Theorem uniioombllem3a 23352

Description: Lemma for uniioombl 23357. (Contributed by Mario Carneiro, 8-May-2015.)

Hypotheses

Ref	Expression
uniioombl.1	⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
uniioombl.2	⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))
uniioombl.3	⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
uniioombl.a	⊢ 𝐴 = ∪ ran ((,) ∘ 𝐹)
uniioombl.e	⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)
uniioombl.c	⊢ (𝜑 → 𝐶 ∈ ℝ+)
uniioombl.g	⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
uniioombl.s	⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐺))
uniioombl.t	⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
uniioombl.v	⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶))
uniioombl.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
uniioombl.m2	⊢ (𝜑 → (abs‘((𝑇‘𝑀) − sup(ran 𝑇, ℝ*, < ))) < 𝐶)
uniioombl.k	⊢ 𝐾 = ∪ (((,) ∘ 𝐺) “ (1...𝑀))

Assertion

Ref	Expression
uniioombllem3a	⊢ (𝜑 → (𝐾 = ∪ 𝑗 ∈ (1...𝑀)((,)‘(𝐺‘𝑗)) ∧ (vol*‘𝐾) ∈ ℝ))

Distinct variable groups:   𝑥,𝑗,𝐹   𝑗,𝐺,𝑥   𝑗,𝐾,𝑥   𝐴,𝑗,𝑥   𝐶,𝑗,𝑥   𝑗,𝑀,𝑥   𝜑,𝑗,𝑥   𝑇,𝑗,𝑥

Allowed substitution hints:   𝑆(𝑥,𝑗)   𝐸(𝑥,𝑗)

Proof of Theorem uniioombllem3a

Step	Hyp	Ref	Expression
1		uniioombl.k	. . 3 ⊢ 𝐾 = ∪ (((,) ∘ 𝐺) “ (1...𝑀))
2		ioof 12271	. . . . . 6 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
3		uniioombl.g	. . . . . . 7 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
4		inss2 3834	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
5		rexpssxrxp 10084	. . . . . . . 8 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
6	4, 5	sstri 3612	. . . . . . 7 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
7		fss 6056	. . . . . . 7 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐺:ℕ⟶(ℝ* × ℝ*))
8	3, 6, 7	sylancl 694	. . . . . 6 ⊢ (𝜑 → 𝐺:ℕ⟶(ℝ* × ℝ*))
9		fco 6058	. . . . . 6 ⊢ (((,):(ℝ* × ℝ*)⟶𝒫 ℝ ∧ 𝐺:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
10	2, 8, 9	sylancr 695	. . . . 5 ⊢ (𝜑 → ((,) ∘ 𝐺):ℕ⟶𝒫 ℝ)
11		ffun 6048	. . . . 5 ⊢ (((,) ∘ 𝐺):ℕ⟶𝒫 ℝ → Fun ((,) ∘ 𝐺))
12		funiunfv 6506	. . . . 5 ⊢ (Fun ((,) ∘ 𝐺) → ∪ 𝑗 ∈ (1...𝑀)(((,) ∘ 𝐺)‘𝑗) = ∪ (((,) ∘ 𝐺) “ (1...𝑀)))
13	10, 11, 12	3syl 18	. . . 4 ⊢ (𝜑 → ∪ 𝑗 ∈ (1...𝑀)(((,) ∘ 𝐺)‘𝑗) = ∪ (((,) ∘ 𝐺) “ (1...𝑀)))
14		elfznn 12370	. . . . . 6 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℕ)
15		fvco3 6275	. . . . . 6 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (((,) ∘ 𝐺)‘𝑗) = ((,)‘(𝐺‘𝑗)))
16	3, 14, 15	syl2an 494	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (((,) ∘ 𝐺)‘𝑗) = ((,)‘(𝐺‘𝑗)))
17	16	iuneq2dv 4542	. . . 4 ⊢ (𝜑 → ∪ 𝑗 ∈ (1...𝑀)(((,) ∘ 𝐺)‘𝑗) = ∪ 𝑗 ∈ (1...𝑀)((,)‘(𝐺‘𝑗)))
18	13, 17	eqtr3d 2658	. . 3 ⊢ (𝜑 → ∪ (((,) ∘ 𝐺) “ (1...𝑀)) = ∪ 𝑗 ∈ (1...𝑀)((,)‘(𝐺‘𝑗)))
19	1, 18	syl5eq 2668	. 2 ⊢ (𝜑 → 𝐾 = ∪ 𝑗 ∈ (1...𝑀)((,)‘(𝐺‘𝑗)))
20		ffvelrn 6357	. . . . . . . . . . . 12 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → (𝐺‘𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
21	3, 14, 20	syl2an 494	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (𝐺‘𝑗) ∈ ( ≤ ∩ (ℝ × ℝ)))
22	4, 21	sseldi 3601	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (𝐺‘𝑗) ∈ (ℝ × ℝ))
23		1st2nd2 7205	. . . . . . . . . 10 ⊢ ((𝐺‘𝑗) ∈ (ℝ × ℝ) → (𝐺‘𝑗) = ⟨(1st ‘(𝐺‘𝑗)), (2nd ‘(𝐺‘𝑗))⟩)
24	22, 23	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (𝐺‘𝑗) = ⟨(1st ‘(𝐺‘𝑗)), (2nd ‘(𝐺‘𝑗))⟩)
25	24	fveq2d 6195	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺‘𝑗)) = ((,)‘⟨(1st ‘(𝐺‘𝑗)), (2nd ‘(𝐺‘𝑗))⟩))
26		df-ov 6653	. . . . . . . 8 ⊢ ((1st ‘(𝐺‘𝑗))(,)(2nd ‘(𝐺‘𝑗))) = ((,)‘⟨(1st ‘(𝐺‘𝑗)), (2nd ‘(𝐺‘𝑗))⟩)
27	25, 26	syl6eqr 2674	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺‘𝑗)) = ((1st ‘(𝐺‘𝑗))(,)(2nd ‘(𝐺‘𝑗))))
28		ioossre 12235	. . . . . . 7 ⊢ ((1st ‘(𝐺‘𝑗))(,)(2nd ‘(𝐺‘𝑗))) ⊆ ℝ
29	27, 28	syl6eqss 3655	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((,)‘(𝐺‘𝑗)) ⊆ ℝ)
30	29	ralrimiva 2966	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ (1...𝑀)((,)‘(𝐺‘𝑗)) ⊆ ℝ)
31		iunss 4561	. . . . 5 ⊢ (∪ 𝑗 ∈ (1...𝑀)((,)‘(𝐺‘𝑗)) ⊆ ℝ ↔ ∀𝑗 ∈ (1...𝑀)((,)‘(𝐺‘𝑗)) ⊆ ℝ)
32	30, 31	sylibr 224	. . . 4 ⊢ (𝜑 → ∪ 𝑗 ∈ (1...𝑀)((,)‘(𝐺‘𝑗)) ⊆ ℝ)
33	19, 32	eqsstrd 3639	. . 3 ⊢ (𝜑 → 𝐾 ⊆ ℝ)
34		fzfid 12772	. . . 4 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
35	27	fveq2d 6195	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (vol*‘((,)‘(𝐺‘𝑗))) = (vol*‘((1st ‘(𝐺‘𝑗))(,)(2nd ‘(𝐺‘𝑗)))))
36		ovolfcl 23235	. . . . . . . 8 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑗 ∈ ℕ) → ((1st ‘(𝐺‘𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺‘𝑗)) ∈ ℝ ∧ (1st ‘(𝐺‘𝑗)) ≤ (2nd ‘(𝐺‘𝑗))))
37	3, 14, 36	syl2an 494	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((1st ‘(𝐺‘𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺‘𝑗)) ∈ ℝ ∧ (1st ‘(𝐺‘𝑗)) ≤ (2nd ‘(𝐺‘𝑗))))
38		ovolioo 23336	. . . . . . 7 ⊢ (((1st ‘(𝐺‘𝑗)) ∈ ℝ ∧ (2nd ‘(𝐺‘𝑗)) ∈ ℝ ∧ (1st ‘(𝐺‘𝑗)) ≤ (2nd ‘(𝐺‘𝑗))) → (vol*‘((1st ‘(𝐺‘𝑗))(,)(2nd ‘(𝐺‘𝑗)))) = ((2nd ‘(𝐺‘𝑗)) − (1st ‘(𝐺‘𝑗))))
39	37, 38	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (vol*‘((1st ‘(𝐺‘𝑗))(,)(2nd ‘(𝐺‘𝑗)))) = ((2nd ‘(𝐺‘𝑗)) − (1st ‘(𝐺‘𝑗))))
40	35, 39	eqtrd 2656	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (vol*‘((,)‘(𝐺‘𝑗))) = ((2nd ‘(𝐺‘𝑗)) − (1st ‘(𝐺‘𝑗))))
41	37	simp2d 1074	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (2nd ‘(𝐺‘𝑗)) ∈ ℝ)
42	37	simp1d 1073	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (1st ‘(𝐺‘𝑗)) ∈ ℝ)
43	41, 42	resubcld 10458	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → ((2nd ‘(𝐺‘𝑗)) − (1st ‘(𝐺‘𝑗))) ∈ ℝ)
44	40, 43	eqeltrd 2701	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (vol*‘((,)‘(𝐺‘𝑗))) ∈ ℝ)
45	34, 44	fsumrecl 14465	. . 3 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑀)(vol*‘((,)‘(𝐺‘𝑗))) ∈ ℝ)
46	19	fveq2d 6195	. . . 4 ⊢ (𝜑 → (vol*‘𝐾) = (vol*‘∪ 𝑗 ∈ (1...𝑀)((,)‘(𝐺‘𝑗))))
47	29, 44	jca 554	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑀)) → (((,)‘(𝐺‘𝑗)) ⊆ ℝ ∧ (vol*‘((,)‘(𝐺‘𝑗))) ∈ ℝ))
48	47	ralrimiva 2966	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ (1...𝑀)(((,)‘(𝐺‘𝑗)) ⊆ ℝ ∧ (vol*‘((,)‘(𝐺‘𝑗))) ∈ ℝ))
49		ovolfiniun 23269	. . . . 5 ⊢ (((1...𝑀) ∈ Fin ∧ ∀𝑗 ∈ (1...𝑀)(((,)‘(𝐺‘𝑗)) ⊆ ℝ ∧ (vol*‘((,)‘(𝐺‘𝑗))) ∈ ℝ)) → (vol*‘∪ 𝑗 ∈ (1...𝑀)((,)‘(𝐺‘𝑗))) ≤ Σ𝑗 ∈ (1...𝑀)(vol*‘((,)‘(𝐺‘𝑗))))
50	34, 48, 49	syl2anc 693	. . . 4 ⊢ (𝜑 → (vol*‘∪ 𝑗 ∈ (1...𝑀)((,)‘(𝐺‘𝑗))) ≤ Σ𝑗 ∈ (1...𝑀)(vol*‘((,)‘(𝐺‘𝑗))))
51	46, 50	eqbrtrd 4675	. . 3 ⊢ (𝜑 → (vol*‘𝐾) ≤ Σ𝑗 ∈ (1...𝑀)(vol*‘((,)‘(𝐺‘𝑗))))
52		ovollecl 23251	. . 3 ⊢ ((𝐾 ⊆ ℝ ∧ Σ𝑗 ∈ (1...𝑀)(vol*‘((,)‘(𝐺‘𝑗))) ∈ ℝ ∧ (vol*‘𝐾) ≤ Σ𝑗 ∈ (1...𝑀)(vol*‘((,)‘(𝐺‘𝑗)))) → (vol*‘𝐾) ∈ ℝ)
53	33, 45, 51, 52	syl3anc 1326	. 2 ⊢ (𝜑 → (vol*‘𝐾) ∈ ℝ)
54	19, 53	jca 554	1 ⊢ (𝜑 → (𝐾 = ∪ 𝑗 ∈ (1...𝑀)((,)‘(𝐺‘𝑗)) ∧ (vol*‘𝐾) ∈ ℝ))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  ⟨cop 4183  ∪ cuni 4436  ∪ ciun 4520  Disj wdisj 4620   class class class wbr 4653   × cxp 5112  ran crn 5115   “ cima 5117   ∘ ccom 5118  Fun wfun 5882  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167  Fincfn 7955  supcsup 8346  ℝcr 9935  1c1 9937   + caddc 9939  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075   − cmin 10266  ℕcn 11020  ℝ⁺crp 11832  (,)cioo 12175  ...cfz 12326  seqcseq 12801  abscabs 13974  Σcsu 14416  vol*covol 23231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233  df-vol 23234

This theorem is referenced by:  uniioombllem3  23353