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Theorem itg1addlem2 23370
Description: Lemma for itg1add 23374. 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 23372 and itg1addlem5 23373. (Contributed by Mario Carneiro, 26-Jun-2014.)
Hypotheses
Ref Expression
i1fadd.1 (𝜑𝐹 ∈ dom ∫1)
i1fadd.2 (𝜑𝐺 ∈ dom ∫1)
itg1add.3 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))))
Assertion
Ref Expression
itg1addlem2 (𝜑𝐼:(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
StepHypRef Expression
1 iffalse 4067 . . . . . . . 8 (¬ (𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) = (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))))
21adantl 482 . . . . . . 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 (𝜑𝐹 ∈ dom ∫1)
4 i1fima 23351 . . . . . . . . . . 11 (𝐹 ∈ dom ∫1 → (𝐹 “ {𝑖}) ∈ dom vol)
53, 4syl 17 . . . . . . . . . 10 (𝜑 → (𝐹 “ {𝑖}) ∈ dom vol)
6 i1fadd.2 . . . . . . . . . . 11 (𝜑𝐺 ∈ dom ∫1)
7 i1fima 23351 . . . . . . . . . . 11 (𝐺 ∈ dom ∫1 → (𝐺 “ {𝑗}) ∈ dom vol)
86, 7syl 17 . . . . . . . . . 10 (𝜑 → (𝐺 “ {𝑗}) ∈ dom vol)
9 inmbl 23217 . . . . . . . . . 10 (((𝐹 “ {𝑖}) ∈ dom vol ∧ (𝐺 “ {𝑗}) ∈ dom vol) → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ∈ dom vol)
105, 8, 9syl2anc 692 . . . . . . . . 9 (𝜑 → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ∈ dom vol)
1110ad2antrr 761 . . . . . . . 8 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ∈ dom vol)
12 mblvol 23205 . . . . . . . 8 (((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ∈ dom vol → (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) = (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))))
1311, 12syl 17 . . . . . . 7 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) = (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))))
142, 13eqtrd 2655 . . . . . 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 2884 . . . . . . 7 ((𝑖 ≠ 0 ∨ 𝑗 ≠ 0) ↔ ¬ (𝑖 = 0 ∧ 𝑗 = 0))
16 inss1 3811 . . . . . . . . . 10 ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐹 “ {𝑖})
1716a1i 11 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐹 “ {𝑖}))
185ad2antrr 761 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (𝐹 “ {𝑖}) ∈ dom vol)
19 mblss 23206 . . . . . . . . . 10 ((𝐹 “ {𝑖}) ∈ dom vol → (𝐹 “ {𝑖}) ⊆ ℝ)
2018, 19syl 17 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (𝐹 “ {𝑖}) ⊆ ℝ)
21 mblvol 23205 . . . . . . . . . . 11 ((𝐹 “ {𝑖}) ∈ dom vol → (vol‘(𝐹 “ {𝑖})) = (vol*‘(𝐹 “ {𝑖})))
2218, 21syl 17 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol‘(𝐹 “ {𝑖})) = (vol*‘(𝐹 “ {𝑖})))
233ad2antrr 761 . . . . . . . . . . 11 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝐹 ∈ dom ∫1)
24 simplrl 799 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝑖 ∈ ℝ)
25 simpr 477 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝑖 ≠ 0)
26 eldifsn 4287 . . . . . . . . . . . 12 (𝑖 ∈ (ℝ ∖ {0}) ↔ (𝑖 ∈ ℝ ∧ 𝑖 ≠ 0))
2724, 25, 26sylanbrc 697 . . . . . . . . . . 11 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝑖 ∈ (ℝ ∖ {0}))
28 i1fima2sn 23353 . . . . . . . . . . 11 ((𝐹 ∈ dom ∫1𝑖 ∈ (ℝ ∖ {0})) → (vol‘(𝐹 “ {𝑖})) ∈ ℝ)
2923, 27, 28syl2anc 692 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol‘(𝐹 “ {𝑖})) ∈ ℝ)
3022, 29eqeltrrd 2699 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol*‘(𝐹 “ {𝑖})) ∈ ℝ)
31 ovolsscl 23161 . . . . . . . . 9 ((((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐹 “ {𝑖}) ∧ (𝐹 “ {𝑖}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {𝑖})) ∈ ℝ) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
3217, 20, 30, 31syl3anc 1323 . . . . . . . 8 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
33 inss2 3812 . . . . . . . . . 10 ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐺 “ {𝑗})
3433a1i 11 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐺 “ {𝑗}))
356adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → 𝐺 ∈ dom ∫1)
3635, 7syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝐺 “ {𝑗}) ∈ dom vol)
3736adantr 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (𝐺 “ {𝑗}) ∈ dom vol)
38 mblss 23206 . . . . . . . . . 10 ((𝐺 “ {𝑗}) ∈ dom vol → (𝐺 “ {𝑗}) ⊆ ℝ)
3937, 38syl 17 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (𝐺 “ {𝑗}) ⊆ ℝ)
40 mblvol 23205 . . . . . . . . . . 11 ((𝐺 “ {𝑗}) ∈ dom vol → (vol‘(𝐺 “ {𝑗})) = (vol*‘(𝐺 “ {𝑗})))
4137, 40syl 17 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol‘(𝐺 “ {𝑗})) = (vol*‘(𝐺 “ {𝑗})))
426ad2antrr 761 . . . . . . . . . . 11 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝐺 ∈ dom ∫1)
43 simplrr 800 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝑗 ∈ ℝ)
44 simpr 477 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝑗 ≠ 0)
45 eldifsn 4287 . . . . . . . . . . . 12 (𝑗 ∈ (ℝ ∖ {0}) ↔ (𝑗 ∈ ℝ ∧ 𝑗 ≠ 0))
4643, 44, 45sylanbrc 697 . . . . . . . . . . 11 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝑗 ∈ (ℝ ∖ {0}))
47 i1fima2sn 23353 . . . . . . . . . . 11 ((𝐺 ∈ dom ∫1𝑗 ∈ (ℝ ∖ {0})) → (vol‘(𝐺 “ {𝑗})) ∈ ℝ)
4842, 46, 47syl2anc 692 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol‘(𝐺 “ {𝑗})) ∈ ℝ)
4941, 48eqeltrrd 2699 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol*‘(𝐺 “ {𝑗})) ∈ ℝ)
50 ovolsscl 23161 . . . . . . . . 9 ((((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐺 “ {𝑗}) ∧ (𝐺 “ {𝑗}) ⊆ ℝ ∧ (vol*‘(𝐺 “ {𝑗})) ∈ ℝ) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
5134, 39, 49, 50syl3anc 1323 . . . . . . . 8 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
5232, 51jaodan 825 . . . . . . 7 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ (𝑖 ≠ 0 ∨ 𝑗 ≠ 0)) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
5315, 52sylan2br 493 . . . . . 6 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
5414, 53eqeltrd 2698 . . . . 5 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) ∈ ℝ)
5554ex 450 . . . 4 ((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (¬ (𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) ∈ ℝ))
56 iftrue 4064 . . . . 5 ((𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) = 0)
57 0re 9984 . . . . 5 0 ∈ ℝ
5856, 57syl6eqel 2706 . . . 4 ((𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) ∈ ℝ)
5955, 58pm2.61d2 172 . . 3 ((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) ∈ ℝ)
6059ralrimivva 2965 . 2 (𝜑 → ∀𝑖 ∈ ℝ ∀𝑗 ∈ ℝ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) ∈ ℝ)
61 itg1add.3 . . 3 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))))
6261fmpt2 7182 . 2 (∀𝑖 ∈ ℝ ∀𝑗 ∈ ℝ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) ∈ ℝ ↔ 𝐼:(ℝ × ℝ)⟶ℝ)
6360, 62sylib 208 1 (𝜑𝐼:(ℝ × ℝ)⟶ℝ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  cdif 3552  cin 3554  wss 3555  ifcif 4058  {csn 4148   × cxp 5072  ccnv 5073  dom cdm 5074  cima 5077  wf 5843  cfv 5847  cmpt2 6606  cr 9879  0cc0 9880  vol*covol 23138  volcvol 23139  1citg1 23290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-q 11733  df-rp 11777  df-xadd 11891  df-ioo 12121  df-ico 12123  df-icc 12124  df-fz 12269  df-fzo 12407  df-fl 12533  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-sum 14351  df-xmet 19658  df-met 19659  df-ovol 23140  df-vol 23141  df-mbf 23294  df-itg1 23295
This theorem is referenced by:  itg1addlem4  23372  itg1addlem5  23373
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