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Theorem itg1addlem2 23663
Description: Lemma for itg1add 23667. 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 23665 and itg1addlem5 23666. (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 4239 . . . . . . . 8 (¬ (𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) = (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))))
21adantl 473 . . . . . . 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 23644 . . . . . . . . . . 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 23644 . . . . . . . . . . 11 (𝐺 ∈ dom ∫1 → (𝐺 “ {𝑗}) ∈ dom vol)
86, 7syl 17 . . . . . . . . . 10 (𝜑 → (𝐺 “ {𝑗}) ∈ dom vol)
9 inmbl 23510 . . . . . . . . . 10 (((𝐹 “ {𝑖}) ∈ dom vol ∧ (𝐺 “ {𝑗}) ∈ dom vol) → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ∈ dom vol)
105, 8, 9syl2anc 696 . . . . . . . . 9 (𝜑 → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ∈ dom vol)
1110ad2antrr 764 . . . . . . . 8 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ∈ dom vol)
12 mblvol 23498 . . . . . . . 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 2794 . . . . . 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 3026 . . . . . . 7 ((𝑖 ≠ 0 ∨ 𝑗 ≠ 0) ↔ ¬ (𝑖 = 0 ∧ 𝑗 = 0))
16 inss1 3976 . . . . . . . . . 10 ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐹 “ {𝑖})
1716a1i 11 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐹 “ {𝑖}))
185ad2antrr 764 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (𝐹 “ {𝑖}) ∈ dom vol)
19 mblss 23499 . . . . . . . . . 10 ((𝐹 “ {𝑖}) ∈ dom vol → (𝐹 “ {𝑖}) ⊆ ℝ)
2018, 19syl 17 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (𝐹 “ {𝑖}) ⊆ ℝ)
21 mblvol 23498 . . . . . . . . . . 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 764 . . . . . . . . . . 11 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝐹 ∈ dom ∫1)
24 simplrl 819 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝑖 ∈ ℝ)
25 simpr 479 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝑖 ≠ 0)
26 eldifsn 4462 . . . . . . . . . . . 12 (𝑖 ∈ (ℝ ∖ {0}) ↔ (𝑖 ∈ ℝ ∧ 𝑖 ≠ 0))
2724, 25, 26sylanbrc 701 . . . . . . . . . . 11 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → 𝑖 ∈ (ℝ ∖ {0}))
28 i1fima2sn 23646 . . . . . . . . . . 11 ((𝐹 ∈ dom ∫1𝑖 ∈ (ℝ ∖ {0})) → (vol‘(𝐹 “ {𝑖})) ∈ ℝ)
2923, 27, 28syl2anc 696 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol‘(𝐹 “ {𝑖})) ∈ ℝ)
3022, 29eqeltrrd 2840 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol*‘(𝐹 “ {𝑖})) ∈ ℝ)
31 ovolsscl 23454 . . . . . . . . 9 ((((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐹 “ {𝑖}) ∧ (𝐹 “ {𝑖}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {𝑖})) ∈ ℝ) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
3217, 20, 30, 31syl3anc 1477 . . . . . . . 8 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑖 ≠ 0) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
33 inss2 3977 . . . . . . . . . 10 ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐺 “ {𝑗})
3433a1i 11 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐺 “ {𝑗}))
356adantr 472 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → 𝐺 ∈ dom ∫1)
3635, 7syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝐺 “ {𝑗}) ∈ dom vol)
3736adantr 472 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (𝐺 “ {𝑗}) ∈ dom vol)
38 mblss 23499 . . . . . . . . . 10 ((𝐺 “ {𝑗}) ∈ dom vol → (𝐺 “ {𝑗}) ⊆ ℝ)
3937, 38syl 17 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (𝐺 “ {𝑗}) ⊆ ℝ)
40 mblvol 23498 . . . . . . . . . . 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 764 . . . . . . . . . . 11 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝐺 ∈ dom ∫1)
43 simplrr 820 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝑗 ∈ ℝ)
44 simpr 479 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝑗 ≠ 0)
45 eldifsn 4462 . . . . . . . . . . . 12 (𝑗 ∈ (ℝ ∖ {0}) ↔ (𝑗 ∈ ℝ ∧ 𝑗 ≠ 0))
4643, 44, 45sylanbrc 701 . . . . . . . . . . 11 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → 𝑗 ∈ (ℝ ∖ {0}))
47 i1fima2sn 23646 . . . . . . . . . . 11 ((𝐺 ∈ dom ∫1𝑗 ∈ (ℝ ∖ {0})) → (vol‘(𝐺 “ {𝑗})) ∈ ℝ)
4842, 46, 47syl2anc 696 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol‘(𝐺 “ {𝑗})) ∈ ℝ)
4941, 48eqeltrrd 2840 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol*‘(𝐺 “ {𝑗})) ∈ ℝ)
50 ovolsscl 23454 . . . . . . . . 9 ((((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) ⊆ (𝐺 “ {𝑗}) ∧ (𝐺 “ {𝑗}) ⊆ ℝ ∧ (vol*‘(𝐺 “ {𝑗})) ∈ ℝ) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
5134, 39, 49, 50syl3anc 1477 . . . . . . . 8 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ 𝑗 ≠ 0) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
5232, 51jaodan 861 . . . . . . 7 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ (𝑖 ≠ 0 ∨ 𝑗 ≠ 0)) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
5315, 52sylan2br 494 . . . . . 6 (((𝜑 ∧ (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ)) ∧ ¬ (𝑖 = 0 ∧ 𝑗 = 0)) → (vol*‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) ∈ ℝ)
5414, 53eqeltrd 2839 . . . . 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 4236 . . . . 5 ((𝑖 = 0 ∧ 𝑗 = 0) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) = 0)
57 0re 10232 . . . . 5 0 ∈ ℝ
5856, 57syl6eqel 2847 . . . 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 3109 . 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 7405 . 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 382  wa 383   = wceq 1632  wcel 2139  wne 2932  wral 3050  cdif 3712  cin 3714  wss 3715  ifcif 4230  {csn 4321   × cxp 5264  ccnv 5265  dom cdm 5266  cima 5269  wf 6045  cfv 6049  cmpt2 6815  cr 10127  0cc0 10128  vol*covol 23431  volcvol 23432  1citg1 23583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-inf 8514  df-oi 8580  df-card 8955  df-cda 9182  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-z 11570  df-uz 11880  df-q 11982  df-rp 12026  df-xadd 12140  df-ioo 12372  df-ico 12374  df-icc 12375  df-fz 12520  df-fzo 12660  df-fl 12787  df-seq 12996  df-exp 13055  df-hash 13312  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-clim 14418  df-sum 14616  df-xmet 19941  df-met 19942  df-ovol 23433  df-vol 23434  df-mbf 23587  df-itg1 23588
This theorem is referenced by:  itg1addlem4  23665  itg1addlem5  23666
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