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Mirrors > Home > MPE Home > Th. List > ftc1lem2 | Structured version Visualization version GIF version |
Description: Lemma for ftc1 25204. (Contributed by Mario Carneiro, 12-Aug-2014.) |
Ref | Expression |
---|---|
ftc1.g | ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) |
ftc1.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ftc1.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ftc1.le | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
ftc1.s | ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷) |
ftc1.d | ⊢ (𝜑 → 𝐷 ⊆ ℝ) |
ftc1.i | ⊢ (𝜑 → 𝐹 ∈ 𝐿1) |
ftc1a.f | ⊢ (𝜑 → 𝐹:𝐷⟶ℂ) |
Ref | Expression |
---|---|
ftc1lem2 | ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvexd 6786 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑡 ∈ (𝐴(,)𝑥)) → (𝐹‘𝑡) ∈ V) | |
2 | ftc1.b | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | 2 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ) |
4 | 3 | rexrd 11026 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ*) |
5 | ftc1.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
6 | elicc2 13143 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))) | |
7 | 5, 2, 6 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))) |
8 | 7 | biimpa 477 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)) |
9 | 8 | simp3d 1143 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵) |
10 | iooss2 13114 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑥 ≤ 𝐵) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵)) | |
11 | 4, 9, 10 | syl2anc 584 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵)) |
12 | ftc1.s | . . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷) | |
13 | 12 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝐵) ⊆ 𝐷) |
14 | 11, 13 | sstrd 3936 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ⊆ 𝐷) |
15 | ioombl 24727 | . . . . 5 ⊢ (𝐴(,)𝑥) ∈ dom vol | |
16 | 15 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ∈ dom vol) |
17 | fvexd 6786 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑡 ∈ 𝐷) → (𝐹‘𝑡) ∈ V) | |
18 | ftc1a.f | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝐷⟶ℂ) | |
19 | 18 | feqmptd 6834 | . . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝐹‘𝑡))) |
20 | ftc1.i | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝐿1) | |
21 | 19, 20 | eqeltrrd 2842 | . . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝐷 ↦ (𝐹‘𝑡)) ∈ 𝐿1) |
22 | 21 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ 𝐷 ↦ (𝐹‘𝑡)) ∈ 𝐿1) |
23 | 14, 16, 17, 22 | iblss 24967 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝑥) ↦ (𝐹‘𝑡)) ∈ 𝐿1) |
24 | 1, 23 | itgcl 24946 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡 ∈ ℂ) |
25 | ftc1.g | . 2 ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) | |
26 | 24, 25 | fmptd 6985 | 1 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 Vcvv 3431 ⊆ wss 3892 class class class wbr 5079 ↦ cmpt 5162 dom cdm 5590 ⟶wf 6428 ‘cfv 6432 (class class class)co 7271 ℂcc 10870 ℝcr 10871 ℝ*cxr 11009 ≤ cle 11011 (,)cioo 13078 [,]cicc 13081 volcvol 24625 𝐿1cibl 24779 ∫citg 24780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-inf2 9377 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-ofr 7528 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-2o 8289 df-er 8481 df-map 8600 df-pm 8601 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-sup 9179 df-inf 9180 df-oi 9247 df-dju 9660 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-n0 12234 df-z 12320 df-uz 12582 df-q 12688 df-rp 12730 df-xadd 12848 df-ioo 13082 df-ico 13084 df-icc 13085 df-fz 13239 df-fzo 13382 df-fl 13510 df-mod 13588 df-seq 13720 df-exp 13781 df-hash 14043 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-clim 15195 df-rlim 15196 df-sum 15396 df-xmet 20588 df-met 20589 df-ovol 24626 df-vol 24627 df-mbf 24781 df-itg1 24782 df-itg2 24783 df-ibl 24784 df-itg 24785 |
This theorem is referenced by: ftc1a 25199 ftc1lem5 25202 ftc1lem6 25203 ftc1 25204 ftc1cn 25205 ftc1cnnc 35845 ftc1anc 35854 |
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