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Mirrors > Home > MPE Home > Th. List > iblposlem | Structured version Visualization version GIF version |
Description: Lemma for iblpos 24396. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
Ref | Expression |
---|---|
iblrelem.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
iblpos.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵) |
Ref | Expression |
---|---|
iblposlem | ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iblpos.2 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵) | |
2 | iblrelem.1 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
3 | 2 | le0neg2d 11201 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ 𝐵 ↔ -𝐵 ≤ 0)) |
4 | 1, 3 | mpbid 235 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ≤ 0) |
5 | 4 | adantrr 716 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ≤ 0) |
6 | simprr 772 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → 0 ≤ -𝐵) | |
7 | 2 | adantrr 716 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → 𝐵 ∈ ℝ) |
8 | 7 | renegcld 11056 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ ℝ) |
9 | 0re 10632 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
10 | letri3 10715 | . . . . . . . . 9 ⊢ ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → (-𝐵 = 0 ↔ (-𝐵 ≤ 0 ∧ 0 ≤ -𝐵))) | |
11 | 8, 9, 10 | sylancl 589 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → (-𝐵 = 0 ↔ (-𝐵 ≤ 0 ∧ 0 ≤ -𝐵))) |
12 | 5, 6, 11 | mpbir2and 712 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 = 0) |
13 | 12 | ifeq1da 4455 | . . . . . 6 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), 0, 0)) |
14 | ifid 4464 | . . . . . 6 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), 0, 0) = 0 | |
15 | 13, 14 | eqtrdi 2849 | . . . . 5 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0) = 0) |
16 | 15 | mpteq2dv 5126 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)) = (𝑥 ∈ ℝ ↦ 0)) |
17 | fconstmpt 5578 | . . . 4 ⊢ (ℝ × {0}) = (𝑥 ∈ ℝ ↦ 0) | |
18 | 16, 17 | eqtr4di 2851 | . . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)) = (ℝ × {0})) |
19 | 18 | fveq2d 6649 | . 2 ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) = (∫2‘(ℝ × {0}))) |
20 | itg20 24341 | . 2 ⊢ (∫2‘(ℝ × {0})) = 0 | |
21 | 19, 20 | eqtrdi 2849 | 1 ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ifcif 4425 {csn 4525 class class class wbr 5030 ↦ cmpt 5110 × cxp 5517 ‘cfv 6324 ℝcr 10525 0cc0 10526 ≤ cle 10665 -cneg 10860 ∫2citg2 24220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-ofr 7390 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xadd 12496 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-xmet 20084 df-met 20085 df-ovol 24068 df-vol 24069 df-mbf 24223 df-itg1 24224 df-itg2 24225 df-0p 24274 |
This theorem is referenced by: iblpos 24396 itgposval 24399 |
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