itg2i1fseq2 - Metamath Proof Explorer

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Theorem itg2i1fseq2 24921

Description: In an extension to the results of itg2i1fseq 24920, if there is an upper bound on the integrals of the simple functions approaching 𝐹, then ∫₂𝐹 is real and the standard limit relation applies. (Contributed by Mario Carneiro, 17-Aug-2014.)

**Hypotheses**
Ref	Expression
itg2i1fseq.1	⊢ (𝜑 → 𝐹 ∈ MblFn)
itg2i1fseq.2	⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
itg2i1fseq.3	⊢ (𝜑 → 𝑃:ℕ⟶dom ∫₁)
itg2i1fseq.4	⊢ (𝜑 → ∀𝑛 ∈ ℕ (0_𝑝 ∘_r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1))))
itg2i1fseq.5	⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))
itg2i1fseq.6	⊢ 𝑆 = (𝑚 ∈ ℕ ↦ (∫₁‘(𝑃‘𝑚)))
itg2i1fseq2.7	⊢ (𝜑 → 𝑀 ∈ ℝ)
itg2i1fseq2.8	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∫₁‘(𝑃‘𝑘)) ≤ 𝑀)

**Assertion**
Ref	Expression
itg2i1fseq2	⊢ (𝜑 → 𝑆 ⇝ (∫₂‘𝐹))

Distinct variable groups: 𝑘,𝑚,𝑛,𝑥,𝐹 𝑘,𝑀,𝑛 𝑃,𝑘,𝑚,𝑛,𝑥 𝜑,𝑘,𝑚 𝑆,𝑘 Allowed substitution hints: 𝜑(𝑥,𝑛) 𝑆(𝑥,𝑚,𝑛) 𝑀(𝑥,𝑚)

Proof of Theorem itg2i1fseq2 Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 12621	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12351	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
3		itg2i1fseq.3	. . . . . 6 ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫₁)
4	3	ffvelrnda 6961	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∈ dom ∫₁)
5		itg1cl 24849	. . . . 5 ⊢ ((𝑃‘𝑚) ∈ dom ∫₁ → (∫₁‘(𝑃‘𝑚)) ∈ ℝ)
6	4, 5	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∫₁‘(𝑃‘𝑚)) ∈ ℝ)
7		itg2i1fseq.6	. . . 4 ⊢ 𝑆 = (𝑚 ∈ ℕ ↦ (∫₁‘(𝑃‘𝑚)))
8	6, 7	fmptd 6988	. . 3 ⊢ (𝜑 → 𝑆:ℕ⟶ℝ)
9	3	ffvelrnda 6961	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘) ∈ dom ∫₁)
10		peano2nn 11985	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
11		ffvelrn 6959	. . . . . 6 ⊢ ((𝑃:ℕ⟶dom ∫₁ ∧ (𝑘 + 1) ∈ ℕ) → (𝑃‘(𝑘 + 1)) ∈ dom ∫₁)
12	3, 10, 11	syl2an 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃‘(𝑘 + 1)) ∈ dom ∫₁)
13		itg2i1fseq.4	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0_𝑝 ∘_r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1))))
14		simpr 485	. . . . . . . 8 ⊢ ((0_𝑝 ∘_r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1))) → (𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1)))
15	14	ralimi 3087	. . . . . . 7 ⊢ (∀𝑛 ∈ ℕ (0_𝑝 ∘_r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1))) → ∀𝑛 ∈ ℕ (𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1)))
16	13, 15	syl 17	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1)))
17		fveq2 6774	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑃‘𝑛) = (𝑃‘𝑘))
18		fvoveq1 7298	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑘 + 1)))
19	17, 18	breq12d 5087	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ((𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1)) ↔ (𝑃‘𝑘) ∘_r ≤ (𝑃‘(𝑘 + 1))))
20	19	rspccva 3560	. . . . . 6 ⊢ ((∀𝑛 ∈ ℕ (𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1)) ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘) ∘_r ≤ (𝑃‘(𝑘 + 1)))
21	16, 20	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘) ∘_r ≤ (𝑃‘(𝑘 + 1)))
22		itg1le 24878	. . . . 5 ⊢ (((𝑃‘𝑘) ∈ dom ∫₁ ∧ (𝑃‘(𝑘 + 1)) ∈ dom ∫₁ ∧ (𝑃‘𝑘) ∘_r ≤ (𝑃‘(𝑘 + 1))) → (∫₁‘(𝑃‘𝑘)) ≤ (∫₁‘(𝑃‘(𝑘 + 1))))
23	9, 12, 21, 22	syl3anc 1370	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∫₁‘(𝑃‘𝑘)) ≤ (∫₁‘(𝑃‘(𝑘 + 1))))
24		2fveq3 6779	. . . . . 6 ⊢ (𝑚 = 𝑘 → (∫₁‘(𝑃‘𝑚)) = (∫₁‘(𝑃‘𝑘)))
25		fvex 6787	. . . . . 6 ⊢ (∫₁‘(𝑃‘𝑘)) ∈ V
26	24, 7, 25	fvmpt 6875	. . . . 5 ⊢ (𝑘 ∈ ℕ → (𝑆‘𝑘) = (∫₁‘(𝑃‘𝑘)))
27	26	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) = (∫₁‘(𝑃‘𝑘)))
28		2fveq3 6779	. . . . . . 7 ⊢ (𝑚 = (𝑘 + 1) → (∫₁‘(𝑃‘𝑚)) = (∫₁‘(𝑃‘(𝑘 + 1))))
29		fvex 6787	. . . . . . 7 ⊢ (∫₁‘(𝑃‘(𝑘 + 1))) ∈ V
30	28, 7, 29	fvmpt 6875	. . . . . 6 ⊢ ((𝑘 + 1) ∈ ℕ → (𝑆‘(𝑘 + 1)) = (∫₁‘(𝑃‘(𝑘 + 1))))
31	10, 30	syl 17	. . . . 5 ⊢ (𝑘 ∈ ℕ → (𝑆‘(𝑘 + 1)) = (∫₁‘(𝑃‘(𝑘 + 1))))
32	31	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘(𝑘 + 1)) = (∫₁‘(𝑃‘(𝑘 + 1))))
33	23, 27, 32	3brtr4d 5106	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ≤ (𝑆‘(𝑘 + 1)))
34		itg2i1fseq2.7	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℝ)
35		itg2i1fseq2.8	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∫₁‘(𝑃‘𝑘)) ≤ 𝑀)
36	27, 35	eqbrtrd 5096	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ≤ 𝑀)
37	36	ralrimiva 3103	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑀)
38		brralrspcev 5134	. . . 4 ⊢ ((𝑀 ∈ ℝ ∧ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑀) → ∃𝑧 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑧)
39	34, 37, 38	syl2anc 584	. . 3 ⊢ (𝜑 → ∃𝑧 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑧)
40	1, 2, 8, 33, 39	climsup 15381	. 2 ⊢ (𝜑 → 𝑆 ⇝ sup(ran 𝑆, ℝ, < ))
41		itg2i1fseq.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ MblFn)
42		itg2i1fseq.2	. . . 4 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
43		itg2i1fseq.5	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))
44	41, 42, 3, 13, 43, 7	itg2i1fseq 24920	. . 3 ⊢ (𝜑 → (∫₂‘𝐹) = sup(ran 𝑆, ℝ^*, < ))
45	8	frnd 6608	. . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ)
46	7, 6	dmmptd 6578	. . . . . 6 ⊢ (𝜑 → dom 𝑆 = ℕ)
47		1nn 11984	. . . . . . 7 ⊢ 1 ∈ ℕ
48		ne0i 4268	. . . . . . 7 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
49	47, 48	mp1i 13	. . . . . 6 ⊢ (𝜑 → ℕ ≠ ∅)
50	46, 49	eqnetrd 3011	. . . . 5 ⊢ (𝜑 → dom 𝑆 ≠ ∅)
51		dm0rn0 5834	. . . . . 6 ⊢ (dom 𝑆 = ∅ ↔ ran 𝑆 = ∅)
52	51	necon3bii 2996	. . . . 5 ⊢ (dom 𝑆 ≠ ∅ ↔ ran 𝑆 ≠ ∅)
53	50, 52	sylib 217	. . . 4 ⊢ (𝜑 → ran 𝑆 ≠ ∅)
54		ffn 6600	. . . . . . 7 ⊢ (𝑆:ℕ⟶ℝ → 𝑆 Fn ℕ)
55		breq1 5077	. . . . . . . 8 ⊢ (𝑦 = (𝑆‘𝑘) → (𝑦 ≤ 𝑧 ↔ (𝑆‘𝑘) ≤ 𝑧))
56	55	ralrn 6964	. . . . . . 7 ⊢ (𝑆 Fn ℕ → (∀𝑦 ∈ ran 𝑆 𝑦 ≤ 𝑧 ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑧))
57	8, 54, 56	3syl 18	. . . . . 6 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝑆 𝑦 ≤ 𝑧 ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑧))
58	57	rexbidv 3226	. . . . 5 ⊢ (𝜑 → (∃𝑧 ∈ ℝ ∀𝑦 ∈ ran 𝑆 𝑦 ≤ 𝑧 ↔ ∃𝑧 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑧))
59	39, 58	mpbird 256	. . . 4 ⊢ (𝜑 → ∃𝑧 ∈ ℝ ∀𝑦 ∈ ran 𝑆 𝑦 ≤ 𝑧)
60		supxrre 13061	. . . 4 ⊢ ((ran 𝑆 ⊆ ℝ ∧ ran 𝑆 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑦 ∈ ran 𝑆 𝑦 ≤ 𝑧) → sup(ran 𝑆, ℝ^*, < ) = sup(ran 𝑆, ℝ, < ))
61	45, 53, 59, 60	syl3anc 1370	. . 3 ⊢ (𝜑 → sup(ran 𝑆, ℝ^*, < ) = sup(ran 𝑆, ℝ, < ))
62	44, 61	eqtrd 2778	. 2 ⊢ (𝜑 → (∫₂‘𝐹) = sup(ran 𝑆, ℝ, < ))
63	40, 62	breqtrrd 5102	1 ⊢ (𝜑 → 𝑆 ⇝ (∫₂‘𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 ∅c0 4256 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 ran crn 5590 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ∘_r cofr 7532 supcsup 9199 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 +∞cpnf 11006 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 ℕcn 11973 [,)cico 13081 ⇝ cli 15193 MblFncmbf 24778 ∫₁citg1 24779 ∫₂citg2 24780 0_𝑝c0p 24833

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cc 10191 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-disj 5040 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-ofr 7534 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-omul 8302 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-acn 9700 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-rlim 15198 df-sum 15398 df-rest 17133 df-topgen 17154 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-top 22043 df-topon 22060 df-bases 22096 df-cmp 22538 df-ovol 24628 df-vol 24629 df-mbf 24783 df-itg1 24784 df-itg2 24785 df-0p 24834

This theorem is referenced by: itg2i1fseq3 24922 itg2addlem 24923

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