| Mathbox for Glauco Siliprandi |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > hoimbl | Structured version Visualization version GIF version | ||
| Description: Any n-dimensional half-open interval is Lebesgue measurable. This is a substep of Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
| Ref | Expression |
|---|---|
| hoimbl.x | ⊢ (𝜑 → 𝑋 ∈ Fin) |
| hoimbl.s | ⊢ 𝑆 = dom (voln‘𝑋) |
| hoimbl.a | ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) |
| hoimbl.b | ⊢ (𝜑 → 𝐵:𝑋⟶ℝ) |
| Ref | Expression |
|---|---|
| hoimbl | ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hoimbl.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
| 2 | 1 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑋 ∈ Fin) |
| 3 | 2 | rrnmbl 46629 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑m 𝑋) ∈ dom (voln‘𝑋)) |
| 4 | reex 11246 | . . . . . . . . 9 ⊢ ℝ ∈ V | |
| 5 | mapdm0 8882 | . . . . . . . . 9 ⊢ (ℝ ∈ V → (ℝ ↑m ∅) = {∅}) | |
| 6 | 4, 5 | ax-mp 5 | . . . . . . . 8 ⊢ (ℝ ↑m ∅) = {∅} |
| 7 | 6 | eqcomi 2746 | . . . . . . 7 ⊢ {∅} = (ℝ ↑m ∅) |
| 8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝑋 = ∅ → {∅} = (ℝ ↑m ∅)) |
| 9 | id 22 | . . . . . . . 8 ⊢ (𝑋 = ∅ → 𝑋 = ∅) | |
| 10 | 9 | ixpeq1d 8949 | . . . . . . 7 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = X𝑖 ∈ ∅ ((𝐴‘𝑖)[,)(𝐵‘𝑖))) |
| 11 | ixp0x 8966 | . . . . . . . 8 ⊢ X𝑖 ∈ ∅ ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = {∅} | |
| 12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝑋 = ∅ → X𝑖 ∈ ∅ ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = {∅}) |
| 13 | 10, 12 | eqtrd 2777 | . . . . . 6 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = {∅}) |
| 14 | oveq2 7439 | . . . . . 6 ⊢ (𝑋 = ∅ → (ℝ ↑m 𝑋) = (ℝ ↑m ∅)) | |
| 15 | 8, 13, 14 | 3eqtr4d 2787 | . . . . 5 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = (ℝ ↑m 𝑋)) |
| 16 | 15 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = (ℝ ↑m 𝑋)) |
| 17 | hoimbl.s | . . . . 5 ⊢ 𝑆 = dom (voln‘𝑋) | |
| 18 | 17 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑆 = dom (voln‘𝑋)) |
| 19 | 16, 18 | eleq12d 2835 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆 ↔ (ℝ ↑m 𝑋) ∈ dom (voln‘𝑋))) |
| 20 | 3, 19 | mpbird 257 | . 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆) |
| 21 | 1 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin) |
| 22 | 9 | necon3bi 2967 | . . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) |
| 23 | 22 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅) |
| 24 | hoimbl.a | . . . 4 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) | |
| 25 | 24 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐴:𝑋⟶ℝ) |
| 26 | hoimbl.b | . . . 4 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ) | |
| 27 | 26 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐵:𝑋⟶ℝ) |
| 28 | id 22 | . . . . . 6 ⊢ (𝑤 = 𝑥 → 𝑤 = 𝑥) | |
| 29 | eqidd 2738 | . . . . . 6 ⊢ (𝑤 = 𝑥 → ℝ = ℝ) | |
| 30 | 28 | ixpeq1d 8949 | . . . . . . 7 ⊢ (𝑤 = 𝑥 → X𝑗 ∈ 𝑤 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ) = X𝑗 ∈ 𝑥 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ)) |
| 31 | eqeq1 2741 | . . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝑗 = ℎ ↔ 𝑖 = ℎ)) | |
| 32 | 31 | ifbid 4549 | . . . . . . . . 9 ⊢ (𝑗 = 𝑖 → if(𝑗 = ℎ, (-∞(,)𝑧), ℝ) = if(𝑖 = ℎ, (-∞(,)𝑧), ℝ)) |
| 33 | 32 | cbvixpv 8955 | . . . . . . . 8 ⊢ X𝑗 ∈ 𝑥 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ) = X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ) |
| 34 | 33 | a1i 11 | . . . . . . 7 ⊢ (𝑤 = 𝑥 → X𝑗 ∈ 𝑥 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ) = X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ)) |
| 35 | 30, 34 | eqtrd 2777 | . . . . . 6 ⊢ (𝑤 = 𝑥 → X𝑗 ∈ 𝑤 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ) = X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ)) |
| 36 | 28, 29, 35 | mpoeq123dv 7508 | . . . . 5 ⊢ (𝑤 = 𝑥 → (ℎ ∈ 𝑤, 𝑧 ∈ ℝ ↦ X𝑗 ∈ 𝑤 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ)) = (ℎ ∈ 𝑥, 𝑧 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ))) |
| 37 | eqeq2 2749 | . . . . . . . . 9 ⊢ (ℎ = 𝑙 → (𝑖 = ℎ ↔ 𝑖 = 𝑙)) | |
| 38 | 37 | ifbid 4549 | . . . . . . . 8 ⊢ (ℎ = 𝑙 → if(𝑖 = ℎ, (-∞(,)𝑧), ℝ) = if(𝑖 = 𝑙, (-∞(,)𝑧), ℝ)) |
| 39 | 38 | ixpeq2dv 8953 | . . . . . . 7 ⊢ (ℎ = 𝑙 → X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ) = X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑧), ℝ)) |
| 40 | oveq2 7439 | . . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (-∞(,)𝑧) = (-∞(,)𝑦)) | |
| 41 | 40 | ifeq1d 4545 | . . . . . . . 8 ⊢ (𝑧 = 𝑦 → if(𝑖 = 𝑙, (-∞(,)𝑧), ℝ) = if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ)) |
| 42 | 41 | ixpeq2dv 8953 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑧), ℝ) = X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ)) |
| 43 | 39, 42 | cbvmpov 7528 | . . . . . 6 ⊢ (ℎ ∈ 𝑥, 𝑧 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ)) = (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ)) |
| 44 | 43 | a1i 11 | . . . . 5 ⊢ (𝑤 = 𝑥 → (ℎ ∈ 𝑥, 𝑧 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = ℎ, (-∞(,)𝑧), ℝ)) = (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ))) |
| 45 | 36, 44 | eqtrd 2777 | . . . 4 ⊢ (𝑤 = 𝑥 → (ℎ ∈ 𝑤, 𝑧 ∈ ℝ ↦ X𝑗 ∈ 𝑤 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ)) = (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ))) |
| 46 | 45 | cbvmptv 5255 | . . 3 ⊢ (𝑤 ∈ Fin ↦ (ℎ ∈ 𝑤, 𝑧 ∈ ℝ ↦ X𝑗 ∈ 𝑤 if(𝑗 = ℎ, (-∞(,)𝑧), ℝ))) = (𝑥 ∈ Fin ↦ (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ))) |
| 47 | 21, 23, 17, 25, 27, 46 | hoimbllem 46645 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆) |
| 48 | 20, 47 | pm2.61dan 813 | 1 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∅c0 4333 ifcif 4525 {csn 4626 ↦ cmpt 5225 dom cdm 5685 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ↑m cmap 8866 Xcixp 8937 Fincfn 8985 ℝcr 11154 -∞cmnf 11293 (,)cioo 13387 [,)cico 13389 volncvoln 46553 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cc 10475 ax-ac2 10503 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-acn 9982 df-ac 10156 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-prod 15940 df-rest 17467 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-bases 22953 df-cmp 23395 df-ovol 25499 df-vol 25500 df-salg 46324 df-sumge0 46378 df-mea 46465 df-ome 46505 df-caragen 46507 df-ovoln 46552 df-voln 46554 |
| This theorem is referenced by: opnvonmbllem2 46648 hoimbl2 46680 vonhoi 46682 vonioolem1 46695 vonioolem2 46696 vonicclem1 46698 vonicclem2 46699 |
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