Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  itg1addlem2 Structured version   Visualization version   GIF version

 Description: Lemma for itg1add 23274. 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 23272 and itg1addlem5 23273. (Contributed by Mario Carneiro, 26-Jun-2014.)
Hypotheses
Ref Expression
itg1add.3 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))))
Assertion
Ref Expression
Distinct variable groups:   𝑖,𝑗,𝐹   𝑖,𝐺,𝑗   𝜑,𝑖,𝑗
Allowed substitution hints:   𝐼(𝑖,𝑗)

StepHypRef Expression
1 iffalse 4045 . . . . . . . 8 (¬ (𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) = (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))))
21adantl 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 (𝜑𝐹 ∈ dom ∫1)
4 i1fima 23251 . . . . . . . . . . 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 23251 . . . . . . . . . . 11 (𝐺 ∈ dom ∫1 → (𝐺 “ {𝑗}) ∈ dom vol)
86, 7syl 17 . . . . . . . . . 10 (𝜑 → (𝐺 “ {𝑗}) ∈ dom vol)
9 inmbl 23117 . . . . . . . . . 10 (((𝐹 “ {𝑖}) ∈ dom vol ∧ (𝐺 “ {𝑗}) ∈ dom vol) → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ∈ dom vol)
105, 8, 9syl2anc 691 . . . . . . . . 9 (𝜑 → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ∈ dom vol)
1110ad2antrr 758 . . . . . . . 8 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ∈ dom vol)
12 mblvol 23105 . . . . . . . 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 2644 . . . . . 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 2876 . . . . . . 7 ((𝑖 ≠ 0 ∨ 𝑗 ≠ 0) ↔ ¬ (𝑖 = 0 ∧ 𝑗 = 0))
16 inss1 3795 . . . . . . . . . 10 ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐹 “ {𝑖})
1716a1i 11 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐹 “ {𝑖}))
185ad2antrr 758 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (𝐹 “ {𝑖}) ∈ dom vol)
19 mblss 23106 . . . . . . . . . 10 ((𝐹 “ {𝑖}) ∈ dom vol → (𝐹 “ {𝑖}) ⊆ ℝ)
2018, 19syl 17 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (𝐹 “ {𝑖}) ⊆ ℝ)
21 mblvol 23105 . . . . . . . . . . 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 758 . . . . . . . . . . 11 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝐹 ∈ dom ∫1)
24 simplrl 796 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝑖 ∈ ℝ)
25 simpr 476 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝑖 ≠ 0)
26 eldifsn 4260 . . . . . . . . . . . 12 (𝑖 ∈ (ℝ ∖ {0}) ↔ (𝑖 ∈ ℝ ∧ 𝑖 ≠ 0))
2724, 25, 26sylanbrc 695 . . . . . . . . . . 11 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝑖 ∈ (ℝ ∖ {0}))
28 i1fima2sn 23253 . . . . . . . . . . 11 ((𝐹 ∈ dom ∫1𝑖 ∈ (ℝ ∖ {0})) → (vol‘(𝐹 “ {𝑖})) ∈ ℝ)
2923, 27, 28syl2anc 691 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol‘(𝐹 “ {𝑖})) ∈ ℝ)
3022, 29eqeltrrd 2689 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol*‘(𝐹 “ {𝑖})) ∈ ℝ)
31 ovolsscl 23061 . . . . . . . . 9 ((((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐹 “ {𝑖}) ∧ (𝐹 “ {𝑖}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {𝑖})) ∈ ℝ) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
3217, 20, 30, 31syl3anc 1318 . . . . . . . 8 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
33 inss2 3796 . . . . . . . . . 10 ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐺 “ {𝑗})
3433a1i 11 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐺 “ {𝑗}))
356adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → 𝐺 ∈ dom ∫1)
3635, 7syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝐺 “ {𝑗}) ∈ dom vol)
3736adantr 480 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (𝐺 “ {𝑗}) ∈ dom vol)
38 mblss 23106 . . . . . . . . . 10 ((𝐺 “ {𝑗}) ∈ dom vol → (𝐺 “ {𝑗}) ⊆ ℝ)
3937, 38syl 17 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (𝐺 “ {𝑗}) ⊆ ℝ)
40 mblvol 23105 . . . . . . . . . . 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 758 . . . . . . . . . . 11 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝐺 ∈ dom ∫1)
43 simplrr 797 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝑗 ∈ ℝ)
44 simpr 476 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝑗 ≠ 0)
45 eldifsn 4260 . . . . . . . . . . . 12 (𝑗 ∈ (ℝ ∖ {0}) ↔ (𝑗 ∈ ℝ ∧ 𝑗 ≠ 0))
4643, 44, 45sylanbrc 695 . . . . . . . . . . 11 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝑗 ∈ (ℝ ∖ {0}))
47 i1fima2sn 23253 . . . . . . . . . . 11 ((𝐺 ∈ dom ∫1𝑗 ∈ (ℝ ∖ {0})) → (vol‘(𝐺 “ {𝑗})) ∈ ℝ)
4842, 46, 47syl2anc 691 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol‘(𝐺 “ {𝑗})) ∈ ℝ)
4941, 48eqeltrrd 2689 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol*‘(𝐺 “ {𝑗})) ∈ ℝ)
50 ovolsscl 23061 . . . . . . . . 9 ((((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐺 “ {𝑗}) ∧ (𝐺 “ {𝑗}) ⊆ ℝ ∧ (vol*‘(𝐺 “ {𝑗})) ∈ ℝ) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
5134, 39, 49, 50syl3anc 1318 . . . . . . . 8 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
5232, 51jaodan 822 . . . . . . 7 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ (𝑖 ≠ 0 ∨ 𝑗 ≠ 0)) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
5315, 52sylan2br 492 . . . . . 6 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
5414, 53eqeltrd 2688 . . . . 5 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) ∈ ℝ)
5554ex 449 . . . 4 ((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (¬ (𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) ∈ ℝ))
56 iftrue 4042 . . . . 5 ((𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) = 0)
57 0re 9919 . . . . 5 0 ∈ ℝ
5856, 57syl6eqel 2696 . . . 4 ((𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) ∈ ℝ)
5955, 58pm2.61d2 171 . . 3 ((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) ∈ ℝ)
6059ralrimivva 2954 . 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 7126 . 2 (∀𝑖 ∈ ℝ ∀𝑗 ∈ ℝ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) ∈ ℝ ↔ 𝐼:(ℝ × ℝ)⟶ℝ)
6360, 62sylib 207 1 (𝜑𝐼:(ℝ × ℝ)⟶ℝ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ifcif 4036  {csn 4125   × cxp 5036  ◡ccnv 5037  dom cdm 5038   “ cima 5041  ⟶wf 5800  ‘cfv 5804   ↦ cmpt2 6551  ℝcr 9814  0cc0 9815  vol*covol 23038  volcvol 23039  ∫1citg1 23190 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xadd 11823  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-xmet 19560  df-met 19561  df-ovol 23040  df-vol 23041  df-mbf 23194  df-itg1 23195 This theorem is referenced by:  itg1addlem4  23272  itg1addlem5  23273
 Copyright terms: Public domain W3C validator