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Mirrors > Home > MPE Home > Th. List > iblposlem | Structured version Visualization version GIF version |
Description: Lemma for iblpos 25807. (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 11824 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ 𝐵 ↔ -𝐵 ≤ 0)) |
4 | 1, 3 | mpbid 231 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ≤ 0) |
5 | 4 | adantrr 715 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ≤ 0) |
6 | simprr 771 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → 0 ≤ -𝐵) | |
7 | 2 | adantrr 715 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → 𝐵 ∈ ℝ) |
8 | 7 | renegcld 11679 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ ℝ) |
9 | 0re 11254 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
10 | letri3 11337 | . . . . . . . . 9 ⊢ ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → (-𝐵 = 0 ↔ (-𝐵 ≤ 0 ∧ 0 ≤ -𝐵))) | |
11 | 8, 9, 10 | sylancl 584 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → (-𝐵 = 0 ↔ (-𝐵 ≤ 0 ∧ 0 ≤ -𝐵))) |
12 | 5, 6, 11 | mpbir2and 711 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 = 0) |
13 | 12 | ifeq1da 4554 | . . . . . 6 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), 0, 0)) |
14 | ifid 4563 | . . . . . 6 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), 0, 0) = 0 | |
15 | 13, 14 | eqtrdi 2782 | . . . . 5 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0) = 0) |
16 | 15 | mpteq2dv 5245 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)) = (𝑥 ∈ ℝ ↦ 0)) |
17 | fconstmpt 5734 | . . . 4 ⊢ (ℝ × {0}) = (𝑥 ∈ ℝ ↦ 0) | |
18 | 16, 17 | eqtr4di 2784 | . . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)) = (ℝ × {0})) |
19 | 18 | fveq2d 6894 | . 2 ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) = (∫2‘(ℝ × {0}))) |
20 | itg20 25752 | . 2 ⊢ (∫2‘(ℝ × {0})) = 0 | |
21 | 19, 20 | eqtrdi 2782 | 1 ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ifcif 4523 {csn 4623 class class class wbr 5143 ↦ cmpt 5226 × cxp 5670 ‘cfv 6543 ℝcr 11145 0cc0 11146 ≤ cle 11287 -cneg 11483 ∫2citg2 25630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-inf2 9674 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-ofr 7680 df-om 7866 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8723 df-map 8846 df-pm 8847 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-sup 9475 df-inf 9476 df-oi 9543 df-dju 9934 df-card 9972 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12256 df-2 12318 df-3 12319 df-n0 12516 df-z 12602 df-uz 12866 df-q 12976 df-rp 13020 df-xadd 13138 df-ioo 13373 df-ico 13375 df-icc 13376 df-fz 13530 df-fzo 13673 df-fl 13803 df-seq 14013 df-exp 14073 df-hash 14340 df-cj 15096 df-re 15097 df-im 15098 df-sqrt 15232 df-abs 15233 df-clim 15482 df-sum 15683 df-xmet 21329 df-met 21330 df-ovol 25478 df-vol 25479 df-mbf 25633 df-itg1 25634 df-itg2 25635 df-0p 25684 |
This theorem is referenced by: iblpos 25807 itgposval 25810 |
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