![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > itg2monolem2 | Structured version Visualization version GIF version |
Description: Lemma for itg2mono 24047. (Contributed by Mario Carneiro, 16-Aug-2014.) |
Ref | Expression |
---|---|
itg2mono.1 | ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) |
itg2mono.2 | ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn) |
itg2mono.3 | ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,)+∞)) |
itg2mono.4 | ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘𝑟 ≤ (𝐹‘(𝑛 + 1))) |
itg2mono.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) |
itg2mono.6 | ⊢ 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) |
itg2monolem2.7 | ⊢ (𝜑 → 𝑃 ∈ dom ∫1) |
itg2monolem2.8 | ⊢ (𝜑 → 𝑃 ∘𝑟 ≤ 𝐺) |
itg2monolem2.9 | ⊢ (𝜑 → ¬ (∫1‘𝑃) ≤ 𝑆) |
Ref | Expression |
---|---|
itg2monolem2 | ⊢ (𝜑 → 𝑆 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itg2mono.6 | . . 3 ⊢ 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) | |
2 | itg2mono.3 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,)+∞)) | |
3 | icossicc 12633 | . . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
4 | fss 6351 | . . . . . . . 8 ⊢ (((𝐹‘𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹‘𝑛):ℝ⟶(0[,]+∞)) | |
5 | 2, 3, 4 | sylancl 577 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,]+∞)) |
6 | itg2cl 24026 | . . . . . . 7 ⊢ ((𝐹‘𝑛):ℝ⟶(0[,]+∞) → (∫2‘(𝐹‘𝑛)) ∈ ℝ*) | |
7 | 5, 6 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∫2‘(𝐹‘𝑛)) ∈ ℝ*) |
8 | 7 | fmpttd 6696 | . . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))):ℕ⟶ℝ*) |
9 | 8 | frnd 6345 | . . . 4 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) ⊆ ℝ*) |
10 | supxrcl 12517 | . . . 4 ⊢ (ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) ⊆ ℝ* → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) ∈ ℝ*) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) ∈ ℝ*) |
12 | 1, 11 | syl5eqel 2864 | . 2 ⊢ (𝜑 → 𝑆 ∈ ℝ*) |
13 | itg2monolem2.7 | . . 3 ⊢ (𝜑 → 𝑃 ∈ dom ∫1) | |
14 | itg1cl 23979 | . . 3 ⊢ (𝑃 ∈ dom ∫1 → (∫1‘𝑃) ∈ ℝ) | |
15 | 13, 14 | syl 17 | . 2 ⊢ (𝜑 → (∫1‘𝑃) ∈ ℝ) |
16 | mnfxr 10490 | . . . 4 ⊢ -∞ ∈ ℝ* | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → -∞ ∈ ℝ*) |
18 | fveq2 6493 | . . . . . 6 ⊢ (𝑛 = 1 → (𝐹‘𝑛) = (𝐹‘1)) | |
19 | 18 | feq1d 6323 | . . . . 5 ⊢ (𝑛 = 1 → ((𝐹‘𝑛):ℝ⟶(0[,]+∞) ↔ (𝐹‘1):ℝ⟶(0[,]+∞))) |
20 | 5 | ralrimiva 3126 | . . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛):ℝ⟶(0[,]+∞)) |
21 | 1nn 11444 | . . . . . 6 ⊢ 1 ∈ ℕ | |
22 | 21 | a1i 11 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℕ) |
23 | 19, 20, 22 | rspcdva 3535 | . . . 4 ⊢ (𝜑 → (𝐹‘1):ℝ⟶(0[,]+∞)) |
24 | itg2cl 24026 | . . . 4 ⊢ ((𝐹‘1):ℝ⟶(0[,]+∞) → (∫2‘(𝐹‘1)) ∈ ℝ*) | |
25 | 23, 24 | syl 17 | . . 3 ⊢ (𝜑 → (∫2‘(𝐹‘1)) ∈ ℝ*) |
26 | itg2ge0 24029 | . . . . 5 ⊢ ((𝐹‘1):ℝ⟶(0[,]+∞) → 0 ≤ (∫2‘(𝐹‘1))) | |
27 | 23, 26 | syl 17 | . . . 4 ⊢ (𝜑 → 0 ≤ (∫2‘(𝐹‘1))) |
28 | mnflt0 12330 | . . . . 5 ⊢ -∞ < 0 | |
29 | 0xr 10479 | . . . . . 6 ⊢ 0 ∈ ℝ* | |
30 | xrltletr 12360 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ* ∧ (∫2‘(𝐹‘1)) ∈ ℝ*) → ((-∞ < 0 ∧ 0 ≤ (∫2‘(𝐹‘1))) → -∞ < (∫2‘(𝐹‘1)))) | |
31 | 16, 29, 25, 30 | mp3an12i 1444 | . . . . 5 ⊢ (𝜑 → ((-∞ < 0 ∧ 0 ≤ (∫2‘(𝐹‘1))) → -∞ < (∫2‘(𝐹‘1)))) |
32 | 28, 31 | mpani 683 | . . . 4 ⊢ (𝜑 → (0 ≤ (∫2‘(𝐹‘1)) → -∞ < (∫2‘(𝐹‘1)))) |
33 | 27, 32 | mpd 15 | . . 3 ⊢ (𝜑 → -∞ < (∫2‘(𝐹‘1))) |
34 | 2fveq3 6498 | . . . . . . . 8 ⊢ (𝑛 = 1 → (∫2‘(𝐹‘𝑛)) = (∫2‘(𝐹‘1))) | |
35 | eqid 2772 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) | |
36 | fvex 6506 | . . . . . . . 8 ⊢ (∫2‘(𝐹‘1)) ∈ V | |
37 | 34, 35, 36 | fvmpt 6589 | . . . . . . 7 ⊢ (1 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘1) = (∫2‘(𝐹‘1))) |
38 | 21, 37 | ax-mp 5 | . . . . . 6 ⊢ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘1) = (∫2‘(𝐹‘1)) |
39 | 8 | ffnd 6339 | . . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) Fn ℕ) |
40 | fnfvelrn 6667 | . . . . . . 7 ⊢ (((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) Fn ℕ ∧ 1 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘1) ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))) | |
41 | 39, 21, 40 | sylancl 577 | . . . . . 6 ⊢ (𝜑 → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘1) ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))) |
42 | 38, 41 | syl5eqelr 2865 | . . . . 5 ⊢ (𝜑 → (∫2‘(𝐹‘1)) ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))) |
43 | supxrub 12526 | . . . . 5 ⊢ ((ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) ⊆ ℝ* ∧ (∫2‘(𝐹‘1)) ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))) → (∫2‘(𝐹‘1)) ≤ sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < )) | |
44 | 9, 42, 43 | syl2anc 576 | . . . 4 ⊢ (𝜑 → (∫2‘(𝐹‘1)) ≤ sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < )) |
45 | 44, 1 | syl6breqr 4965 | . . 3 ⊢ (𝜑 → (∫2‘(𝐹‘1)) ≤ 𝑆) |
46 | 17, 25, 12, 33, 45 | xrltletrd 12364 | . 2 ⊢ (𝜑 → -∞ < 𝑆) |
47 | 15 | rexrd 10482 | . . 3 ⊢ (𝜑 → (∫1‘𝑃) ∈ ℝ*) |
48 | itg2monolem2.9 | . . . 4 ⊢ (𝜑 → ¬ (∫1‘𝑃) ≤ 𝑆) | |
49 | xrltnle 10500 | . . . . 5 ⊢ ((𝑆 ∈ ℝ* ∧ (∫1‘𝑃) ∈ ℝ*) → (𝑆 < (∫1‘𝑃) ↔ ¬ (∫1‘𝑃) ≤ 𝑆)) | |
50 | 12, 47, 49 | syl2anc 576 | . . . 4 ⊢ (𝜑 → (𝑆 < (∫1‘𝑃) ↔ ¬ (∫1‘𝑃) ≤ 𝑆)) |
51 | 48, 50 | mpbird 249 | . . 3 ⊢ (𝜑 → 𝑆 < (∫1‘𝑃)) |
52 | 12, 47, 51 | xrltled 12353 | . 2 ⊢ (𝜑 → 𝑆 ≤ (∫1‘𝑃)) |
53 | xrre 12372 | . 2 ⊢ (((𝑆 ∈ ℝ* ∧ (∫1‘𝑃) ∈ ℝ) ∧ (-∞ < 𝑆 ∧ 𝑆 ≤ (∫1‘𝑃))) → 𝑆 ∈ ℝ) | |
54 | 12, 15, 46, 52, 53 | syl22anc 826 | 1 ⊢ (𝜑 → 𝑆 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ∀wral 3082 ∃wrex 3083 ⊆ wss 3825 class class class wbr 4923 ↦ cmpt 5002 dom cdm 5400 ran crn 5401 Fn wfn 6177 ⟶wf 6178 ‘cfv 6182 (class class class)co 6970 ∘𝑟 cofr 7220 supcsup 8691 ℝcr 10326 0cc0 10327 1c1 10328 + caddc 10330 +∞cpnf 10463 -∞cmnf 10464 ℝ*cxr 10465 < clt 10466 ≤ cle 10467 ℕcn 11431 [,)cico 12549 [,]cicc 12550 MblFncmbf 23908 ∫1citg1 23909 ∫2citg2 23910 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-inf2 8890 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-se 5360 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-ofr 7222 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-2o 7898 df-oadd 7901 df-er 8081 df-map 8200 df-pm 8201 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-sup 8693 df-inf 8694 df-oi 8761 df-dju 9116 df-card 9154 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-n0 11701 df-z 11787 df-uz 12052 df-q 12156 df-rp 12198 df-xadd 12318 df-ioo 12551 df-ico 12553 df-icc 12554 df-fz 12702 df-fzo 12843 df-fl 12970 df-seq 13178 df-exp 13238 df-hash 13499 df-cj 14309 df-re 14310 df-im 14311 df-sqrt 14445 df-abs 14446 df-clim 14696 df-sum 14894 df-xmet 20230 df-met 20231 df-ovol 23758 df-vol 23759 df-mbf 23913 df-itg1 23914 df-itg2 23915 |
This theorem is referenced by: itg2monolem3 24046 |
Copyright terms: Public domain | W3C validator |