itg2i1fseq2 - Metamath Proof Explorer

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Theorem itg2i1fseq2 26082

Description: In an extension to the results of itg2i1fseq 26081, 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 13007	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12730	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
3		itg2i1fseq.3	. . . . . 6 ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫₁)
4	3	ffvelcdmda 7084	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑃‘𝑚) ∈ dom ∫₁)
5		itg1cl 26011	. . . . 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 7114	. . 3 ⊢ (𝜑 → 𝑆:ℕ⟶ℝ)
9	3	ffvelcdmda 7084	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘) ∈ dom ∫₁)
10		peano2nn 12350	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
11		ffvelcdm 7081	. . . . . 6 ⊢ ((𝑃:ℕ⟶dom ∫₁ ∧ (𝑘 + 1) ∈ ℕ) → (𝑃‘(𝑘 + 1)) ∈ dom ∫₁)
12	3, 10, 11	syl2an 608	. . . . 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 490	. . . . . . . 8 ⊢ ((0_𝑝 ∘_r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1))) → (𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1)))
15	14	ralimi 3099	. . . . . . 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 6885	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑃‘𝑛) = (𝑃‘𝑘))
18		fvoveq1 7443	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑘 + 1)))
19	17, 18	breq12d 5115	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ((𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1)) ↔ (𝑃‘𝑘) ∘_r ≤ (𝑃‘(𝑘 + 1))))
20	19	rspccva 3575	. . . . . 6 ⊢ ((∀𝑛 ∈ ℕ (𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1)) ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘) ∘_r ≤ (𝑃‘(𝑘 + 1)))
21	16, 20	sylan 592	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃‘𝑘) ∘_r ≤ (𝑃‘(𝑘 + 1)))
22		itg1le 26039	. . . . 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 6890	. . . . . 6 ⊢ (𝑚 = 𝑘 → (∫₁‘(𝑃‘𝑚)) = (∫₁‘(𝑃‘𝑘)))
25		fvex 6898	. . . . . 6 ⊢ (∫₁‘(𝑃‘𝑘)) ∈ V
26	24, 7, 25	fvmpt 6993	. . . . 5 ⊢ (𝑘 ∈ ℕ → (𝑆‘𝑘) = (∫₁‘(𝑃‘𝑘)))
27	26	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) = (∫₁‘(𝑃‘𝑘)))
28		2fveq3 6890	. . . . . . 7 ⊢ (𝑚 = (𝑘 + 1) → (∫₁‘(𝑃‘𝑚)) = (∫₁‘(𝑃‘(𝑘 + 1))))
29		fvex 6898	. . . . . . 7 ⊢ (∫₁‘(𝑃‘(𝑘 + 1))) ∈ V
30	28, 7, 29	fvmpt 6993	. . . . . 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 487	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘(𝑘 + 1)) = (∫₁‘(𝑃‘(𝑘 + 1))))
33	23, 27, 32	3brtr4d 5136	. . 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 5126	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ≤ 𝑀)
37	36	ralrimiva 3154	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑀)
38		brralrspcev 5164	. . . 4 ⊢ ((𝑀 ∈ ℝ ∧ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑀) → ∃𝑧 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑧)
39	34, 37, 38	syl2anc 596	. . 3 ⊢ (𝜑 → ∃𝑧 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑧)
40	1, 2, 8, 33, 39	climsup 15842	. 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 26081	. . 3 ⊢ (𝜑 → (∫₂‘𝐹) = sup(ran 𝑆, ℝ^*, < ))
45	8	frnd 6718	. . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ)
46	7, 6	dmmptd 6684	. . . . . 6 ⊢ (𝜑 → dom 𝑆 = ℕ)
47		1nn 12349	. . . . . . 7 ⊢ 1 ∈ ℕ
48		ne0i 4286	. . . . . . 7 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
49	47, 48	mp1i 14	. . . . . 6 ⊢ (𝜑 → ℕ ≠ ∅)
50	46, 49	eqnetrd 3022	. . . . 5 ⊢ (𝜑 → dom 𝑆 ≠ ∅)
51		dm0rn0 5905	. . . . . 6 ⊢ (dom 𝑆 = ∅ ↔ ran 𝑆 = ∅)
52	51	necon3bii 3007	. . . . 5 ⊢ (dom 𝑆 ≠ ∅ ↔ ran 𝑆 ≠ ∅)
53	50, 52	sylib 221	. . . 4 ⊢ (𝜑 → ran 𝑆 ≠ ∅)
54		ffn 6709	. . . . . . 7 ⊢ (𝑆:ℕ⟶ℝ → 𝑆 Fn ℕ)
55		breq1 5105	. . . . . . . 8 ⊢ (𝑦 = (𝑆‘𝑘) → (𝑦 ≤ 𝑧 ↔ (𝑆‘𝑘) ≤ 𝑧))
56	55	ralrn 7088	. . . . . . 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 3186	. . . . 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 13460	. . . 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 2795	. 2 ⊢ (𝜑 → (∫₂‘𝐹) = sup(ran 𝑆, ℝ, < ))
63	40, 62	breqtrrd 5132	1 ⊢ (𝜑 → 𝑆 ⇝ (∫₂‘𝐹))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 ⊆ wss 3895 ∅c0 4278 class class class wbr 5102 ↦ cmpt 5185 dom cdm 5650 ran crn 5651 Fn wfn 6533 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ∘_r cofr 7692 supcsup 9435 ℝcr 11202 0cc0 11203 1c1 11204 + caddc 11206 +∞cpnf 11343 ℝ^*cxr 11345 < clt 11346 ≤ cle 11347 ℕcn 12338 [,)cico 13481 ⇝ cli 15656 MblFncmbf 25940 ∫₁citg1 25941 ∫₂citg2 25942 0_𝑝c0p 25995

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9645 ax-cc 10516 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280 ax-pre-sup 11281 ax-addf 11282

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-disj 5070 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 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 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-ofr 7694 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-2o 8480 df-oadd 8483 df-omul 8484 df-er 8720 df-map 8852 df-pm 8853 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9507 df-dju 9985 df-card 10023 df-acn 10026 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-div 11977 df-nn 12339 df-2 12408 df-3 12409 df-n0 12610 df-z 12697 df-uz 12969 df-q 13079 df-rp 13124 df-xneg 13244 df-xadd 13245 df-xmul 13246 df-ioo 13483 df-ioc 13484 df-ico 13485 df-icc 13486 df-fz 13643 df-fzo 13792 df-fl 13935 df-seq 14148 df-exp 14208 df-hash 14478 df-cj 15271 df-re 15272 df-im 15273 df-sqrt 15407 df-abs 15408 df-clim 15660 df-rlim 15661 df-sum 15859 df-rest 17598 df-topgen 17619 df-psmet 21675 df-xmet 21676 df-met 21677 df-bl 21678 df-mopn 21679 df-top 23217 df-topon 23234 df-bases 23269 df-cmp 23710 df-ovol 25790 df-vol 25791 df-mbf 25945 df-itg1 25946 df-itg2 25947 df-0p 25996

This theorem is used by: itg2i1fseq3 26083 itg2addlem 26084

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