itg2i1fseq2 - Metamath Proof Explorer

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Theorem itg2i1fseq2 25733

Description: In an extension to the results of itg2i1fseq 25732, 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 12818	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12549	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
3		itg2i1fseq.3	. . . . . 6 ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫₁)
4	3	ffvelcdmda 7030	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∈ dom ∫₁)
5		itg1cl 25662	. . . . 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 7060	. . 3 ⊢ (𝜑 → 𝑆:ℕ⟶ℝ)
9	3	ffvelcdmda 7030	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘) ∈ dom ∫₁)
10		peano2nn 12177	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
11		ffvelcdm 7027	. . . . . 6 ⊢ ((𝑃:ℕ⟶dom ∫₁ ∧ (𝑘 + 1) ∈ ℕ) → (𝑃‘(𝑘 + 1)) ∈ dom ∫₁)
12	3, 10, 11	syl2an 597	. . . . 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 484	. . . . . . . 8 ⊢ ((0_𝑝 ∘_r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1))) → (𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1)))
15	14	ralimi 3075	. . . . . . 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 6834	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑃‘𝑛) = (𝑃‘𝑘))
18		fvoveq1 7383	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑘 + 1)))
19	17, 18	breq12d 5099	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ((𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1)) ↔ (𝑃‘𝑘) ∘_r ≤ (𝑃‘(𝑘 + 1))))
20	19	rspccva 3564	. . . . . 6 ⊢ ((∀𝑛 ∈ ℕ (𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1)) ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘) ∘_r ≤ (𝑃‘(𝑘 + 1)))
21	16, 20	sylan 581	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘) ∘_r ≤ (𝑃‘(𝑘 + 1)))
22		itg1le 25690	. . . . 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 1374	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∫₁‘(𝑃‘𝑘)) ≤ (∫₁‘(𝑃‘(𝑘 + 1))))
24		2fveq3 6839	. . . . . 6 ⊢ (𝑚 = 𝑘 → (∫₁‘(𝑃‘𝑚)) = (∫₁‘(𝑃‘𝑘)))
25		fvex 6847	. . . . . 6 ⊢ (∫₁‘(𝑃‘𝑘)) ∈ V
26	24, 7, 25	fvmpt 6941	. . . . 5 ⊢ (𝑘 ∈ ℕ → (𝑆‘𝑘) = (∫₁‘(𝑃‘𝑘)))
27	26	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) = (∫₁‘(𝑃‘𝑘)))
28		2fveq3 6839	. . . . . . 7 ⊢ (𝑚 = (𝑘 + 1) → (∫₁‘(𝑃‘𝑚)) = (∫₁‘(𝑃‘(𝑘 + 1))))
29		fvex 6847	. . . . . . 7 ⊢ (∫₁‘(𝑃‘(𝑘 + 1))) ∈ V
30	28, 7, 29	fvmpt 6941	. . . . . 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 481	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘(𝑘 + 1)) = (∫₁‘(𝑃‘(𝑘 + 1))))
33	23, 27, 32	3brtr4d 5118	. . 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 5108	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ≤ 𝑀)
37	36	ralrimiva 3130	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑀)
38		brralrspcev 5146	. . . 4 ⊢ ((𝑀 ∈ ℝ ∧ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑀) → ∃𝑧 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑧)
39	34, 37, 38	syl2anc 585	. . 3 ⊢ (𝜑 → ∃𝑧 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑧)
40	1, 2, 8, 33, 39	climsup 15623	. 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 25732	. . 3 ⊢ (𝜑 → (∫₂‘𝐹) = sup(ran 𝑆, ℝ^*, < ))
45	8	frnd 6670	. . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ)
46	7, 6	dmmptd 6637	. . . . . 6 ⊢ (𝜑 → dom 𝑆 = ℕ)
47		1nn 12176	. . . . . . 7 ⊢ 1 ∈ ℕ
48		ne0i 4282	. . . . . . 7 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
49	47, 48	mp1i 13	. . . . . 6 ⊢ (𝜑 → ℕ ≠ ∅)
50	46, 49	eqnetrd 3000	. . . . 5 ⊢ (𝜑 → dom 𝑆 ≠ ∅)
51		dm0rn0 5873	. . . . . 6 ⊢ (dom 𝑆 = ∅ ↔ ran 𝑆 = ∅)
52	51	necon3bii 2985	. . . . 5 ⊢ (dom 𝑆 ≠ ∅ ↔ ran 𝑆 ≠ ∅)
53	50, 52	sylib 218	. . . 4 ⊢ (𝜑 → ran 𝑆 ≠ ∅)
54		ffn 6662	. . . . . . 7 ⊢ (𝑆:ℕ⟶ℝ → 𝑆 Fn ℕ)
55		breq1 5089	. . . . . . . 8 ⊢ (𝑦 = (𝑆‘𝑘) → (𝑦 ≤ 𝑧 ↔ (𝑆‘𝑘) ≤ 𝑧))
56	55	ralrn 7034	. . . . . . 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 3162	. . . . 5 ⊢ (𝜑 → (∃𝑧 ∈ ℝ ∀𝑦 ∈ ran 𝑆 𝑦 ≤ 𝑧 ↔ ∃𝑧 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑧))
59	39, 58	mpbird 257	. . . 4 ⊢ (𝜑 → ∃𝑧 ∈ ℝ ∀𝑦 ∈ ran 𝑆 𝑦 ≤ 𝑧)
60		supxrre 13270	. . . 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 1374	. . 3 ⊢ (𝜑 → sup(ran 𝑆, ℝ^*, < ) = sup(ran 𝑆, ℝ, < ))
62	44, 61	eqtrd 2772	. 2 ⊢ (𝜑 → (∫₂‘𝐹) = sup(ran 𝑆, ℝ, < ))
63	40, 62	breqtrrd 5114	1 ⊢ (𝜑 → 𝑆 ⇝ (∫₂‘𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 ⊆ wss 3890 ∅c0 4274 class class class wbr 5086 ↦ cmpt 5167 dom cdm 5624 ran crn 5625 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 ∘_r cofr 7623 supcsup 9346 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 +∞cpnf 11167 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 ℕcn 12165 [,)cico 13291 ⇝ cli 15437 MblFncmbf 25591 ∫₁citg1 25592 ∫₂citg2 25593 0_𝑝c0p 25646

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-inf2 9553 ax-cc 10348 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-ofr 7625 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-oadd 8402 df-omul 8403 df-er 8636 df-map 8768 df-pm 8769 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fi 9317 df-sup 9348 df-inf 9349 df-oi 9418 df-dju 9816 df-card 9854 df-acn 9857 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-n0 12429 df-z 12516 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-rlim 15442 df-sum 15640 df-rest 17376 df-topgen 17397 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-top 22869 df-topon 22886 df-bases 22921 df-cmp 23362 df-ovol 25441 df-vol 25442 df-mbf 25596 df-itg1 25597 df-itg2 25598 df-0p 25647

This theorem is referenced by: itg2i1fseq3 25734 itg2addlem 25735

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