![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > itg2monolem2 | Structured version Visualization version GIF version |
Description: Lemma for itg2mono 25803. (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 13473 | . . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
4 | fss 6753 | . . . . . . . 8 ⊢ (((𝐹‘𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹‘𝑛):ℝ⟶(0[,]+∞)) | |
5 | 2, 3, 4 | sylancl 586 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,]+∞)) |
6 | itg2cl 25782 | . . . . . . 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 7135 | . . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))):ℕ⟶ℝ*) |
9 | 8 | frnd 6745 | . . . 4 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) ⊆ ℝ*) |
10 | supxrcl 13354 | . . . 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 2843 | . 2 ⊢ (𝜑 → 𝑆 ∈ ℝ*) |
13 | itg2monolem2.7 | . . 3 ⊢ (𝜑 → 𝑃 ∈ dom ∫1) | |
14 | itg1cl 25734 | . . 3 ⊢ (𝑃 ∈ dom ∫1 → (∫1‘𝑃) ∈ ℝ) | |
15 | 13, 14 | syl 17 | . 2 ⊢ (𝜑 → (∫1‘𝑃) ∈ ℝ) |
16 | mnfxr 11316 | . . . 4 ⊢ -∞ ∈ ℝ* | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → -∞ ∈ ℝ*) |
18 | fveq2 6907 | . . . . . 6 ⊢ (𝑛 = 1 → (𝐹‘𝑛) = (𝐹‘1)) | |
19 | 18 | feq1d 6721 | . . . . 5 ⊢ (𝑛 = 1 → ((𝐹‘𝑛):ℝ⟶(0[,]+∞) ↔ (𝐹‘1):ℝ⟶(0[,]+∞))) |
20 | 5 | ralrimiva 3144 | . . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛):ℝ⟶(0[,]+∞)) |
21 | 1nn 12275 | . . . . . 6 ⊢ 1 ∈ ℕ | |
22 | 21 | a1i 11 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℕ) |
23 | 19, 20, 22 | rspcdva 3623 | . . . 4 ⊢ (𝜑 → (𝐹‘1):ℝ⟶(0[,]+∞)) |
24 | itg2cl 25782 | . . . 4 ⊢ ((𝐹‘1):ℝ⟶(0[,]+∞) → (∫2‘(𝐹‘1)) ∈ ℝ*) | |
25 | 23, 24 | syl 17 | . . 3 ⊢ (𝜑 → (∫2‘(𝐹‘1)) ∈ ℝ*) |
26 | itg2ge0 25785 | . . . . 5 ⊢ ((𝐹‘1):ℝ⟶(0[,]+∞) → 0 ≤ (∫2‘(𝐹‘1))) | |
27 | 23, 26 | syl 17 | . . . 4 ⊢ (𝜑 → 0 ≤ (∫2‘(𝐹‘1))) |
28 | mnflt0 13165 | . . . . 5 ⊢ -∞ < 0 | |
29 | 0xr 11306 | . . . . . 6 ⊢ 0 ∈ ℝ* | |
30 | xrltletr 13196 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ* ∧ (∫2‘(𝐹‘1)) ∈ ℝ*) → ((-∞ < 0 ∧ 0 ≤ (∫2‘(𝐹‘1))) → -∞ < (∫2‘(𝐹‘1)))) | |
31 | 16, 29, 25, 30 | mp3an12i 1464 | . . . . 5 ⊢ (𝜑 → ((-∞ < 0 ∧ 0 ≤ (∫2‘(𝐹‘1))) → -∞ < (∫2‘(𝐹‘1)))) |
32 | 28, 31 | mpani 696 | . . . 4 ⊢ (𝜑 → (0 ≤ (∫2‘(𝐹‘1)) → -∞ < (∫2‘(𝐹‘1)))) |
33 | 27, 32 | mpd 15 | . . 3 ⊢ (𝜑 → -∞ < (∫2‘(𝐹‘1))) |
34 | 2fveq3 6912 | . . . . . . . 8 ⊢ (𝑛 = 1 → (∫2‘(𝐹‘𝑛)) = (∫2‘(𝐹‘1))) | |
35 | eqid 2735 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) | |
36 | fvex 6920 | . . . . . . . 8 ⊢ (∫2‘(𝐹‘1)) ∈ V | |
37 | 34, 35, 36 | fvmpt 7016 | . . . . . . 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 6738 | . . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) Fn ℕ) |
40 | fnfvelrn 7100 | . . . . . . 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 586 | . . . . . 6 ⊢ (𝜑 → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘1) ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))) |
42 | 38, 41 | eqeltrrid 2844 | . . . . 5 ⊢ (𝜑 → (∫2‘(𝐹‘1)) ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))) |
43 | supxrub 13363 | . . . . 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 584 | . . . 4 ⊢ (𝜑 → (∫2‘(𝐹‘1)) ≤ sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < )) |
45 | 44, 1 | breqtrrdi 5190 | . . 3 ⊢ (𝜑 → (∫2‘(𝐹‘1)) ≤ 𝑆) |
46 | 17, 25, 12, 33, 45 | xrltletrd 13200 | . 2 ⊢ (𝜑 → -∞ < 𝑆) |
47 | 15 | rexrd 11309 | . . 3 ⊢ (𝜑 → (∫1‘𝑃) ∈ ℝ*) |
48 | itg2monolem2.9 | . . . 4 ⊢ (𝜑 → ¬ (∫1‘𝑃) ≤ 𝑆) | |
49 | xrltnle 11326 | . . . . 5 ⊢ ((𝑆 ∈ ℝ* ∧ (∫1‘𝑃) ∈ ℝ*) → (𝑆 < (∫1‘𝑃) ↔ ¬ (∫1‘𝑃) ≤ 𝑆)) | |
50 | 12, 47, 49 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑆 < (∫1‘𝑃) ↔ ¬ (∫1‘𝑃) ≤ 𝑆)) |
51 | 48, 50 | mpbird 257 | . . 3 ⊢ (𝜑 → 𝑆 < (∫1‘𝑃)) |
52 | 12, 47, 51 | xrltled 13189 | . 2 ⊢ (𝜑 → 𝑆 ≤ (∫1‘𝑃)) |
53 | xrre 13208 | . 2 ⊢ (((𝑆 ∈ ℝ* ∧ (∫1‘𝑃) ∈ ℝ) ∧ (-∞ < 𝑆 ∧ 𝑆 ≤ (∫1‘𝑃))) → 𝑆 ∈ ℝ) | |
54 | 12, 15, 46, 52, 53 | syl22anc 839 | 1 ⊢ (𝜑 → 𝑆 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ran crn 5690 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∘r cofr 7696 supcsup 9478 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 +∞cpnf 11290 -∞cmnf 11291 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 ℕcn 12264 [,)cico 13386 [,]cicc 13387 MblFncmbf 25663 ∫1citg1 25664 ∫2citg2 25665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-xadd 13153 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-xmet 21375 df-met 21376 df-ovol 25513 df-vol 25514 df-mbf 25668 df-itg1 25669 df-itg2 25670 |
This theorem is referenced by: itg2monolem3 25802 |
Copyright terms: Public domain | W3C validator |