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Mirrors > Home > MPE Home > Th. List > itg2monolem2 | Structured version Visualization version GIF version |
Description: Lemma for itg2mono 25727. (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 13448 | . . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
4 | fss 6739 | . . . . . . . 8 ⊢ (((𝐹‘𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹‘𝑛):ℝ⟶(0[,]+∞)) | |
5 | 2, 3, 4 | sylancl 584 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,]+∞)) |
6 | itg2cl 25706 | . . . . . . 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 7124 | . . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))):ℕ⟶ℝ*) |
9 | 8 | frnd 6731 | . . . 4 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) ⊆ ℝ*) |
10 | supxrcl 13329 | . . . 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 2829 | . 2 ⊢ (𝜑 → 𝑆 ∈ ℝ*) |
13 | itg2monolem2.7 | . . 3 ⊢ (𝜑 → 𝑃 ∈ dom ∫1) | |
14 | itg1cl 25658 | . . 3 ⊢ (𝑃 ∈ dom ∫1 → (∫1‘𝑃) ∈ ℝ) | |
15 | 13, 14 | syl 17 | . 2 ⊢ (𝜑 → (∫1‘𝑃) ∈ ℝ) |
16 | mnfxr 11303 | . . . 4 ⊢ -∞ ∈ ℝ* | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → -∞ ∈ ℝ*) |
18 | fveq2 6896 | . . . . . 6 ⊢ (𝑛 = 1 → (𝐹‘𝑛) = (𝐹‘1)) | |
19 | 18 | feq1d 6708 | . . . . 5 ⊢ (𝑛 = 1 → ((𝐹‘𝑛):ℝ⟶(0[,]+∞) ↔ (𝐹‘1):ℝ⟶(0[,]+∞))) |
20 | 5 | ralrimiva 3135 | . . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛):ℝ⟶(0[,]+∞)) |
21 | 1nn 12256 | . . . . . 6 ⊢ 1 ∈ ℕ | |
22 | 21 | a1i 11 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℕ) |
23 | 19, 20, 22 | rspcdva 3607 | . . . 4 ⊢ (𝜑 → (𝐹‘1):ℝ⟶(0[,]+∞)) |
24 | itg2cl 25706 | . . . 4 ⊢ ((𝐹‘1):ℝ⟶(0[,]+∞) → (∫2‘(𝐹‘1)) ∈ ℝ*) | |
25 | 23, 24 | syl 17 | . . 3 ⊢ (𝜑 → (∫2‘(𝐹‘1)) ∈ ℝ*) |
26 | itg2ge0 25709 | . . . . 5 ⊢ ((𝐹‘1):ℝ⟶(0[,]+∞) → 0 ≤ (∫2‘(𝐹‘1))) | |
27 | 23, 26 | syl 17 | . . . 4 ⊢ (𝜑 → 0 ≤ (∫2‘(𝐹‘1))) |
28 | mnflt0 13140 | . . . . 5 ⊢ -∞ < 0 | |
29 | 0xr 11293 | . . . . . 6 ⊢ 0 ∈ ℝ* | |
30 | xrltletr 13171 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ* ∧ (∫2‘(𝐹‘1)) ∈ ℝ*) → ((-∞ < 0 ∧ 0 ≤ (∫2‘(𝐹‘1))) → -∞ < (∫2‘(𝐹‘1)))) | |
31 | 16, 29, 25, 30 | mp3an12i 1461 | . . . . 5 ⊢ (𝜑 → ((-∞ < 0 ∧ 0 ≤ (∫2‘(𝐹‘1))) → -∞ < (∫2‘(𝐹‘1)))) |
32 | 28, 31 | mpani 694 | . . . 4 ⊢ (𝜑 → (0 ≤ (∫2‘(𝐹‘1)) → -∞ < (∫2‘(𝐹‘1)))) |
33 | 27, 32 | mpd 15 | . . 3 ⊢ (𝜑 → -∞ < (∫2‘(𝐹‘1))) |
34 | 2fveq3 6901 | . . . . . . . 8 ⊢ (𝑛 = 1 → (∫2‘(𝐹‘𝑛)) = (∫2‘(𝐹‘1))) | |
35 | eqid 2725 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) | |
36 | fvex 6909 | . . . . . . . 8 ⊢ (∫2‘(𝐹‘1)) ∈ V | |
37 | 34, 35, 36 | fvmpt 7004 | . . . . . . 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 6724 | . . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) Fn ℕ) |
40 | fnfvelrn 7089 | . . . . . . 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 584 | . . . . . 6 ⊢ (𝜑 → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘1) ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))) |
42 | 38, 41 | eqeltrrid 2830 | . . . . 5 ⊢ (𝜑 → (∫2‘(𝐹‘1)) ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))) |
43 | supxrub 13338 | . . . . 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 582 | . . . 4 ⊢ (𝜑 → (∫2‘(𝐹‘1)) ≤ sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < )) |
45 | 44, 1 | breqtrrdi 5191 | . . 3 ⊢ (𝜑 → (∫2‘(𝐹‘1)) ≤ 𝑆) |
46 | 17, 25, 12, 33, 45 | xrltletrd 13175 | . 2 ⊢ (𝜑 → -∞ < 𝑆) |
47 | 15 | rexrd 11296 | . . 3 ⊢ (𝜑 → (∫1‘𝑃) ∈ ℝ*) |
48 | itg2monolem2.9 | . . . 4 ⊢ (𝜑 → ¬ (∫1‘𝑃) ≤ 𝑆) | |
49 | xrltnle 11313 | . . . . 5 ⊢ ((𝑆 ∈ ℝ* ∧ (∫1‘𝑃) ∈ ℝ*) → (𝑆 < (∫1‘𝑃) ↔ ¬ (∫1‘𝑃) ≤ 𝑆)) | |
50 | 12, 47, 49 | syl2anc 582 | . . . 4 ⊢ (𝜑 → (𝑆 < (∫1‘𝑃) ↔ ¬ (∫1‘𝑃) ≤ 𝑆)) |
51 | 48, 50 | mpbird 256 | . . 3 ⊢ (𝜑 → 𝑆 < (∫1‘𝑃)) |
52 | 12, 47, 51 | xrltled 13164 | . 2 ⊢ (𝜑 → 𝑆 ≤ (∫1‘𝑃)) |
53 | xrre 13183 | . 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 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ∃wrex 3059 ⊆ wss 3944 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5678 ran crn 5679 Fn wfn 6544 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ∘r cofr 7684 supcsup 9465 ℝcr 11139 0cc0 11140 1c1 11141 + caddc 11143 +∞cpnf 11277 -∞cmnf 11278 ℝ*cxr 11279 < clt 11280 ≤ cle 11281 ℕcn 12245 [,)cico 13361 [,]cicc 13362 MblFncmbf 25587 ∫1citg1 25588 ∫2citg2 25589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-ofr 7686 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9467 df-inf 9468 df-oi 9535 df-dju 9926 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-q 12966 df-rp 13010 df-xadd 13128 df-ioo 13363 df-ico 13365 df-icc 13366 df-fz 13520 df-fzo 13663 df-fl 13793 df-seq 14003 df-exp 14063 df-hash 14326 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-clim 15468 df-sum 15669 df-xmet 21289 df-met 21290 df-ovol 25437 df-vol 25438 df-mbf 25592 df-itg1 25593 df-itg2 25594 |
This theorem is referenced by: itg2monolem3 25726 |
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