Proof of Theorem sibfinima
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | sitgval.2 | . . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ ∪ ran
measures) | 
| 2 |  | measbasedom 34204 | . . . . . . . 8
⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀)) | 
| 3 | 1, 2 | sylib 218 | . . . . . . 7
⊢ (𝜑 → 𝑀 ∈ (measures‘dom 𝑀)) | 
| 4 | 3 | 3ad2ant1 1133 | . . . . . 6
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑀 ∈ (measures‘dom 𝑀)) | 
| 5 |  | dmmeas 34203 | . . . . . . . . 9
⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran
sigAlgebra) | 
| 6 | 1, 5 | syl 17 | . . . . . . . 8
⊢ (𝜑 → dom 𝑀 ∈ ∪ ran
sigAlgebra) | 
| 7 | 6 | 3ad2ant1 1133 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → dom 𝑀 ∈ ∪ ran
sigAlgebra) | 
| 8 |  | sitgval.s | . . . . . . . . . 10
⊢ 𝑆 = (sigaGen‘𝐽) | 
| 9 |  | sibfinima.j | . . . . . . . . . . 11
⊢ (𝜑 → 𝐽 ∈ Fre) | 
| 10 | 9 | sgsiga 34144 | . . . . . . . . . 10
⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra) | 
| 11 | 8, 10 | eqeltrid 2844 | . . . . . . . . 9
⊢ (𝜑 → 𝑆 ∈ ∪ ran
sigAlgebra) | 
| 12 | 11 | 3ad2ant1 1133 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑆 ∈ ∪ ran
sigAlgebra) | 
| 13 |  | sitgval.b | . . . . . . . . . 10
⊢ 𝐵 = (Base‘𝑊) | 
| 14 |  | sitgval.j | . . . . . . . . . 10
⊢ 𝐽 = (TopOpen‘𝑊) | 
| 15 |  | sitgval.0 | . . . . . . . . . 10
⊢  0 =
(0g‘𝑊) | 
| 16 |  | sitgval.x | . . . . . . . . . 10
⊢  · = (
·𝑠 ‘𝑊) | 
| 17 |  | sitgval.h | . . . . . . . . . 10
⊢ 𝐻 =
(ℝHom‘(Scalar‘𝑊)) | 
| 18 |  | sitgval.1 | . . . . . . . . . 10
⊢ (𝜑 → 𝑊 ∈ 𝑉) | 
| 19 |  | sibfmbl.1 | . . . . . . . . . 10
⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀)) | 
| 20 | 13, 14, 8, 15, 16, 17, 18, 1, 19 | sibfmbl 34338 | . . . . . . . . 9
⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆)) | 
| 21 | 20 | 3ad2ant1 1133 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝐹 ∈ (dom 𝑀MblFnM𝑆)) | 
| 22 |  | sibfinima.w | . . . . . . . . . . . 12
⊢ (𝜑 → 𝑊 ∈ TopSp) | 
| 23 | 14 | tpstop 22944 | . . . . . . . . . . . 12
⊢ (𝑊 ∈ TopSp → 𝐽 ∈ Top) | 
| 24 |  | cldssbrsiga 34189 | . . . . . . . . . . . 12
⊢ (𝐽 ∈ Top →
(Clsd‘𝐽) ⊆
(sigaGen‘𝐽)) | 
| 25 | 22, 23, 24 | 3syl 18 | . . . . . . . . . . 11
⊢ (𝜑 → (Clsd‘𝐽) ⊆ (sigaGen‘𝐽)) | 
| 26 | 25, 8 | sseqtrrdi 4024 | . . . . . . . . . 10
⊢ (𝜑 → (Clsd‘𝐽) ⊆ 𝑆) | 
| 27 | 26 | 3ad2ant1 1133 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (Clsd‘𝐽) ⊆ 𝑆) | 
| 28 | 9 | 3ad2ant1 1133 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝐽 ∈ Fre) | 
| 29 | 13, 14, 8, 15, 16, 17, 18, 1, 19 | sibff 34339 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝐽) | 
| 30 | 29 | frnd 6743 | . . . . . . . . . . . 12
⊢ (𝜑 → ran 𝐹 ⊆ ∪ 𝐽) | 
| 31 | 30 | 3ad2ant1 1133 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → ran 𝐹 ⊆ ∪ 𝐽) | 
| 32 |  | simp2 1137 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑋 ∈ ran 𝐹) | 
| 33 | 31, 32 | sseldd 3983 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑋 ∈ ∪ 𝐽) | 
| 34 |  | eqid 2736 | . . . . . . . . . . 11
⊢ ∪ 𝐽 =
∪ 𝐽 | 
| 35 | 34 | t1sncld 23335 | . . . . . . . . . 10
⊢ ((𝐽 ∈ Fre ∧ 𝑋 ∈ ∪ 𝐽)
→ {𝑋} ∈
(Clsd‘𝐽)) | 
| 36 | 28, 33, 35 | syl2anc 584 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → {𝑋} ∈ (Clsd‘𝐽)) | 
| 37 | 27, 36 | sseldd 3983 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → {𝑋} ∈ 𝑆) | 
| 38 | 7, 12, 21, 37 | mbfmcnvima 34258 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (◡𝐹 “ {𝑋}) ∈ dom 𝑀) | 
| 39 |  | sibfinima.g | . . . . . . . . . 10
⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀)) | 
| 40 | 13, 14, 8, 15, 16, 17, 18, 1, 39 | sibfmbl 34338 | . . . . . . . . 9
⊢ (𝜑 → 𝐺 ∈ (dom 𝑀MblFnM𝑆)) | 
| 41 | 40 | 3ad2ant1 1133 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝐺 ∈ (dom 𝑀MblFnM𝑆)) | 
| 42 | 13, 14, 8, 15, 16, 17, 18, 1, 39 | sibff 34339 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝐺:∪ dom 𝑀⟶∪ 𝐽) | 
| 43 | 42 | frnd 6743 | . . . . . . . . . . . 12
⊢ (𝜑 → ran 𝐺 ⊆ ∪ 𝐽) | 
| 44 | 43 | 3ad2ant1 1133 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → ran 𝐺 ⊆ ∪ 𝐽) | 
| 45 |  | simp3 1138 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑌 ∈ ran 𝐺) | 
| 46 | 44, 45 | sseldd 3983 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 𝑌 ∈ ∪ 𝐽) | 
| 47 | 34 | t1sncld 23335 | . . . . . . . . . 10
⊢ ((𝐽 ∈ Fre ∧ 𝑌 ∈ ∪ 𝐽)
→ {𝑌} ∈
(Clsd‘𝐽)) | 
| 48 | 28, 46, 47 | syl2anc 584 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → {𝑌} ∈ (Clsd‘𝐽)) | 
| 49 | 27, 48 | sseldd 3983 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → {𝑌} ∈ 𝑆) | 
| 50 | 7, 12, 41, 49 | mbfmcnvima 34258 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (◡𝐺 “ {𝑌}) ∈ dom 𝑀) | 
| 51 |  | inelsiga 34137 | . . . . . . 7
⊢ ((dom
𝑀 ∈ ∪ ran sigAlgebra ∧ (◡𝐹 “ {𝑋}) ∈ dom 𝑀 ∧ (◡𝐺 “ {𝑌}) ∈ dom 𝑀) → ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ∈ dom 𝑀) | 
| 52 | 7, 38, 50, 51 | syl3anc 1372 | . . . . . 6
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ∈ dom 𝑀) | 
| 53 |  | measvxrge0 34207 | . . . . . 6
⊢ ((𝑀 ∈ (measures‘dom
𝑀) ∧ ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ∈ dom 𝑀) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,]+∞)) | 
| 54 | 4, 52, 53 | syl2anc 584 | . . . . 5
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,]+∞)) | 
| 55 |  | elxrge0 13498 | . . . . . 6
⊢ ((𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,]+∞) ↔ ((𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ* ∧ 0 ≤
(𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))))) | 
| 56 | 55 | simplbi 497 | . . . . 5
⊢ ((𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,]+∞) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈
ℝ*) | 
| 57 | 54, 56 | syl 17 | . . . 4
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈
ℝ*) | 
| 58 | 57 | adantr 480 | . . 3
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈
ℝ*) | 
| 59 |  | 0re 11264 | . . . 4
⊢ 0 ∈
ℝ | 
| 60 | 59 | a1i 11 | . . 3
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → 0 ∈
ℝ) | 
| 61 | 55 | simprbi 496 | . . . . 5
⊢ ((𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,]+∞) → 0 ≤
(𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})))) | 
| 62 | 54, 61 | syl 17 | . . . 4
⊢ ((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) → 0 ≤ (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})))) | 
| 63 | 62 | adantr 480 | . . 3
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → 0 ≤ (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})))) | 
| 64 | 57 | adantr 480 | . . . . 5
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈
ℝ*) | 
| 65 | 4 | adantr 480 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → 𝑀 ∈ (measures‘dom 𝑀)) | 
| 66 | 38 | adantr 480 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (◡𝐹 “ {𝑋}) ∈ dom 𝑀) | 
| 67 |  | measvxrge0 34207 | . . . . . . 7
⊢ ((𝑀 ∈ (measures‘dom
𝑀) ∧ (◡𝐹 “ {𝑋}) ∈ dom 𝑀) → (𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,]+∞)) | 
| 68 | 65, 66, 67 | syl2anc 584 | . . . . . 6
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,]+∞)) | 
| 69 |  | elxrge0 13498 | . . . . . . 7
⊢ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,]+∞) ↔ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ ℝ* ∧ 0 ≤
(𝑀‘(◡𝐹 “ {𝑋})))) | 
| 70 | 69 | simplbi 497 | . . . . . 6
⊢ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,]+∞) → (𝑀‘(◡𝐹 “ {𝑋})) ∈
ℝ*) | 
| 71 | 68, 70 | syl 17 | . . . . 5
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘(◡𝐹 “ {𝑋})) ∈
ℝ*) | 
| 72 |  | pnfxr 11316 | . . . . . 6
⊢ +∞
∈ ℝ* | 
| 73 | 72 | a1i 11 | . . . . 5
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → +∞ ∈
ℝ*) | 
| 74 | 52 | adantr 480 | . . . . . 6
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ∈ dom 𝑀) | 
| 75 |  | inss1 4236 | . . . . . . 7
⊢ ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ⊆ (◡𝐹 “ {𝑋}) | 
| 76 | 75 | a1i 11 | . . . . . 6
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ⊆ (◡𝐹 “ {𝑋})) | 
| 77 | 65, 74, 66, 76 | measssd 34217 | . . . . 5
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ≤ (𝑀‘(◡𝐹 “ {𝑋}))) | 
| 78 |  | simpl1 1191 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → 𝜑) | 
| 79 | 32 | anim1i 615 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑋 ∈ ran 𝐹 ∧ 𝑋 ≠ 0 )) | 
| 80 |  | eldifsn 4785 | . . . . . . . 8
⊢ (𝑋 ∈ (ran 𝐹 ∖ { 0 }) ↔ (𝑋 ∈ ran 𝐹 ∧ 𝑋 ≠ 0 )) | 
| 81 | 79, 80 | sylibr 234 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ (ran 𝐹 ∖ { 0 })) | 
| 82 | 13, 14, 8, 15, 16, 17, 18, 1, 19 | sibfima 34341 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,)+∞)) | 
| 83 | 78, 81, 82 | syl2anc 584 | . . . . . 6
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,)+∞)) | 
| 84 |  | elico2 13452 | . . . . . . . 8
⊢ ((0
∈ ℝ ∧ +∞ ∈ ℝ*) → ((𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,)+∞) ↔ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ ℝ ∧ 0 ≤ (𝑀‘(◡𝐹 “ {𝑋})) ∧ (𝑀‘(◡𝐹 “ {𝑋})) < +∞))) | 
| 85 | 59, 72, 84 | mp2an 692 | . . . . . . 7
⊢ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,)+∞) ↔ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ ℝ ∧ 0 ≤ (𝑀‘(◡𝐹 “ {𝑋})) ∧ (𝑀‘(◡𝐹 “ {𝑋})) < +∞)) | 
| 86 | 85 | simp3bi 1147 | . . . . . 6
⊢ ((𝑀‘(◡𝐹 “ {𝑋})) ∈ (0[,)+∞) → (𝑀‘(◡𝐹 “ {𝑋})) < +∞) | 
| 87 | 83, 86 | syl 17 | . . . . 5
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘(◡𝐹 “ {𝑋})) < +∞) | 
| 88 | 64, 71, 73, 77, 87 | xrlelttrd 13203 | . . . 4
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑋 ≠ 0 ) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) < +∞) | 
| 89 | 57 | adantr 480 | . . . . 5
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈
ℝ*) | 
| 90 | 4 | adantr 480 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → 𝑀 ∈ (measures‘dom 𝑀)) | 
| 91 | 50 | adantr 480 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (◡𝐺 “ {𝑌}) ∈ dom 𝑀) | 
| 92 |  | measvxrge0 34207 | . . . . . . 7
⊢ ((𝑀 ∈ (measures‘dom
𝑀) ∧ (◡𝐺 “ {𝑌}) ∈ dom 𝑀) → (𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,]+∞)) | 
| 93 | 90, 91, 92 | syl2anc 584 | . . . . . 6
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,]+∞)) | 
| 94 |  | elxrge0 13498 | . . . . . . 7
⊢ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,]+∞) ↔ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ ℝ* ∧ 0 ≤
(𝑀‘(◡𝐺 “ {𝑌})))) | 
| 95 | 94 | simplbi 497 | . . . . . 6
⊢ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,]+∞) → (𝑀‘(◡𝐺 “ {𝑌})) ∈
ℝ*) | 
| 96 | 93, 95 | syl 17 | . . . . 5
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘(◡𝐺 “ {𝑌})) ∈
ℝ*) | 
| 97 | 72 | a1i 11 | . . . . 5
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → +∞ ∈
ℝ*) | 
| 98 | 52 | adantr 480 | . . . . . 6
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ∈ dom 𝑀) | 
| 99 |  | inss2 4237 | . . . . . . 7
⊢ ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ⊆ (◡𝐺 “ {𝑌}) | 
| 100 | 99 | a1i 11 | . . . . . 6
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → ((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌})) ⊆ (◡𝐺 “ {𝑌})) | 
| 101 | 90, 98, 91, 100 | measssd 34217 | . . . . 5
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ≤ (𝑀‘(◡𝐺 “ {𝑌}))) | 
| 102 |  | simpl1 1191 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → 𝜑) | 
| 103 | 45 | anim1i 615 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑌 ∈ ran 𝐺 ∧ 𝑌 ≠ 0 )) | 
| 104 |  | eldifsn 4785 | . . . . . . . 8
⊢ (𝑌 ∈ (ran 𝐺 ∖ { 0 }) ↔ (𝑌 ∈ ran 𝐺 ∧ 𝑌 ≠ 0 )) | 
| 105 | 103, 104 | sylibr 234 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → 𝑌 ∈ (ran 𝐺 ∖ { 0 })) | 
| 106 | 13, 14, 8, 15, 16, 17, 18, 1, 39 | sibfima 34341 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑌 ∈ (ran 𝐺 ∖ { 0 })) → (𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,)+∞)) | 
| 107 | 102, 105,
106 | syl2anc 584 | . . . . . 6
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,)+∞)) | 
| 108 |  | elico2 13452 | . . . . . . . 8
⊢ ((0
∈ ℝ ∧ +∞ ∈ ℝ*) → ((𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,)+∞) ↔ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ ℝ ∧ 0 ≤ (𝑀‘(◡𝐺 “ {𝑌})) ∧ (𝑀‘(◡𝐺 “ {𝑌})) < +∞))) | 
| 109 | 59, 72, 108 | mp2an 692 | . . . . . . 7
⊢ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,)+∞) ↔ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ ℝ ∧ 0 ≤ (𝑀‘(◡𝐺 “ {𝑌})) ∧ (𝑀‘(◡𝐺 “ {𝑌})) < +∞)) | 
| 110 | 109 | simp3bi 1147 | . . . . . 6
⊢ ((𝑀‘(◡𝐺 “ {𝑌})) ∈ (0[,)+∞) → (𝑀‘(◡𝐺 “ {𝑌})) < +∞) | 
| 111 | 107, 110 | syl 17 | . . . . 5
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘(◡𝐺 “ {𝑌})) < +∞) | 
| 112 | 89, 96, 97, 101, 111 | xrlelttrd 13203 | . . . 4
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ 𝑌 ≠ 0 ) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) < +∞) | 
| 113 | 88, 112 | jaodan 959 | . . 3
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) < +∞) | 
| 114 |  | xrre3 13214 | . . 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) | 
| 115 | 58, 60, 63, 113, 114 | syl22anc 838 | . 2
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ ℝ) | 
| 116 |  | elico2 13452 | . . 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}))) < +∞))) | 
| 117 | 59, 72, 116 | mp2an 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}))) < +∞)) | 
| 118 | 115, 63, 113, 117 | syl3anbrc 1343 | 1
⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,)+∞)) |