Theorem itg1addlem2 25614

Description: Lemma for itg1add 25618. 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 25616 and itg1addlem5 25617. (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 4487	. . . . . . . 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 25595	. . . . . . . . . . 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 25595	. . . . . . . . . . 11 ⊢ (𝐺 ∈ dom ∫1 → (◡𝐺 “ {𝑗}) ∈ dom vol)
8	6, 7	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (◡𝐺 “ {𝑗}) ∈ dom vol)
9		inmbl 25459	. . . . . . . . . 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 25447	. . . . . . . 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 2764	. . . . . 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 3020	. . . . . . 7 ⊢ ((𝑖 ≠ 0 ∨ 𝑗 ≠ 0) ↔ ¬ (𝑖 = 0 ∧ 𝑗 = 0))
16		inss1 4190	. . . . . . . . 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 25448	. . . . . . . . . 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 25447	. . . . . . . . . . 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 4740	. . . . . . . . . . . 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 25597	. . . . . . . . . . 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 2829	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol*‘(◡𝐹 “ {𝑖})) ∈ ℝ)
30		ovolsscl 25403	. . . . . . . . 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 4191	. . . . . . . . 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 25448	. . . . . . . . . 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 25447	. . . . . . . . . . 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 4740	. . . . . . . . . . . 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 25597	. . . . . . . . . . 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 2829	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol*‘(◡𝐺 “ {𝑗})) ∈ ℝ)
48		ovolsscl 25403	. . . . . . . . 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 2828	. . . . 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 4484	. . . . 5 ⊢ ((𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))) = 0)
55		0re 11136	. . . . 5 ⊢ 0 ∈ ℝ
56	54, 55	eqeltrdi 2836	. . . 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 3172	. 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 8010	. 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 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  ifcif 4478  {csn 4579   × cxp 5621  ^◡ccnv 5622  dom cdm 5623   “ cima 5626  ⟶wf 6482  ‘cfv 6486   ∈ cmpo 7355  ℝcr 11027  0cc0 11028  vol*covol 25379  volcvol 25380  ∫₁citg1 25532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-inf2 9556  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-of 7617  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8632  df-map 8762  df-pm 8763  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-sup 9351  df-inf 9352  df-oi 9421  df-dju 9816  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-div 11796  df-nn 12147  df-2 12209  df-3 12210  df-n0 12403  df-z 12490  df-uz 12754  df-q 12868  df-rp 12912  df-xadd 13033  df-ioo 13270  df-ico 13272  df-icc 13273  df-fz 13429  df-fzo 13576  df-fl 13714  df-seq 13927  df-exp 13987  df-hash 14256  df-cj 15024  df-re 15025  df-im 15026  df-sqrt 15160  df-abs 15161  df-clim 15413  df-sum 15612  df-xmet 21272  df-met 21273  df-ovol 25381  df-vol 25382  df-mbf 25536  df-itg1 25537

This theorem is referenced by:  itg1addlem4  25616  itg1addlem5  25617