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Theorem itg1addlem2 23684
Description: Lemma for itg1add 23688. 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 23686 and itg1addlem5 23687. (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 4234 . . . . . . . 8 (¬ (𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) = (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))))
21adantl 467 . . . . . . 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 23665 . . . . . . . . . . 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 23665 . . . . . . . . . . 11 (𝐺 ∈ dom ∫1 → (𝐺 “ {𝑗}) ∈ dom vol)
86, 7syl 17 . . . . . . . . . 10 (𝜑 → (𝐺 “ {𝑗}) ∈ dom vol)
9 inmbl 23530 . . . . . . . . . 10 (((𝐹 “ {𝑖}) ∈ dom vol ∧ (𝐺 “ {𝑗}) ∈ dom vol) → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ∈ dom vol)
105, 8, 9syl2anc 573 . . . . . . . . 9 (𝜑 → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ∈ dom vol)
1110ad2antrr 705 . . . . . . . 8 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ∈ dom vol)
12 mblvol 23518 . . . . . . . 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 2805 . . . . . 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 3037 . . . . . . 7 ((𝑖 ≠ 0 ∨ 𝑗 ≠ 0) ↔ ¬ (𝑖 = 0 ∧ 𝑗 = 0))
16 inss1 3981 . . . . . . . . . 10 ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐹 “ {𝑖})
1716a1i 11 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐹 “ {𝑖}))
185ad2antrr 705 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (𝐹 “ {𝑖}) ∈ dom vol)
19 mblss 23519 . . . . . . . . . 10 ((𝐹 “ {𝑖}) ∈ dom vol → (𝐹 “ {𝑖}) ⊆ ℝ)
2018, 19syl 17 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (𝐹 “ {𝑖}) ⊆ ℝ)
21 mblvol 23518 . . . . . . . . . . 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 705 . . . . . . . . . . 11 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝐹 ∈ dom ∫1)
24 simplrl 762 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝑖 ∈ ℝ)
25 simpr 471 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝑖 ≠ 0)
26 eldifsn 4453 . . . . . . . . . . . 12 (𝑖 ∈ (ℝ ∖ {0}) ↔ (𝑖 ∈ ℝ ∧ 𝑖 ≠ 0))
2724, 25, 26sylanbrc 572 . . . . . . . . . . 11 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝑖 ∈ (ℝ ∖ {0}))
28 i1fima2sn 23667 . . . . . . . . . . 11 ((𝐹 ∈ dom ∫1𝑖 ∈ (ℝ ∖ {0})) → (vol‘(𝐹 “ {𝑖})) ∈ ℝ)
2923, 27, 28syl2anc 573 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol‘(𝐹 “ {𝑖})) ∈ ℝ)
3022, 29eqeltrrd 2851 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol*‘(𝐹 “ {𝑖})) ∈ ℝ)
31 ovolsscl 23474 . . . . . . . . 9 ((((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐹 “ {𝑖}) ∧ (𝐹 “ {𝑖}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {𝑖})) ∈ ℝ) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
3217, 20, 30, 31syl3anc 1476 . . . . . . . 8 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
33 inss2 3982 . . . . . . . . . 10 ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐺 “ {𝑗})
3433a1i 11 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐺 “ {𝑗}))
356adantr 466 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → 𝐺 ∈ dom ∫1)
3635, 7syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝐺 “ {𝑗}) ∈ dom vol)
3736adantr 466 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (𝐺 “ {𝑗}) ∈ dom vol)
38 mblss 23519 . . . . . . . . . 10 ((𝐺 “ {𝑗}) ∈ dom vol → (𝐺 “ {𝑗}) ⊆ ℝ)
3937, 38syl 17 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (𝐺 “ {𝑗}) ⊆ ℝ)
40 mblvol 23518 . . . . . . . . . . 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 705 . . . . . . . . . . 11 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝐺 ∈ dom ∫1)
43 simplrr 763 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝑗 ∈ ℝ)
44 simpr 471 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝑗 ≠ 0)
45 eldifsn 4453 . . . . . . . . . . . 12 (𝑗 ∈ (ℝ ∖ {0}) ↔ (𝑗 ∈ ℝ ∧ 𝑗 ≠ 0))
4643, 44, 45sylanbrc 572 . . . . . . . . . . 11 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝑗 ∈ (ℝ ∖ {0}))
47 i1fima2sn 23667 . . . . . . . . . . 11 ((𝐺 ∈ dom ∫1𝑗 ∈ (ℝ ∖ {0})) → (vol‘(𝐺 “ {𝑗})) ∈ ℝ)
4842, 46, 47syl2anc 573 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol‘(𝐺 “ {𝑗})) ∈ ℝ)
4941, 48eqeltrrd 2851 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol*‘(𝐺 “ {𝑗})) ∈ ℝ)
50 ovolsscl 23474 . . . . . . . . 9 ((((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐺 “ {𝑗}) ∧ (𝐺 “ {𝑗}) ⊆ ℝ ∧ (vol*‘(𝐺 “ {𝑗})) ∈ ℝ) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
5134, 39, 49, 50syl3anc 1476 . . . . . . . 8 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
5232, 51jaodan 942 . . . . . . 7 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ (𝑖 ≠ 0 ∨ 𝑗 ≠ 0)) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
5315, 52sylan2br 582 . . . . . 6 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
5414, 53eqeltrd 2850 . . . . 5 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) ∈ ℝ)
5554ex 397 . . . 4 ((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (¬ (𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) ∈ ℝ))
56 iftrue 4231 . . . . 5 ((𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) = 0)
57 0re 10242 . . . . 5 0 ∈ ℝ
5856, 57syl6eqel 2858 . . . 4 ((𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) ∈ ℝ)
5955, 58pm2.61d2 173 . . 3 ((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) ∈ ℝ)
6059ralrimivva 3120 . 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 7387 . 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  wa 382  wo 836   = wceq 1631  wcel 2145  wne 2943  wral 3061  cdif 3720  cin 3722  wss 3723  ifcif 4225  {csn 4316   × cxp 5247  ccnv 5248  dom cdm 5249  cima 5252  wf 6027  cfv 6031  cmpt2 6795  cr 10137  0cc0 10138  vol*covol 23450  volcvol 23451  1citg1 23603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8504  df-inf 8505  df-oi 8571  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-z 11580  df-uz 11889  df-q 11992  df-rp 12036  df-xadd 12152  df-ioo 12384  df-ico 12386  df-icc 12387  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-clim 14427  df-sum 14625  df-xmet 19954  df-met 19955  df-ovol 23452  df-vol 23453  df-mbf 23607  df-itg1 23608
This theorem is referenced by:  itg1addlem4  23686  itg1addlem5  23687
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