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Mirrors > Home > MPE Home > Th. List > itg20 | Structured version Visualization version GIF version |
Description: The integral of the zero function. (Contributed by Mario Carneiro, 28-Jun-2014.) |
Ref | Expression |
---|---|
itg20 | ⊢ (∫2‘(ℝ × {0})) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | i1f0 25186 | . . 3 ⊢ (ℝ × {0}) ∈ dom ∫1 | |
2 | reex 11197 | . . . . . . 7 ⊢ ℝ ∈ V | |
3 | 2 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℝ ∈ V) |
4 | i1ff 25175 | . . . . . . 7 ⊢ ((ℝ × {0}) ∈ dom ∫1 → (ℝ × {0}):ℝ⟶ℝ) | |
5 | 1, 4 | mp1i 13 | . . . . . 6 ⊢ (⊤ → (ℝ × {0}):ℝ⟶ℝ) |
6 | leid 11306 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ≤ 𝑥) | |
7 | 6 | adantl 483 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 𝑥 ≤ 𝑥) |
8 | 3, 5, 7 | caofref 7694 | . . . . 5 ⊢ (⊤ → (ℝ × {0}) ∘r ≤ (ℝ × {0})) |
9 | ax-resscn 11163 | . . . . . . 7 ⊢ ℝ ⊆ ℂ | |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℝ ⊆ ℂ) |
11 | 5 | ffnd 6715 | . . . . . 6 ⊢ (⊤ → (ℝ × {0}) Fn ℝ) |
12 | 10, 11 | 0pledm 25172 | . . . . 5 ⊢ (⊤ → (0𝑝 ∘r ≤ (ℝ × {0}) ↔ (ℝ × {0}) ∘r ≤ (ℝ × {0}))) |
13 | 8, 12 | mpbird 257 | . . . 4 ⊢ (⊤ → 0𝑝 ∘r ≤ (ℝ × {0})) |
14 | 13 | mptru 1549 | . . 3 ⊢ 0𝑝 ∘r ≤ (ℝ × {0}) |
15 | itg2itg1 25236 | . . 3 ⊢ (((ℝ × {0}) ∈ dom ∫1 ∧ 0𝑝 ∘r ≤ (ℝ × {0})) → (∫2‘(ℝ × {0})) = (∫1‘(ℝ × {0}))) | |
16 | 1, 14, 15 | mp2an 691 | . 2 ⊢ (∫2‘(ℝ × {0})) = (∫1‘(ℝ × {0})) |
17 | itg10 25187 | . 2 ⊢ (∫1‘(ℝ × {0})) = 0 | |
18 | 16, 17 | eqtri 2761 | 1 ⊢ (∫2‘(ℝ × {0})) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 Vcvv 3475 ⊆ wss 3947 {csn 4627 class class class wbr 5147 × cxp 5673 dom cdm 5675 ⟶wf 6536 ‘cfv 6540 ∘r cofr 7664 ℂcc 11104 ℝcr 11105 0cc0 11106 ≤ cle 11245 ∫1citg1 25114 ∫2citg2 25115 0𝑝c0p 25168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7665 df-ofr 7666 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-xadd 13089 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-xmet 20922 df-met 20923 df-ovol 24963 df-vol 24964 df-mbf 25118 df-itg1 25119 df-itg2 25120 df-0p 25169 |
This theorem is referenced by: itg2mulc 25247 itg0 25279 itgz 25280 itgvallem3 25285 iblposlem 25291 bddmulibl 25338 iblempty 44616 |
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