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Mirrors > Home > MPE Home > Th. List > i1f0rn | Structured version Visualization version GIF version |
Description: Any simple function takes the value zero on a set of unbounded measure, so in particular this set is not empty. (Contributed by Mario Carneiro, 18-Jun-2014.) |
Ref | Expression |
---|---|
i1f0rn | ⊢ (𝐹 ∈ dom ∫1 → 0 ∈ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfnre 11331 | . . 3 ⊢ +∞ ∉ ℝ | |
2 | 1 | neli 3054 | . 2 ⊢ ¬ +∞ ∈ ℝ |
3 | rembl 25594 | . . . . . 6 ⊢ ℝ ∈ dom vol | |
4 | mblvol 25584 | . . . . . 6 ⊢ (ℝ ∈ dom vol → (vol‘ℝ) = (vol*‘ℝ)) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ (vol‘ℝ) = (vol*‘ℝ) |
6 | ovolre 25579 | . . . . 5 ⊢ (vol*‘ℝ) = +∞ | |
7 | 5, 6 | eqtri 2768 | . . . 4 ⊢ (vol‘ℝ) = +∞ |
8 | cnvimarndm 6112 | . . . . . . 7 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹 | |
9 | i1ff 25730 | . . . . . . . . 9 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ) | |
10 | 9 | fdmd 6757 | . . . . . . . 8 ⊢ (𝐹 ∈ dom ∫1 → dom 𝐹 = ℝ) |
11 | 10 | adantr 480 | . . . . . . 7 ⊢ ((𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ ran 𝐹) → dom 𝐹 = ℝ) |
12 | 8, 11 | eqtrid 2792 | . . . . . 6 ⊢ ((𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ ran 𝐹) → (◡𝐹 “ ran 𝐹) = ℝ) |
13 | 12 | fveq2d 6924 | . . . . 5 ⊢ ((𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ ran 𝐹) → (vol‘(◡𝐹 “ ran 𝐹)) = (vol‘ℝ)) |
14 | i1fima2 25733 | . . . . 5 ⊢ ((𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ ran 𝐹) → (vol‘(◡𝐹 “ ran 𝐹)) ∈ ℝ) | |
15 | 13, 14 | eqeltrrd 2845 | . . . 4 ⊢ ((𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ ran 𝐹) → (vol‘ℝ) ∈ ℝ) |
16 | 7, 15 | eqeltrrid 2849 | . . 3 ⊢ ((𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ ran 𝐹) → +∞ ∈ ℝ) |
17 | 16 | ex 412 | . 2 ⊢ (𝐹 ∈ dom ∫1 → (¬ 0 ∈ ran 𝐹 → +∞ ∈ ℝ)) |
18 | 2, 17 | mt3i 149 | 1 ⊢ (𝐹 ∈ dom ∫1 → 0 ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ◡ccnv 5699 dom cdm 5700 ran crn 5701 “ cima 5703 ‘cfv 6573 ℝcr 11183 0cc0 11184 +∞cpnf 11321 vol*covol 25516 volcvol 25517 ∫1citg1 25669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-sum 15735 df-rest 17482 df-topgen 17503 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-top 22921 df-topon 22938 df-bases 22974 df-cmp 23416 df-ovol 25518 df-vol 25519 df-mbf 25673 df-itg1 25674 |
This theorem is referenced by: i1fres 25760 itg1climres 25769 itg2addnclem2 37632 ftc1anclem7 37659 ftc1anc 37661 |
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