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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 25615 | . . 3 ⊢ (ℝ × {0}) ∈ dom ∫1 | |
2 | reex 11229 | . . . . . . 7 ⊢ ℝ ∈ V | |
3 | 2 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℝ ∈ V) |
4 | i1ff 25604 | . . . . . . 7 ⊢ ((ℝ × {0}) ∈ dom ∫1 → (ℝ × {0}):ℝ⟶ℝ) | |
5 | 1, 4 | mp1i 13 | . . . . . 6 ⊢ (⊤ → (ℝ × {0}):ℝ⟶ℝ) |
6 | leid 11340 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → 𝑥 ≤ 𝑥) | |
7 | 6 | adantl 481 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 𝑥 ≤ 𝑥) |
8 | 3, 5, 7 | caofref 7714 | . . . . 5 ⊢ (⊤ → (ℝ × {0}) ∘r ≤ (ℝ × {0})) |
9 | ax-resscn 11195 | . . . . . . 7 ⊢ ℝ ⊆ ℂ | |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (⊤ → ℝ ⊆ ℂ) |
11 | 5 | ffnd 6723 | . . . . . 6 ⊢ (⊤ → (ℝ × {0}) Fn ℝ) |
12 | 10, 11 | 0pledm 25601 | . . . . 5 ⊢ (⊤ → (0𝑝 ∘r ≤ (ℝ × {0}) ↔ (ℝ × {0}) ∘r ≤ (ℝ × {0}))) |
13 | 8, 12 | mpbird 257 | . . . 4 ⊢ (⊤ → 0𝑝 ∘r ≤ (ℝ × {0})) |
14 | 13 | mptru 1541 | . . 3 ⊢ 0𝑝 ∘r ≤ (ℝ × {0}) |
15 | itg2itg1 25665 | . . 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 25616 | . 2 ⊢ (∫1‘(ℝ × {0})) = 0 | |
18 | 16, 17 | eqtri 2756 | 1 ⊢ (∫2‘(ℝ × {0})) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ⊤wtru 1535 ∈ wcel 2099 Vcvv 3471 ⊆ wss 3947 {csn 4629 class class class wbr 5148 × cxp 5676 dom cdm 5678 ⟶wf 6544 ‘cfv 6548 ∘r cofr 7684 ℂcc 11136 ℝcr 11137 0cc0 11138 ≤ cle 11279 ∫1citg1 25543 ∫2citg2 25544 0𝑝c0p 25597 |
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 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-ofr 7686 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-map 8846 df-pm 8847 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-sup 9465 df-inf 9466 df-oi 9533 df-dju 9924 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-q 12963 df-rp 13007 df-xadd 13125 df-ioo 13360 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-sum 15665 df-xmet 21271 df-met 21272 df-ovol 25392 df-vol 25393 df-mbf 25547 df-itg1 25548 df-itg2 25549 df-0p 25598 |
This theorem is referenced by: itg2mulc 25676 itg0 25708 itgz 25709 itgvallem3 25714 iblposlem 25720 bddmulibl 25767 iblempty 45353 |
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