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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 24282 | . . 3 ⊢ (ℝ × {0}) ∈ dom ∫1 | |
2 | reex 10622 | . . . . . . 7 ⊢ ℝ ∈ V | |
3 | 2 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℝ ∈ V) |
4 | i1ff 24271 | . . . . . . 7 ⊢ ((ℝ × {0}) ∈ dom ∫1 → (ℝ × {0}):ℝ⟶ℝ) | |
5 | 1, 4 | mp1i 13 | . . . . . 6 ⊢ (⊤ → (ℝ × {0}):ℝ⟶ℝ) |
6 | leid 10730 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ≤ 𝑥) | |
7 | 6 | adantl 484 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 𝑥 ≤ 𝑥) |
8 | 3, 5, 7 | caofref 7429 | . . . . 5 ⊢ (⊤ → (ℝ × {0}) ∘r ≤ (ℝ × {0})) |
9 | ax-resscn 10588 | . . . . . . 7 ⊢ ℝ ⊆ ℂ | |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℝ ⊆ ℂ) |
11 | 5 | ffnd 6510 | . . . . . 6 ⊢ (⊤ → (ℝ × {0}) Fn ℝ) |
12 | 10, 11 | 0pledm 24268 | . . . . 5 ⊢ (⊤ → (0𝑝 ∘r ≤ (ℝ × {0}) ↔ (ℝ × {0}) ∘r ≤ (ℝ × {0}))) |
13 | 8, 12 | mpbird 259 | . . . 4 ⊢ (⊤ → 0𝑝 ∘r ≤ (ℝ × {0})) |
14 | 13 | mptru 1540 | . . 3 ⊢ 0𝑝 ∘r ≤ (ℝ × {0}) |
15 | itg2itg1 24331 | . . 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 24283 | . 2 ⊢ (∫1‘(ℝ × {0})) = 0 | |
18 | 16, 17 | eqtri 2844 | 1 ⊢ (∫2‘(ℝ × {0})) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ⊤wtru 1534 ∈ wcel 2110 Vcvv 3495 ⊆ wss 3936 {csn 4561 class class class wbr 5059 × cxp 5548 dom cdm 5550 ⟶wf 6346 ‘cfv 6350 ∘r cofr 7402 ℂcc 10529 ℝcr 10530 0cc0 10531 ≤ cle 10670 ∫1citg1 24210 ∫2citg2 24211 0𝑝c0p 24264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-disj 5025 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-ofr 7404 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xadd 12502 df-ioo 12736 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-xmet 20532 df-met 20533 df-ovol 24059 df-vol 24060 df-mbf 24214 df-itg1 24215 df-itg2 24216 df-0p 24265 |
This theorem is referenced by: itg2mulc 24342 itg0 24374 itgz 24375 itgvallem3 24380 iblposlem 24386 bddmulibl 24433 iblempty 42242 |
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