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Mirrors > Home > MPE Home > Th. List > itg2monolem2 | Structured version Visualization version GIF version |
Description: Lemma for itg2mono 24357. (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 12814 | . . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
4 | fss 6501 | . . . . . . . 8 ⊢ (((𝐹‘𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹‘𝑛):ℝ⟶(0[,]+∞)) | |
5 | 2, 3, 4 | sylancl 589 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,]+∞)) |
6 | itg2cl 24336 | . . . . . . 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 6856 | . . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))):ℕ⟶ℝ*) |
9 | 8 | frnd 6494 | . . . 4 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) ⊆ ℝ*) |
10 | supxrcl 12696 | . . . 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 | eqeltrid 2894 | . 2 ⊢ (𝜑 → 𝑆 ∈ ℝ*) |
13 | itg2monolem2.7 | . . 3 ⊢ (𝜑 → 𝑃 ∈ dom ∫1) | |
14 | itg1cl 24289 | . . 3 ⊢ (𝑃 ∈ dom ∫1 → (∫1‘𝑃) ∈ ℝ) | |
15 | 13, 14 | syl 17 | . 2 ⊢ (𝜑 → (∫1‘𝑃) ∈ ℝ) |
16 | mnfxr 10687 | . . . 4 ⊢ -∞ ∈ ℝ* | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → -∞ ∈ ℝ*) |
18 | fveq2 6645 | . . . . . 6 ⊢ (𝑛 = 1 → (𝐹‘𝑛) = (𝐹‘1)) | |
19 | 18 | feq1d 6472 | . . . . 5 ⊢ (𝑛 = 1 → ((𝐹‘𝑛):ℝ⟶(0[,]+∞) ↔ (𝐹‘1):ℝ⟶(0[,]+∞))) |
20 | 5 | ralrimiva 3149 | . . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛):ℝ⟶(0[,]+∞)) |
21 | 1nn 11636 | . . . . . 6 ⊢ 1 ∈ ℕ | |
22 | 21 | a1i 11 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℕ) |
23 | 19, 20, 22 | rspcdva 3573 | . . . 4 ⊢ (𝜑 → (𝐹‘1):ℝ⟶(0[,]+∞)) |
24 | itg2cl 24336 | . . . 4 ⊢ ((𝐹‘1):ℝ⟶(0[,]+∞) → (∫2‘(𝐹‘1)) ∈ ℝ*) | |
25 | 23, 24 | syl 17 | . . 3 ⊢ (𝜑 → (∫2‘(𝐹‘1)) ∈ ℝ*) |
26 | itg2ge0 24339 | . . . . 5 ⊢ ((𝐹‘1):ℝ⟶(0[,]+∞) → 0 ≤ (∫2‘(𝐹‘1))) | |
27 | 23, 26 | syl 17 | . . . 4 ⊢ (𝜑 → 0 ≤ (∫2‘(𝐹‘1))) |
28 | mnflt0 12508 | . . . . 5 ⊢ -∞ < 0 | |
29 | 0xr 10677 | . . . . . 6 ⊢ 0 ∈ ℝ* | |
30 | xrltletr 12538 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ* ∧ (∫2‘(𝐹‘1)) ∈ ℝ*) → ((-∞ < 0 ∧ 0 ≤ (∫2‘(𝐹‘1))) → -∞ < (∫2‘(𝐹‘1)))) | |
31 | 16, 29, 25, 30 | mp3an12i 1462 | . . . . 5 ⊢ (𝜑 → ((-∞ < 0 ∧ 0 ≤ (∫2‘(𝐹‘1))) → -∞ < (∫2‘(𝐹‘1)))) |
32 | 28, 31 | mpani 695 | . . . 4 ⊢ (𝜑 → (0 ≤ (∫2‘(𝐹‘1)) → -∞ < (∫2‘(𝐹‘1)))) |
33 | 27, 32 | mpd 15 | . . 3 ⊢ (𝜑 → -∞ < (∫2‘(𝐹‘1))) |
34 | 2fveq3 6650 | . . . . . . . 8 ⊢ (𝑛 = 1 → (∫2‘(𝐹‘𝑛)) = (∫2‘(𝐹‘1))) | |
35 | eqid 2798 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) | |
36 | fvex 6658 | . . . . . . . 8 ⊢ (∫2‘(𝐹‘1)) ∈ V | |
37 | 34, 35, 36 | fvmpt 6745 | . . . . . . 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 6488 | . . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) Fn ℕ) |
40 | fnfvelrn 6825 | . . . . . . 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 589 | . . . . . 6 ⊢ (𝜑 → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘1) ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))) |
42 | 38, 41 | eqeltrrid 2895 | . . . . 5 ⊢ (𝜑 → (∫2‘(𝐹‘1)) ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))) |
43 | supxrub 12705 | . . . . 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 587 | . . . 4 ⊢ (𝜑 → (∫2‘(𝐹‘1)) ≤ sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < )) |
45 | 44, 1 | breqtrrdi 5072 | . . 3 ⊢ (𝜑 → (∫2‘(𝐹‘1)) ≤ 𝑆) |
46 | 17, 25, 12, 33, 45 | xrltletrd 12542 | . 2 ⊢ (𝜑 → -∞ < 𝑆) |
47 | 15 | rexrd 10680 | . . 3 ⊢ (𝜑 → (∫1‘𝑃) ∈ ℝ*) |
48 | itg2monolem2.9 | . . . 4 ⊢ (𝜑 → ¬ (∫1‘𝑃) ≤ 𝑆) | |
49 | xrltnle 10697 | . . . . 5 ⊢ ((𝑆 ∈ ℝ* ∧ (∫1‘𝑃) ∈ ℝ*) → (𝑆 < (∫1‘𝑃) ↔ ¬ (∫1‘𝑃) ≤ 𝑆)) | |
50 | 12, 47, 49 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝑆 < (∫1‘𝑃) ↔ ¬ (∫1‘𝑃) ≤ 𝑆)) |
51 | 48, 50 | mpbird 260 | . . 3 ⊢ (𝜑 → 𝑆 < (∫1‘𝑃)) |
52 | 12, 47, 51 | xrltled 12531 | . 2 ⊢ (𝜑 → 𝑆 ≤ (∫1‘𝑃)) |
53 | xrre 12550 | . 2 ⊢ (((𝑆 ∈ ℝ* ∧ (∫1‘𝑃) ∈ ℝ) ∧ (-∞ < 𝑆 ∧ 𝑆 ≤ (∫1‘𝑃))) → 𝑆 ∈ ℝ) | |
54 | 12, 15, 46, 52, 53 | syl22anc 837 | 1 ⊢ (𝜑 → 𝑆 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ran crn 5520 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ∘r cofr 7388 supcsup 8888 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 +∞cpnf 10661 -∞cmnf 10662 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 ℕcn 11625 [,)cico 12728 [,]cicc 12729 MblFncmbf 24218 ∫1citg1 24219 ∫2citg2 24220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-ofr 7390 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xadd 12496 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-xmet 20084 df-met 20085 df-ovol 24068 df-vol 24069 df-mbf 24223 df-itg1 24224 df-itg2 24225 |
This theorem is referenced by: itg2monolem3 24356 |
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