Theorem itg1addlem2 25619

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 25621 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 4533	. . . . . . . 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 25600	. . . . . . . . . . 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 25600	. . . . . . . . . . 11 ⊢ (𝐺 ∈ dom ∫1 → (◡𝐺 “ {𝑗}) ∈ dom vol)
8	6, 7	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (◡𝐺 “ {𝑗}) ∈ dom vol)
9		inmbl 25464	. . . . . . . . . 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 725	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → ((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) ∈ dom vol)
12		mblvol 25452	. . . . . . . 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 2768	. . . . . 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 3033	. . . . . . 7 ⊢ ((𝑖 ≠ 0 ∨ 𝑗 ≠ 0) ↔ ¬ (𝑖 = 0 ∧ 𝑗 = 0))
16		inss1 4224	. . . . . . . . 9 ⊢ ((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) ⊆ (◡𝐹 “ {𝑖})
17	5	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (◡𝐹 “ {𝑖}) ∈ dom vol)
18		mblss 25453	. . . . . . . . . 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 25452	. . . . . . . . . . 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 725	. . . . . . . . . . 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 4786	. . . . . . . . . . . 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 25602	. . . . . . . . . . 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 2830	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol*‘(◡𝐹 “ {𝑖})) ∈ ℝ)
30		ovolsscl 25408	. . . . . . . . 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 1463	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol*‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))) ∈ ℝ)
32		inss2 4225	. . . . . . . . 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 25453	. . . . . . . . . 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 25452	. . . . . . . . . . 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 725	. . . . . . . . . . 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 4786	. . . . . . . . . . . 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 25602	. . . . . . . . . . 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 2830	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol*‘(◡𝐺 “ {𝑗})) ∈ ℝ)
48		ovolsscl 25408	. . . . . . . . 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 1463	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol*‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))) ∈ ℝ)
50	31, 49	jaodan 956	. . . . . . 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 2829	. . . . 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 4530	. . . . 5 ⊢ ((𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))) = 0)
55		0re 11240	. . . . 5 ⊢ 0 ∈ ℝ
56	54, 55	eqeltrdi 2837	. . . 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 3196	. 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 8066	. 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 846   = wceq 1534   ∈ wcel 2099   ≠ wne 2936  ∀wral 3057   ∖ cdif 3942   ∩ cin 3944   ⊆ wss 3945  ifcif 4524  {csn 4624   × cxp 5670  ^◡ccnv 5671  dom cdm 5672   “ cima 5675  ⟶wf 6538  ‘cfv 6542   ∈ cmpo 7416  ℝcr 11131  0cc0 11132  vol*covol 25384  volcvol 25385  ∫₁citg1 25537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9658  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209  ax-pre-sup 11210

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7679  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8718  df-map 8840  df-pm 8841  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9459  df-inf 9460  df-oi 9527  df-dju 9918  df-card 9956  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-div 11896  df-nn 12237  df-2 12299  df-3 12300  df-n0 12497  df-z 12583  df-uz 12847  df-q 12957  df-rp 13001  df-xadd 13119  df-ioo 13354  df-ico 13356  df-icc 13357  df-fz 13511  df-fzo 13654  df-fl 13783  df-seq 13993  df-exp 14053  df-hash 14316  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15458  df-sum 15659  df-xmet 21265  df-met 21266  df-ovol 25386  df-vol 25387  df-mbf 25541  df-itg1 25542

This theorem is referenced by:  itg1addlem4  25621  itg1addlem4OLD  25622  itg1addlem5  25623