Theorem itg1addlem2 25620

Description: Lemma for itg1add 25624. The function 𝐼 represents the pieces into which we will break up the domain of the sum. Since it is infinite only when both 𝑖 and 𝑗 are zero, we arbitrarily define it to be zero there to simplify the sums that are manipulated in itg1addlem4 25622 and itg1addlem5 25623. (Contributed by Mario Carneiro, 26-Jun-2014.)

Hypotheses

Ref	Expression
i1fadd.1	⊢ (𝜑 → 𝐹 ∈ dom ∫1)
i1fadd.2	⊢ (𝜑 → 𝐺 ∈ dom ∫1)
itg1add.3	⊢ 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))))

Assertion

Ref	Expression
itg1addlem2	⊢ (𝜑 → 𝐼:(ℝ × ℝ)⟶ℝ)

Distinct variable groups:   𝑖,𝑗,𝐹   𝑖,𝐺,𝑗   𝜑,𝑖,𝑗

Allowed substitution hints:   𝐼(𝑖,𝑗)

Proof of Theorem itg1addlem2

Step	Hyp	Ref	Expression
1		iffalse 4479	. . . . . . . 8 ⊢ (¬ (𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))) = (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))))
2	1	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))) = (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))))
3		i1fadd.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ dom ∫1)
4		i1fima 25601	. . . . . . . . . . 11 ⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ {𝑖}) ∈ dom vol)
5	3, 4	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (◡𝐹 “ {𝑖}) ∈ dom vol)
6		i1fadd.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ dom ∫1)
7		i1fima 25601	. . . . . . . . . . 11 ⊢ (𝐺 ∈ dom ∫1 → (◡𝐺 “ {𝑗}) ∈ dom vol)
8	6, 7	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (◡𝐺 “ {𝑗}) ∈ dom vol)
9		inmbl 25465	. . . . . . . . . 10 ⊢ (((◡𝐹 “ {𝑖}) ∈ dom vol ∧ (◡𝐺 “ {𝑗}) ∈ dom vol) → ((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) ∈ dom vol)
10	5, 8, 9	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) ∈ dom vol)
11	10	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → ((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) ∈ dom vol)
12		mblvol 25453	. . . . . . . 8 ⊢ (((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) ∈ dom vol → (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))) = (vol*‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))))
13	11, 12	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))) = (vol*‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))))
14	2, 13	eqtrd 2766	. . . . . 6 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))) = (vol*‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))))
15		neorian 3023	. . . . . . 7 ⊢ ((𝑖 ≠ 0 ∨ 𝑗 ≠ 0) ↔ ¬ (𝑖 = 0 ∧ 𝑗 = 0))
16		inss1 4182	. . . . . . . . 9 ⊢ ((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) ⊆ (◡𝐹 “ {𝑖})
17	5	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (◡𝐹 “ {𝑖}) ∈ dom vol)
18		mblss 25454	. . . . . . . . . 10 ⊢ ((◡𝐹 “ {𝑖}) ∈ dom vol → (◡𝐹 “ {𝑖}) ⊆ ℝ)
19	17, 18	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (◡𝐹 “ {𝑖}) ⊆ ℝ)
20		mblvol 25453	. . . . . . . . . . 11 ⊢ ((◡𝐹 “ {𝑖}) ∈ dom vol → (vol‘(◡𝐹 “ {𝑖})) = (vol*‘(◡𝐹 “ {𝑖})))
21	17, 20	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol‘(◡𝐹 “ {𝑖})) = (vol*‘(◡𝐹 “ {𝑖})))
22	3	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝐹 ∈ dom ∫1)
23		simplrl 776	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝑖 ∈ ℝ)
24		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝑖 ≠ 0)
25		eldifsn 4733	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (ℝ ∖ {0}) ↔ (𝑖 ∈ ℝ ∧ 𝑖 ≠ 0))
26	23, 24, 25	sylanbrc 583	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝑖 ∈ (ℝ ∖ {0}))
27		i1fima2sn 25603	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑖 ∈ (ℝ ∖ {0})) → (vol‘(◡𝐹 “ {𝑖})) ∈ ℝ)
28	22, 26, 27	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol‘(◡𝐹 “ {𝑖})) ∈ ℝ)
29	21, 28	eqeltrrd 2832	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol*‘(◡𝐹 “ {𝑖})) ∈ ℝ)
30		ovolsscl 25409	. . . . . . . . 9 ⊢ ((((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) ⊆ (◡𝐹 “ {𝑖}) ∧ (◡𝐹 “ {𝑖}) ⊆ ℝ ∧ (vol*‘(◡𝐹 “ {𝑖})) ∈ ℝ) → (vol*‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))) ∈ ℝ)
31	16, 19, 29, 30	mp3an2i 1468	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol*‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))) ∈ ℝ)
32		inss2 4183	. . . . . . . . 9 ⊢ ((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) ⊆ (◡𝐺 “ {𝑗})
33	6	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → 𝐺 ∈ dom ∫1)
34	33, 7	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (◡𝐺 “ {𝑗}) ∈ dom vol)
35	34	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (◡𝐺 “ {𝑗}) ∈ dom vol)
36		mblss 25454	. . . . . . . . . 10 ⊢ ((◡𝐺 “ {𝑗}) ∈ dom vol → (◡𝐺 “ {𝑗}) ⊆ ℝ)
37	35, 36	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (◡𝐺 “ {𝑗}) ⊆ ℝ)
38		mblvol 25453	. . . . . . . . . . 11 ⊢ ((◡𝐺 “ {𝑗}) ∈ dom vol → (vol‘(◡𝐺 “ {𝑗})) = (vol*‘(◡𝐺 “ {𝑗})))
39	35, 38	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol‘(◡𝐺 “ {𝑗})) = (vol*‘(◡𝐺 “ {𝑗})))
40	6	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝐺 ∈ dom ∫1)
41		simplrr 777	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝑗 ∈ ℝ)
42		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝑗 ≠ 0)
43		eldifsn 4733	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℝ ∖ {0}) ↔ (𝑗 ∈ ℝ ∧ 𝑗 ≠ 0))
44	41, 42, 43	sylanbrc 583	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝑗 ∈ (ℝ ∖ {0}))
45		i1fima2sn 25603	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝑗 ∈ (ℝ ∖ {0})) → (vol‘(◡𝐺 “ {𝑗})) ∈ ℝ)
46	40, 44, 45	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol‘(◡𝐺 “ {𝑗})) ∈ ℝ)
47	39, 46	eqeltrrd 2832	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol*‘(◡𝐺 “ {𝑗})) ∈ ℝ)
48		ovolsscl 25409	. . . . . . . . 9 ⊢ ((((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) ⊆ (◡𝐺 “ {𝑗}) ∧ (◡𝐺 “ {𝑗}) ⊆ ℝ ∧ (vol*‘(◡𝐺 “ {𝑗})) ∈ ℝ) → (vol*‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))) ∈ ℝ)
49	32, 37, 47, 48	mp3an2i 1468	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol*‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))) ∈ ℝ)
50	31, 49	jaodan 959	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ (𝑖 ≠ 0 ∨ 𝑗 ≠ 0)) → (vol*‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))) ∈ ℝ)
51	15, 50	sylan2br 595	. . . . . 6 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → (vol*‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))) ∈ ℝ)
52	14, 51	eqeltrd 2831	. . . . 5 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))) ∈ ℝ)
53	52	ex 412	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (¬ (𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))) ∈ ℝ))
54		iftrue 4476	. . . . 5 ⊢ ((𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))) = 0)
55		0re 11109	. . . . 5 ⊢ 0 ∈ ℝ
56	54, 55	eqeltrdi 2839	. . . 4 ⊢ ((𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))) ∈ ℝ)
57	53, 56	pm2.61d2 181	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))) ∈ ℝ)
58	57	ralrimivva 3175	. 2 ⊢ (𝜑 → ∀𝑖 ∈ ℝ ∀𝑗 ∈ ℝ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))) ∈ ℝ)
59		itg1add.3	. . 3 ⊢ 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))))
60	59	fmpo 7995	. 2 ⊢ (∀𝑖 ∈ ℝ ∀𝑗 ∈ ℝ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))) ∈ ℝ ↔ 𝐼:(ℝ × ℝ)⟶ℝ)
61	58, 60	sylib 218	1 ⊢ (𝜑 → 𝐼:(ℝ × ℝ)⟶ℝ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047   ∖ cdif 3894   ∩ cin 3896   ⊆ wss 3897  ifcif 4470  {csn 4571   × cxp 5609  ^◡ccnv 5610  dom cdm 5611   “ cima 5614  ⟶wf 6472  ‘cfv 6476   ∈ cmpo 7343  ℝcr 11000  0cc0 11001  vol*covol 25385  volcvol 25386  ∫₁citg1 25538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-inf2 9526  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078  ax-pre-sup 11079

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-se 5565  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-isom 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7605  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-er 8617  df-map 8747  df-pm 8748  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-sup 9321  df-inf 9322  df-oi 9391  df-dju 9789  df-card 9827  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-div 11770  df-nn 12121  df-2 12183  df-3 12184  df-n0 12377  df-z 12464  df-uz 12728  df-q 12842  df-rp 12886  df-xadd 13007  df-ioo 13244  df-ico 13246  df-icc 13247  df-fz 13403  df-fzo 13550  df-fl 13691  df-seq 13904  df-exp 13964  df-hash 14233  df-cj 15001  df-re 15002  df-im 15003  df-sqrt 15137  df-abs 15138  df-clim 15390  df-sum 15589  df-xmet 21279  df-met 21280  df-ovol 25387  df-vol 25388  df-mbf 25542  df-itg1 25543

This theorem is referenced by:  itg1addlem4  25622  itg1addlem5  25623