 (𝜑 → (Clsd‘𝐽) ⊆ 𝑆)
27263ad2ant1 1102 . . . . . . . . 9 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → (Clsd‘𝐽) ⊆ 𝑆)
2893ad2ant1 1102 . . . . . . . . . 10 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝐽 ∈ Fre)
2913, 14, 8, 15, 16, 17, 18, 1, 19sibff 30526 . . . . . . . . . . . . 13 (𝜑𝐹: dom 𝑀 𝐽)
30 frn 6091 . . . . . . . . . . . . 13 (𝐹: dom 𝑀 𝐽 → ran 𝐹 𝐽)
3129, 30syl 17 . . . . . . . . . . . 12 (𝜑 → ran 𝐹 𝐽)
32313ad2ant1 1102 . . . . . . . . . . 11 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → ran 𝐹 𝐽)
33 simp2 1082 . . . . . . . . . . 11 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑋 ∈ ran 𝐹)
3432, 33sseldd 3637 . . . . . . . . . 10 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑋 𝐽)
35 eqid 2651 . . . . . . . . . . 11 𝐽 = 𝐽
3635t1sncld 21178 . . . . . . . . . 10 ((𝐽 ∈ Fre ∧ 𝑋 𝐽) → {𝑋} ∈ (Clsd‘𝐽))
3728, 34, 36syl2anc 694 . . . . . . . . 9 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → {𝑋} ∈ (Clsd‘𝐽))
3827, 37sseldd 3637 . . . . . . . 8 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → {𝑋} ∈ 𝑆)
397, 12, 21, 38mbfmcnvima 30447 . . . . . . 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 30525 . . . . . . . . 9 (𝜑𝐺 ∈ (dom 𝑀MblFnM𝑆))
42413ad2ant1 1102 . . . . . . . 8 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝐺 ∈ (dom 𝑀MblFnM𝑆))
4313, 14, 8, 15, 16, 17, 18, 1, 40sibff 30526 . . . . . . . . . . . . 13 (𝜑𝐺: dom 𝑀 𝐽)
44 frn 6091 . . . . . . . . . . . . 13 (𝐺: dom 𝑀 𝐽 → ran 𝐺 𝐽)
4543, 44syl 17 . . . . . . . . . . . 12 (𝜑 → ran 𝐺 𝐽)
46453ad2ant1 1102 . . . . . . . . . . 11 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → ran 𝐺 𝐽)
47 simp3 1083 . . . . . . . . . . 11 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑌 ∈ ran 𝐺)
4846, 47sseldd 3637 . . . . . . . . . 10 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → 𝑌 𝐽)
4935t1sncld 21178 . . . . . . . . . 10 ((𝐽 ∈ Fre ∧ 𝑌 𝐽) → {𝑌} ∈ (Clsd‘𝐽))
5028, 48, 49syl2anc 694 . . . . . . . . 9 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → {𝑌} ∈ (Clsd‘𝐽))
5127, 50sseldd 3637 . . . . . . . 8 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → {𝑌} ∈ 𝑆)
527, 12, 42, 51mbfmcnvima 30447 . . . . . . 7 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → (𝐺 “ {𝑌}) ∈ dom 𝑀)
53 inelsiga 30326 . . . . . . 7 ((dom 𝑀 ran sigAlgebra ∧ (𝐹 “ {𝑋}) ∈ dom 𝑀 ∧ (𝐺 “ {𝑌}) ∈ dom 𝑀) → ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ∈ dom 𝑀)
547, 39, 52, 53syl3anc 1366 . . . . . 6 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ∈ dom 𝑀)
55 measvxrge0 30396 . . . . . 6 ((𝑀 ∈ (measures‘dom 𝑀) ∧ ((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌})) ∈ dom 𝑀) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,]+∞))
564, 54, 55syl2anc 694 . . . . 5 ((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,]+∞))
57 elxrge0 12319 . . . . . 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 10078 . . . 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 30396 . . . . . . 7 ((𝑀 ∈ (measures‘dom 𝑀) ∧ (𝐹 “ {𝑋}) ∈ dom 𝑀) → (𝑀‘(𝐹 “ {𝑋})) ∈ (0[,]+∞))
7067, 68, 69syl2anc 694 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘(𝐹 “ {𝑋})) ∈ (0[,]+∞))
71 elxrge0 12319 . . . . . . 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 10130 . . . . . 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 3866 . . . . . . 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 30406 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ≤ (𝑀‘(𝐹 “ {𝑋})))
80 simpl1 1084 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → 𝜑)
8133anim1i 591 . . . . . . . 8 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑋 ∈ ran 𝐹𝑋0 ))
82 eldifsn 4350 . . . . . . . 8 (𝑋 ∈ (ran 𝐹 ∖ { 0 }) ↔ (𝑋 ∈ ran 𝐹𝑋0 ))
8381, 82sylibr 224 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → 𝑋 ∈ (ran 𝐹 ∖ { 0 }))
8413, 14, 8, 15, 16, 17, 18, 1, 19sibfima 30528 . . . . . . 7 ((𝜑𝑋 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(𝐹 “ {𝑋})) ∈ (0[,)+∞))
8580, 83, 84syl2anc 694 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑋0 ) → (𝑀‘(𝐹 “ {𝑋})) ∈ (0[,)+∞))
86 elico2 12275 . . . . . . . 8 ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ((𝑀‘(𝐹 “ {𝑋})) ∈ (0[,)+∞) ↔ ((𝑀‘(𝐹 “ {𝑋})) ∈ ℝ ∧ 0 ≤ (𝑀‘(𝐹 “ {𝑋})) ∧ (𝑀‘(𝐹 “ {𝑋})) < +∞)))
8761, 74, 86mp2an 708 . . . . . . 7 ((𝑀‘(𝐹 “ {𝑋})) ∈ (0[,)+∞) ↔ ((𝑀‘(𝐹 “ {𝑋})) ∈ ℝ ∧ 0 ≤ (𝑀‘(𝐹 “ {𝑋})) ∧ (𝑀‘(𝐹 “ {𝑋})) < +∞))
8887simp3bi 1098 . . . . . 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 12029 . . . 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 30396 . . . . . . 7 ((𝑀 ∈ (measures‘dom 𝑀) ∧ (𝐺 “ {𝑌}) ∈ dom 𝑀) → (𝑀‘(𝐺 “ {𝑌})) ∈ (0[,]+∞))
9592, 93, 94syl2anc 694 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘(𝐺 “ {𝑌})) ∈ (0[,]+∞))
96 elxrge0 12319 . . . . . . 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 3867 . . . . . . 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 30406 . . . . 5 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ≤ (𝑀‘(𝐺 “ {𝑌})))
104 simpl1 1084 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → 𝜑)
10547anim1i 591 . . . . . . . 8 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑌 ∈ ran 𝐺𝑌0 ))
106 eldifsn 4350 . . . . . . . 8 (𝑌 ∈ (ran 𝐺 ∖ { 0 }) ↔ (𝑌 ∈ ran 𝐺𝑌0 ))
107105, 106sylibr 224 . . . . . . 7 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → 𝑌 ∈ (ran 𝐺 ∖ { 0 }))
10813, 14, 8, 15, 16, 17, 18, 1, 40sibfima 30528 . . . . . . 7 ((𝜑𝑌 ∈ (ran 𝐺 ∖ { 0 })) → (𝑀‘(𝐺 “ {𝑌})) ∈ (0[,)+∞))
109104, 107, 108syl2anc 694 . . . . . 6 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘(𝐺 “ {𝑌})) ∈ (0[,)+∞))
110 elico2 12275 . . . . . . . 8 ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ((𝑀‘(𝐺 “ {𝑌})) ∈ (0[,)+∞) ↔ ((𝑀‘(𝐺 “ {𝑌})) ∈ ℝ ∧ 0 ≤ (𝑀‘(𝐺 “ {𝑌})) ∧ (𝑀‘(𝐺 “ {𝑌})) < +∞)))
11161, 74, 110mp2an 708 . . . . . . 7 ((𝑀‘(𝐺 “ {𝑌})) ∈ (0[,)+∞) ↔ ((𝑀‘(𝐺 “ {𝑌})) ∈ ℝ ∧ 0 ≤ (𝑀‘(𝐺 “ {𝑌})) ∧ (𝑀‘(𝐺 “ {𝑌})) < +∞))
112111simp3bi 1098 . . . . . 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 12029 . . . 4 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ 𝑌0 ) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) < +∞)
11590, 114jaodan 843 . . 3 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) < +∞)
116 xrre3 12040 . . 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 1367 . 2 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ ℝ)
118 elico2 12275 . . 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 708 . 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 1265 1 (((𝜑𝑋 ∈ ran 𝐹𝑌 ∈ ran 𝐺) ∧ (𝑋0𝑌0 )) → (𝑀‘((𝐹 “ {𝑋}) ∩ (𝐺 “ {𝑌}))) ∈ (0[,)+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  cdif 3604  cin 3606  wss 3607  {csn 4210   cuni 4468   class class class wbr 4685  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146  wf 5922  cfv 5926  (class class class)co 6690  cr 9973  0cc0 9974  +∞cpnf 10109  *cxr 10111   < clt 10112  cle 10113  [,)cico 12215  [,]cicc 12216  Basecbs 15904  Scalarcsca 15991   ·𝑠 cvsca 15992  TopOpenctopn 16129  0gc0g 16147  Topctop 20746  TopSpctps 20784  Clsdccld 20868  Frect1 21159  ℝHomcrrh 30165  sigAlgebracsiga 30298  sigaGencsigagen 30329  measurescmeas 30386  MblFnMcmbfm 30440  sitgcsitg 30519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-ac2 9323  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-acn 8806  df-ac 8977  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-shft 13851  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-sum 14461  df-ef 14842  df-sin 14844  df-cos 14845  df-pi 14847  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-ordt 16208  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-ps 17247  df-tsr 17248  df-plusf 17288  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-subg 17638  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-cring 18596  df-subrg 18826  df-abv 18865  df-lmod 18913  df-scaf 18914  df-sra 19220  df-rgmod 19221  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lp 20988  df-perf 20989  df-cn 21079  df-cnp 21080  df-t1 21166  df-haus 21167  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-tmd 21923  df-tgp 21924  df-tsms 21977  df-trg 22010  df-xms 22172  df-ms 22173  df-tms 22174  df-nm 22434  df-ngp 22435  df-nrg 22437  df-nlm 22438  df-ii 22727  df-cncf 22728  df-limc 23675  df-dv 23676  df-log 24348  df-esum 30218  df-siga 30299  df-sigagen 30330  df-meas 30387  df-mbfm 30441  df-sitg 30520
This theorem is referenced by:  sibfof  30530  sitgaddlemb  30538
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