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Mirrors > Home > MPE Home > Th. List > itgless | Structured version Visualization version GIF version |
Description: Expand the integral of a nonnegative function. (Contributed by Mario Carneiro, 31-Aug-2014.) |
Ref | Expression |
---|---|
itgless.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
itgless.2 | ⊢ (𝜑 → 𝐴 ∈ dom vol) |
itgless.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℝ) |
itgless.4 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 0 ≤ 𝐶) |
itgless.5 | ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿1) |
Ref | Expression |
---|---|
itgless | ⊢ (𝜑 → ∫𝐴𝐶 d𝑥 ≤ ∫𝐵𝐶 d𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itgless.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | itgss2 25863 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵if(𝑥 ∈ 𝐴, 𝐶, 0) d𝑥) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵if(𝑥 ∈ 𝐴, 𝐶, 0) d𝑥) |
4 | itgless.5 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿1) | |
5 | iblmbf 25817 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿1 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn) |
7 | itgless.3 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℝ) | |
8 | 6, 7 | mbfdm2 25686 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ dom vol) |
9 | 1 | sselda 3995 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) |
10 | 9, 7 | syldan 591 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ) |
11 | 0re 11261 | . . . . 5 ⊢ 0 ∈ ℝ | |
12 | ifcl 4576 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑥 ∈ 𝐴, 𝐶, 0) ∈ ℝ) | |
13 | 10, 11, 12 | sylancl 586 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, 𝐶, 0) ∈ ℝ) |
14 | eldifn 4142 | . . . . . 6 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴) | |
15 | 14 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ 𝐴)) → ¬ 𝑥 ∈ 𝐴) |
16 | 15 | iffalsed 4542 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ 𝐴)) → if(𝑥 ∈ 𝐴, 𝐶, 0) = 0) |
17 | iftrue 4537 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, 𝐶, 0) = 𝐶) | |
18 | 17 | mpteq2ia 5251 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐴, 𝐶, 0)) = (𝑥 ∈ 𝐴 ↦ 𝐶) |
19 | itgless.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ dom vol) | |
20 | 1, 19, 7, 4 | iblss 25855 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1) |
21 | 18, 20 | eqeltrid 2843 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐴, 𝐶, 0)) ∈ 𝐿1) |
22 | 1, 8, 13, 16, 21 | iblss2 25856 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ if(𝑥 ∈ 𝐴, 𝐶, 0)) ∈ 𝐿1) |
23 | 7, 11, 12 | sylancl 586 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → if(𝑥 ∈ 𝐴, 𝐶, 0) ∈ ℝ) |
24 | 7 | leidd 11827 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ≤ 𝐶) |
25 | itgless.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 0 ≤ 𝐶) | |
26 | breq1 5151 | . . . . 5 ⊢ (𝐶 = if(𝑥 ∈ 𝐴, 𝐶, 0) → (𝐶 ≤ 𝐶 ↔ if(𝑥 ∈ 𝐴, 𝐶, 0) ≤ 𝐶)) | |
27 | breq1 5151 | . . . . 5 ⊢ (0 = if(𝑥 ∈ 𝐴, 𝐶, 0) → (0 ≤ 𝐶 ↔ if(𝑥 ∈ 𝐴, 𝐶, 0) ≤ 𝐶)) | |
28 | 26, 27 | ifboth 4570 | . . . 4 ⊢ ((𝐶 ≤ 𝐶 ∧ 0 ≤ 𝐶) → if(𝑥 ∈ 𝐴, 𝐶, 0) ≤ 𝐶) |
29 | 24, 25, 28 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → if(𝑥 ∈ 𝐴, 𝐶, 0) ≤ 𝐶) |
30 | 22, 4, 23, 7, 29 | itgle 25860 | . 2 ⊢ (𝜑 → ∫𝐵if(𝑥 ∈ 𝐴, 𝐶, 0) d𝑥 ≤ ∫𝐵𝐶 d𝑥) |
31 | 3, 30 | eqbrtrd 5170 | 1 ⊢ (𝜑 → ∫𝐴𝐶 d𝑥 ≤ ∫𝐵𝐶 d𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∖ cdif 3960 ⊆ wss 3963 ifcif 4531 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ℝcr 11152 0cc0 11153 ≤ cle 11294 volcvol 25512 MblFncmbf 25663 𝐿1cibl 25666 ∫citg 25667 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-xadd 13153 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-xmet 21375 df-met 21376 df-ovol 25513 df-vol 25514 df-mbf 25668 df-itg1 25669 df-itg2 25670 df-ibl 25671 df-itg 25672 df-0p 25719 |
This theorem is referenced by: (None) |
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