Theorem itg1addlem2 24298

Description: Lemma for itg1add 24302. 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 24300 and itg1addlem5 24301. (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 4476	. . . . . . . 8 ⊢ (¬ (𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))) = (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))))
2	1	adantl 484	. . . . . . 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 24279	. . . . . . . . . . 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 24279	. . . . . . . . . . 11 ⊢ (𝐺 ∈ dom ∫1 → (◡𝐺 “ {𝑗}) ∈ dom vol)
8	6, 7	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (◡𝐺 “ {𝑗}) ∈ dom vol)
9		inmbl 24143	. . . . . . . . . 10 ⊢ (((◡𝐹 “ {𝑖}) ∈ dom vol ∧ (◡𝐺 “ {𝑗}) ∈ dom vol) → ((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) ∈ dom vol)
10	5, 8, 9	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → ((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) ∈ dom vol)
11	10	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → ((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) ∈ dom vol)
12		mblvol 24131	. . . . . . . 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 2856	. . . . . 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 3111	. . . . . . 7 ⊢ ((𝑖 ≠ 0 ∨ 𝑗 ≠ 0) ↔ ¬ (𝑖 = 0 ∧ 𝑗 = 0))
16		inss1 4205	. . . . . . . . 9 ⊢ ((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) ⊆ (◡𝐹 “ {𝑖})
17	5	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (◡𝐹 “ {𝑖}) ∈ dom vol)
18		mblss 24132	. . . . . . . . . 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 24131	. . . . . . . . . . 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 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝐹 ∈ dom ∫1)
23		simplrl 775	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝑖 ∈ ℝ)
24		simpr 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝑖 ≠ 0)
25		eldifsn 4719	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (ℝ ∖ {0}) ↔ (𝑖 ∈ ℝ ∧ 𝑖 ≠ 0))
26	23, 24, 25	sylanbrc 585	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝑖 ∈ (ℝ ∖ {0}))
27		i1fima2sn 24281	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑖 ∈ (ℝ ∖ {0})) → (vol‘(◡𝐹 “ {𝑖})) ∈ ℝ)
28	22, 26, 27	syl2anc 586	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol‘(◡𝐹 “ {𝑖})) ∈ ℝ)
29	21, 28	eqeltrrd 2914	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol*‘(◡𝐹 “ {𝑖})) ∈ ℝ)
30		ovolsscl 24087	. . . . . . . . 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 1462	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol*‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))) ∈ ℝ)
32		inss2 4206	. . . . . . . . 9 ⊢ ((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) ⊆ (◡𝐺 “ {𝑗})
33	6	adantr 483	. . . . . . . . . . . 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 483	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (◡𝐺 “ {𝑗}) ∈ dom vol)
36		mblss 24132	. . . . . . . . . 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 24131	. . . . . . . . . . 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 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝐺 ∈ dom ∫1)
41		simplrr 776	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝑗 ∈ ℝ)
42		simpr 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝑗 ≠ 0)
43		eldifsn 4719	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℝ ∖ {0}) ↔ (𝑗 ∈ ℝ ∧ 𝑗 ≠ 0))
44	41, 42, 43	sylanbrc 585	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝑗 ∈ (ℝ ∖ {0}))
45		i1fima2sn 24281	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝑗 ∈ (ℝ ∖ {0})) → (vol‘(◡𝐺 “ {𝑗})) ∈ ℝ)
46	40, 44, 45	syl2anc 586	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol‘(◡𝐺 “ {𝑗})) ∈ ℝ)
47	39, 46	eqeltrrd 2914	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol*‘(◡𝐺 “ {𝑗})) ∈ ℝ)
48		ovolsscl 24087	. . . . . . . . 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 1462	. . . . . . . 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 596	. . . . . 6 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → (vol*‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))) ∈ ℝ)
52	14, 51	eqeltrd 2913	. . . . 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 415	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (¬ (𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))) ∈ ℝ))
54		iftrue 4473	. . . . 5 ⊢ ((𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))) = 0)
55		0re 10643	. . . . 5 ⊢ 0 ∈ ℝ
56	54, 55	eqeltrdi 2921	. . . 4 ⊢ ((𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))) ∈ ℝ)
57	53, 56	pm2.61d2 183	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))) ∈ ℝ)
58	57	ralrimivva 3191	. 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 7766	. 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 220	1 ⊢ (𝜑 → 𝐼:(ℝ × ℝ)⟶ℝ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138   ∖ cdif 3933   ∩ cin 3935   ⊆ wss 3936  ifcif 4467  {csn 4567   × cxp 5553  ^◡ccnv 5554  dom cdm 5555   “ cima 5558  ⟶wf 6351  ‘cfv 6355   ∈ cmpo 7158  ℝcr 10536  0cc0 10537  vol*covol 24063  volcvol 24064  ∫₁citg1 24216

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7409  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-er 8289  df-map 8408  df-pm 8409  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-sup 8906  df-inf 8907  df-oi 8974  df-dju 9330  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11639  df-2 11701  df-3 11702  df-n0 11899  df-z 11983  df-uz 12245  df-q 12350  df-rp 12391  df-xadd 12509  df-ioo 12743  df-ico 12745  df-icc 12746  df-fz 12894  df-fzo 13035  df-fl 13163  df-seq 13371  df-exp 13431  df-hash 13692  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-clim 14845  df-sum 15043  df-xmet 20538  df-met 20539  df-ovol 24065  df-vol 24066  df-mbf 24220  df-itg1 24221

This theorem is referenced by:  itg1addlem4  24300  itg1addlem5  24301