itg2i1fseq2 - Metamath Proof Explorer

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Theorem itg2i1fseq2 25990

Description: In an extension to the results of itg2i1fseq 25989, 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 12930	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12653	. . 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 25919	. . . . 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 12273	. . . . . 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 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 3101	. . . . . . 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 6882	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑃‘𝑛) = (𝑃‘𝑘))
18		fvoveq1 7440	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑘 + 1)))
19	17, 18	breq12d 5120	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ((𝑃‘𝑛) ∘_r ≤ (𝑃‘(𝑛 + 1)) ↔ (𝑃‘𝑘) ∘_r ≤ (𝑃‘(𝑘 + 1))))
20	19	rspccva 3578	. . . . . 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 25947	. . . . 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 6887	. . . . . 6 ⊢ (𝑚 = 𝑘 → (∫₁‘(𝑃‘𝑚)) = (∫₁‘(𝑃‘𝑘)))
25		fvex 6895	. . . . . 6 ⊢ (∫₁‘(𝑃‘𝑘)) ∈ V
26	24, 7, 25	fvmpt 6990	. . . . 5 ⊢ (𝑘 ∈ ℕ → (𝑆‘𝑘) = (∫₁‘(𝑃‘𝑘)))
27	26	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) = (∫₁‘(𝑃‘𝑘)))
28		2fveq3 6887	. . . . . . 7 ⊢ (𝑚 = (𝑘 + 1) → (∫₁‘(𝑃‘𝑚)) = (∫₁‘(𝑃‘(𝑘 + 1))))
29		fvex 6895	. . . . . . 7 ⊢ (∫₁‘(𝑃‘(𝑘 + 1))) ∈ V
30	28, 7, 29	fvmpt 6990	. . . . . 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 5141	. . 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 5131	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ≤ 𝑀)
37	36	ralrimiva 3156	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑀)
38		brralrspcev 5169	. . . 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 15761	. 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 25989	. . 3 ⊢ (𝜑 → (∫₂‘𝐹) = sup(ran 𝑆, ℝ^*, < ))
45	8	frnd 6715	. . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ)
46	7, 6	dmmptd 6681	. . . . . 6 ⊢ (𝜑 → dom 𝑆 = ℕ)
47		1nn 12272	. . . . . . 7 ⊢ 1 ∈ ℕ
48		ne0i 4290	. . . . . . 7 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
49	47, 48	mp1i 14	. . . . . 6 ⊢ (𝜑 → ℕ ≠ ∅)
50	46, 49	eqnetrd 3024	. . . . 5 ⊢ (𝜑 → dom 𝑆 ≠ ∅)
51		dm0rn0 5912	. . . . . 6 ⊢ (dom 𝑆 = ∅ ↔ ran 𝑆 = ∅)
52	51	necon3bii 3009	. . . . 5 ⊢ (dom 𝑆 ≠ ∅ ↔ ran 𝑆 ≠ ∅)
53	50, 52	sylib 221	. . . 4 ⊢ (𝜑 → ran 𝑆 ≠ ∅)
54		ffn 6706	. . . . . . 7 ⊢ (𝑆:ℕ⟶ℝ → 𝑆 Fn ℕ)
55		breq1 5110	. . . . . . . 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 3188	. . . . 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 13383	. . . 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 2797	. 2 ⊢ (𝜑 → (∫₂‘𝐹) = sup(ran 𝑆, ℝ, < ))
63	40, 62	breqtrrd 5137	1 ⊢ (𝜑 → 𝑆 ⇝ (∫₂‘𝐹))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ∀wral 3078 ∃wrex 3088 ⊆ wss 3902 ∅c0 4282 class class class wbr 5107 ↦ cmpt 5190 dom cdm 5659 ran crn 5660 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7417 ∘_r cofr 7681 supcsup 9414 ℝcr 11127 0cc0 11128 1c1 11129 + caddc 11131 +∞cpnf 11268 ℝ^*cxr 11270 < clt 11271 ≤ cle 11272 ℕcn 12261 [,)cico 13404 ⇝ cli 15575 MblFncmbf 25848 ∫₁citg1 25849 ∫₂citg2 25850 0_𝑝c0p 25903

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-inf2 9624 ax-cc 10441 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206 ax-addf 11207

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-disj 5075 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7682 df-ofr 7683 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-oadd 8463 df-omul 8464 df-er 8700 df-map 8832 df-pm 8833 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-fi 9385 df-sup 9416 df-inf 9417 df-oi 9486 df-dju 9910 df-card 9948 df-acn 9951 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-3 12332 df-n0 12533 df-z 12620 df-uz 12892 df-q 13002 df-rp 13047 df-xneg 13167 df-xadd 13168 df-xmul 13169 df-ioo 13406 df-ioc 13407 df-ico 13408 df-icc 13409 df-fz 13566 df-fzo 13714 df-fl 13857 df-seq 14070 df-exp 14130 df-hash 14399 df-cj 15190 df-re 15191 df-im 15192 df-sqrt 15326 df-abs 15327 df-clim 15579 df-rlim 15580 df-sum 15778 df-rest 17513 df-topgen 17534 df-psmet 21583 df-xmet 21584 df-met 21585 df-bl 21586 df-mopn 21587 df-top 23125 df-topon 23142 df-bases 23177 df-cmp 23618 df-ovol 25698 df-vol 25699 df-mbf 25853 df-itg1 25854 df-itg2 25855 df-0p 25904

This theorem is used by: itg2i1fseq3 25991 itg2addlem 25992

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