itg2i1fseq2 - Metamath Proof Explorer

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Theorem itg2i1fseq2 25896

Description: In an extension to the results of itg2i1fseq 25895, 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 12902	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12626	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
3		itg2i1fseq.3	. . . . . 6 ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫₁)
4	3	ffvelcdmda 7081	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∈ dom ∫₁)
5		itg1cl 25825	. . . . 5 ⊢ ((𝑃‘𝑚) ∈ dom ∫₁ → (∫₁‘(𝑃‘𝑚)) ∈ ℝ)
6	4, 5	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∫₁‘(𝑃‘𝑚)) ∈ ℝ)
7		itg2i1fseq.6	. . . 4 ⊢ 𝑆 = (𝑚 ∈ ℕ ↦ (∫₁‘(𝑃‘𝑚)))
8	6, 7	fmptd 7111	. . 3 ⊢ (𝜑 → 𝑆:ℕ⟶ℝ)
9	3	ffvelcdmda 7081	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘) ∈ dom ∫₁)
10		peano2nn 12246	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
11		ffvelcdm 7078	. . . . . 6 ⊢ ((𝑃:ℕ⟶dom ∫₁ ∧ (𝑘 + 1) ∈ ℕ) → (𝑃‘(𝑘 + 1)) ∈ dom ∫₁)
12	3, 10, 11	syl2an 607	. . . . 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 489	. . . . . . . 8 ⊢ ((0_𝑝 ∘_r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1))) → (𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1)))
15	14	ralimi 3102	. . . . . . 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 18	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1)))
17		fveq2 6883	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑃‘𝑛) = (𝑃‘𝑘))
18		fvoveq1 7435	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑘 + 1)))
19	17, 18	breq12d 5123	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ((𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1)) ↔ (𝑃‘𝑘) ∘_r ≤ (𝑃‘(𝑘 + 1))))
20	19	rspccva 3581	. . . . . 6 ⊢ ((∀𝑛 ∈ ℕ (𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1)) ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘) ∘_r ≤ (𝑃‘(𝑘 + 1)))
21	16, 20	sylan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘) ∘_r ≤ (𝑃‘(𝑘 + 1)))
22		itg1le 25853	. . . . 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 1398	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∫₁‘(𝑃‘𝑘)) ≤ (∫₁‘(𝑃‘(𝑘 + 1))))
24		2fveq3 6888	. . . . . 6 ⊢ (𝑚 = 𝑘 → (∫₁‘(𝑃‘𝑚)) = (∫₁‘(𝑃‘𝑘)))
25		fvex 6896	. . . . . 6 ⊢ (∫₁‘(𝑃‘𝑘)) ∈ V
26	24, 7, 25	fvmpt 6991	. . . . 5 ⊢ (𝑘 ∈ ℕ → (𝑆‘𝑘) = (∫₁‘(𝑃‘𝑘)))
27	26	adantl 486	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) = (∫₁‘(𝑃‘𝑘)))
28		2fveq3 6888	. . . . . . 7 ⊢ (𝑚 = (𝑘 + 1) → (∫₁‘(𝑃‘𝑚)) = (∫₁‘(𝑃‘(𝑘 + 1))))
29		fvex 6896	. . . . . . 7 ⊢ (∫₁‘(𝑃‘(𝑘 + 1))) ∈ V
30	28, 7, 29	fvmpt 6991	. . . . . 6 ⊢ ((𝑘 + 1) ∈ ℕ → (𝑆‘(𝑘 + 1)) = (∫₁‘(𝑃‘(𝑘 + 1))))
31	10, 30	syl 18	. . . . 5 ⊢ (𝑘 ∈ ℕ → (𝑆‘(𝑘 + 1)) = (∫₁‘(𝑃‘(𝑘 + 1))))
32	31	adantl 486	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘(𝑘 + 1)) = (∫₁‘(𝑃‘(𝑘 + 1))))
33	23, 27, 32	3brtr4d 5144	. . 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 5134	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ≤ 𝑀)
37	36	ralrimiva 3157	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑀)
38		brralrspcev 5172	. . . 4 ⊢ ((𝑀 ∈ ℝ ∧ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑀) → ∃𝑧 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑧)
39	34, 37, 38	syl2anc 595	. . 3 ⊢ (𝜑 → ∃𝑧 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑧)
40	1, 2, 8, 33, 39	climsup 15723	. 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 25895	. . 3 ⊢ (𝜑 → (∫₂‘𝐹) = sup(ran 𝑆, ℝ^*, < ))
45	8	frnd 6716	. . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ)
46	7, 6	dmmptd 6682	. . . . . 6 ⊢ (𝜑 → dom 𝑆 = ℕ)
47		1nn 12245	. . . . . . 7 ⊢ 1 ∈ ℕ
48		ne0i 4295	. . . . . . 7 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
49	47, 48	mp1i 14	. . . . . 6 ⊢ (𝜑 → ℕ ≠ ∅)
50	46, 49	eqnetrd 3025	. . . . 5 ⊢ (𝜑 → dom 𝑆 ≠ ∅)
51		dm0rn0 5916	. . . . . 6 ⊢ (dom 𝑆 = ∅ ↔ ran 𝑆 = ∅)
52	51	necon3bii 3010	. . . . 5 ⊢ (dom 𝑆 ≠ ∅ ↔ ran 𝑆 ≠ ∅)
53	50, 52	sylib 221	. . . 4 ⊢ (𝜑 → ran 𝑆 ≠ ∅)
54		ffn 6707	. . . . . . 7 ⊢ (𝑆:ℕ⟶ℝ → 𝑆 Fn ℕ)
55		breq1 5113	. . . . . . . 8 ⊢ (𝑦 = (𝑆‘𝑘) → (𝑦 ≤ 𝑧 ↔ (𝑆‘𝑘) ≤ 𝑧))
56	55	ralrn 7085	. . . . . . 7 ⊢ (𝑆 Fn ℕ → (∀𝑦 ∈ ran 𝑆 𝑦 ≤ 𝑧 ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑧))
57	8, 54, 56	3syl 19	. . . . . 6 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝑆 𝑦 ≤ 𝑧 ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑧))
58	57	rexbidv 3189	. . . . 5 ⊢ (𝜑 → (∃𝑧 ∈ ℝ ∀𝑦 ∈ ran 𝑆 𝑦 ≤ 𝑧 ↔ ∃𝑧 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑧))
59	39, 58	mpbird 260	. . . 4 ⊢ (𝜑 → ∃𝑧 ∈ ℝ ∀𝑦 ∈ ran 𝑆 𝑦 ≤ 𝑧)
60		supxrre 13354	. . . 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 1398	. . 3 ⊢ (𝜑 → sup(ran 𝑆, ℝ^*, < ) = sup(ran 𝑆, ℝ, < ))
62	44, 61	eqtrd 2798	. 2 ⊢ (𝜑 → (∫₂‘𝐹) = sup(ran 𝑆, ℝ, < ))
63	40, 62	breqtrrd 5140	1 ⊢ (𝜑 → 𝑆 ⇝ (∫₂‘𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 ⊆ wss 3906 ∅c0 4287 class class class wbr 5110 ↦ cmpt 5193 dom cdm 5663 ran crn 5664 Fn wfn 6533 ⟶wf 6534 ‘cfv 6538 (class class class)co 7412 ∘_r cofr 7675 supcsup 9401 ℝcr 11100 0cc0 11101 1c1 11102 + caddc 11104 +∞cpnf 11241 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 ℕcn 12234 [,)cico 13375 ⇝ cli 15537 MblFncmbf 25754 ∫₁citg1 25755 ∫₂citg2 25756 0_𝑝c0p 25809

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cc 10420 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179 ax-addf 11180

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-disj 5078 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-ofr 7677 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-2o 8455 df-oadd 8458 df-omul 8459 df-er 8695 df-map 8827 df-pm 8828 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-fi 9372 df-sup 9403 df-inf 9404 df-oi 9473 df-dju 9888 df-card 9926 df-acn 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-n0 12506 df-z 12593 df-uz 12864 df-q 12974 df-rp 13018 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-ioo 13377 df-ioc 13378 df-ico 13379 df-icc 13380 df-fz 13537 df-fzo 13685 df-fl 13827 df-seq 14040 df-exp 14100 df-hash 14369 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-clim 15541 df-rlim 15542 df-sum 15740 df-rest 17476 df-topgen 17497 df-psmet 21495 df-xmet 21496 df-met 21497 df-bl 21498 df-mopn 21499 df-top 23032 df-topon 23049 df-bases 23084 df-cmp 23525 df-ovol 25604 df-vol 25605 df-mbf 25759 df-itg1 25760 df-itg2 25761 df-0p 25810

This theorem is referenced by: itg2i1fseq3 25897 itg2addlem 25898

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