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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 25203 | . . 3 ⊢ (ℝ × {0}) ∈ dom ∫1 | |
2 | reex 11200 | . . . . . . 7 ⊢ ℝ ∈ V | |
3 | 2 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℝ ∈ V) |
4 | i1ff 25192 | . . . . . . 7 ⊢ ((ℝ × {0}) ∈ dom ∫1 → (ℝ × {0}):ℝ⟶ℝ) | |
5 | 1, 4 | mp1i 13 | . . . . . 6 ⊢ (⊤ → (ℝ × {0}):ℝ⟶ℝ) |
6 | leid 11309 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ≤ 𝑥) | |
7 | 6 | adantl 482 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 𝑥 ≤ 𝑥) |
8 | 3, 5, 7 | caofref 7698 | . . . . 5 ⊢ (⊤ → (ℝ × {0}) ∘r ≤ (ℝ × {0})) |
9 | ax-resscn 11166 | . . . . . . 7 ⊢ ℝ ⊆ ℂ | |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℝ ⊆ ℂ) |
11 | 5 | ffnd 6718 | . . . . . 6 ⊢ (⊤ → (ℝ × {0}) Fn ℝ) |
12 | 10, 11 | 0pledm 25189 | . . . . 5 ⊢ (⊤ → (0𝑝 ∘r ≤ (ℝ × {0}) ↔ (ℝ × {0}) ∘r ≤ (ℝ × {0}))) |
13 | 8, 12 | mpbird 256 | . . . 4 ⊢ (⊤ → 0𝑝 ∘r ≤ (ℝ × {0})) |
14 | 13 | mptru 1548 | . . 3 ⊢ 0𝑝 ∘r ≤ (ℝ × {0}) |
15 | itg2itg1 25253 | . . 3 ⊢ (((ℝ × {0}) ∈ dom ∫1 ∧ 0𝑝 ∘r ≤ (ℝ × {0})) → (∫2‘(ℝ × {0})) = (∫1‘(ℝ × {0}))) | |
16 | 1, 14, 15 | mp2an 690 | . 2 ⊢ (∫2‘(ℝ × {0})) = (∫1‘(ℝ × {0})) |
17 | itg10 25204 | . 2 ⊢ (∫1‘(ℝ × {0})) = 0 | |
18 | 16, 17 | eqtri 2760 | 1 ⊢ (∫2‘(ℝ × {0})) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3948 {csn 4628 class class class wbr 5148 × cxp 5674 dom cdm 5676 ⟶wf 6539 ‘cfv 6543 ∘r cofr 7668 ℂcc 11107 ℝcr 11108 0cc0 11109 ≤ cle 11248 ∫1citg1 25131 ∫2citg2 25132 0𝑝c0p 25185 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-ofr 7670 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-oi 9504 df-dju 9895 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-q 12932 df-rp 12974 df-xadd 13092 df-ioo 13327 df-ico 13329 df-icc 13330 df-fz 13484 df-fzo 13627 df-fl 13756 df-seq 13966 df-exp 14027 df-hash 14290 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-clim 15431 df-sum 15632 df-xmet 20936 df-met 20937 df-ovol 24980 df-vol 24981 df-mbf 25135 df-itg1 25136 df-itg2 25137 df-0p 25186 |
This theorem is referenced by: itg2mulc 25264 itg0 25296 itgz 25297 itgvallem3 25302 iblposlem 25308 bddmulibl 25355 iblempty 44671 |
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