Theorem itg1addlem2 24766

Description: Lemma for itg1add 24771. 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 24768 and itg1addlem5 24770. (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 4465	. . . . . . . 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 24747	. . . . . . . . . . 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 24747	. . . . . . . . . . 11 ⊢ (𝐺 ∈ dom ∫1 → (◡𝐺 “ {𝑗}) ∈ dom vol)
8	6, 7	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (◡𝐺 “ {𝑗}) ∈ dom vol)
9		inmbl 24611	. . . . . . . . . 10 ⊢ (((◡𝐹 “ {𝑖}) ∈ dom vol ∧ (◡𝐺 “ {𝑗}) ∈ dom vol) → ((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) ∈ dom vol)
10	5, 8, 9	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → ((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) ∈ dom vol)
11	10	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → ((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) ∈ dom vol)
12		mblvol 24599	. . . . . . . 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 2778	. . . . . 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 3038	. . . . . . 7 ⊢ ((𝑖 ≠ 0 ∨ 𝑗 ≠ 0) ↔ ¬ (𝑖 = 0 ∧ 𝑗 = 0))
16		inss1 4159	. . . . . . . . 9 ⊢ ((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) ⊆ (◡𝐹 “ {𝑖})
17	5	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (◡𝐹 “ {𝑖}) ∈ dom vol)
18		mblss 24600	. . . . . . . . . 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 24599	. . . . . . . . . . 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 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝐹 ∈ dom ∫1)
23		simplrl 773	. . . . . . . . . . . 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 4717	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (ℝ ∖ {0}) ↔ (𝑖 ∈ ℝ ∧ 𝑖 ≠ 0))
26	23, 24, 25	sylanbrc 582	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝑖 ∈ (ℝ ∖ {0}))
27		i1fima2sn 24749	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑖 ∈ (ℝ ∖ {0})) → (vol‘(◡𝐹 “ {𝑖})) ∈ ℝ)
28	22, 26, 27	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol‘(◡𝐹 “ {𝑖})) ∈ ℝ)
29	21, 28	eqeltrrd 2840	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol*‘(◡𝐹 “ {𝑖})) ∈ ℝ)
30		ovolsscl 24555	. . . . . . . . 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 1464	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol*‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))) ∈ ℝ)
32		inss2 4160	. . . . . . . . 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 24600	. . . . . . . . . 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 24599	. . . . . . . . . . 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 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝐺 ∈ dom ∫1)
41		simplrr 774	. . . . . . . . . . . 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 4717	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℝ ∖ {0}) ↔ (𝑗 ∈ ℝ ∧ 𝑗 ≠ 0))
44	41, 42, 43	sylanbrc 582	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝑗 ∈ (ℝ ∖ {0}))
45		i1fima2sn 24749	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝑗 ∈ (ℝ ∖ {0})) → (vol‘(◡𝐺 “ {𝑗})) ∈ ℝ)
46	40, 44, 45	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol‘(◡𝐺 “ {𝑗})) ∈ ℝ)
47	39, 46	eqeltrrd 2840	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol*‘(◡𝐺 “ {𝑗})) ∈ ℝ)
48		ovolsscl 24555	. . . . . . . . 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 1464	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol*‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))) ∈ ℝ)
50	31, 49	jaodan 954	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ (𝑖 ≠ 0 ∨ 𝑗 ≠ 0)) → (vol*‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))) ∈ ℝ)
51	15, 50	sylan2br 594	. . . . . 6 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → (vol*‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))) ∈ ℝ)
52	14, 51	eqeltrd 2839	. . . . 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 4462	. . . . 5 ⊢ ((𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))) = 0)
55		0re 10908	. . . . 5 ⊢ 0 ∈ ℝ
56	54, 55	eqeltrdi 2847	. . . 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 3114	. 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 7881	. 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 217	1 ⊢ (𝜑 → 𝐼:(ℝ × ℝ)⟶ℝ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ifcif 4456  {csn 4558   × cxp 5578  ^◡ccnv 5579  dom cdm 5580   “ cima 5583  ⟶wf 6414  ‘cfv 6418   ∈ cmpo 7257  ℝcr 10801  0cc0 10802  vol*covol 24531  volcvol 24532  ∫₁citg1 24684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-inf 9132  df-oi 9199  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-q 12618  df-rp 12660  df-xadd 12778  df-ioo 13012  df-ico 13014  df-icc 13015  df-fz 13169  df-fzo 13312  df-fl 13440  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-sum 15326  df-xmet 20503  df-met 20504  df-ovol 24533  df-vol 24534  df-mbf 24688  df-itg1 24689

This theorem is referenced by:  itg1addlem4  24768  itg1addlem4OLD  24769  itg1addlem5  24770