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Mirrors > Home > MPE Home > Th. List > iblsub | Structured version Visualization version GIF version |
Description: Subtract two integrals over the same domain. (Contributed by Mario Carneiro, 25-Aug-2014.) |
Ref | Expression |
---|---|
itgadd.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
itgadd.2 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) |
itgadd.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉) |
itgadd.4 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1) |
Ref | Expression |
---|---|
iblsub | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) ∈ 𝐿1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itgadd.2 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) | |
2 | iblmbf 25710 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) |
4 | itgadd.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
5 | 3, 4 | mbfmptcl 25578 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
6 | itgadd.4 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1) | |
7 | iblmbf 25710 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn) |
9 | itgadd.3 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉) | |
10 | 8, 9 | mbfmptcl 25578 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ) |
11 | 5, 10 | negsubd 11608 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + -𝐶) = (𝐵 − 𝐶)) |
12 | 11 | mpteq2dva 5248 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + -𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶))) |
13 | 10 | negcld 11589 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐶 ∈ ℂ) |
14 | 9, 6 | iblneg 25745 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -𝐶) ∈ 𝐿1) |
15 | 5, 1, 13, 14 | ibladd 25763 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + -𝐶)) ∈ 𝐿1) |
16 | 12, 15 | eqeltrrd 2830 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) ∈ 𝐿1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 ↦ cmpt 5231 (class class class)co 7420 ℂcc 11137 + caddc 11142 − cmin 11475 -cneg 11476 MblFncmbf 25556 𝐿1cibl 25559 |
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 9665 ax-cc 10459 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 |
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 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-omul 8492 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9435 df-sup 9466 df-inf 9467 df-oi 9534 df-dju 9925 df-card 9963 df-acn 9966 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ioo 13361 df-ioc 13362 df-ico 13363 df-icc 13364 df-fz 13518 df-fzo 13661 df-fl 13790 df-seq 14000 df-exp 14060 df-hash 14323 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-clim 15465 df-rlim 15466 df-sum 15666 df-rest 17404 df-topgen 17425 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-top 22809 df-topon 22826 df-bases 22862 df-cmp 23304 df-ovol 25406 df-vol 25407 df-mbf 25561 df-itg1 25562 df-itg2 25563 df-ibl 25564 df-0p 25612 |
This theorem is referenced by: ftc1lem4 25987 iblulm 26356 itgulm 26357 areaquad 42644 |
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