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Theorem itg1addlem3 23510
Description: Lemma for itg1add 23513. (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
itg1addlem3 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴𝐼𝐵) = (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵}))))
Distinct variable groups:   𝑖,𝑗,𝐴   𝐵,𝑖,𝑗   𝑖,𝐹,𝑗   𝑖,𝐺,𝑗   𝜑,𝑖,𝑗
Allowed substitution hints:   𝐼(𝑖,𝑗)

Proof of Theorem itg1addlem3
StepHypRef Expression
1 eqeq1 2655 . . . . 5 (𝑖 = 𝐴 → (𝑖 = 0 ↔ 𝐴 = 0))
2 eqeq1 2655 . . . . 5 (𝑗 = 𝐵 → (𝑗 = 0 ↔ 𝐵 = 0))
31, 2bi2anan9 935 . . . 4 ((𝑖 = 𝐴𝑗 = 𝐵) → ((𝑖 = 0 ∧ 𝑗 = 0) ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
4 sneq 4220 . . . . . . 7 (𝑖 = 𝐴 → {𝑖} = {𝐴})
54imaeq2d 5501 . . . . . 6 (𝑖 = 𝐴 → (𝐹 “ {𝑖}) = (𝐹 “ {𝐴}))
6 sneq 4220 . . . . . . 7 (𝑗 = 𝐵 → {𝑗} = {𝐵})
76imaeq2d 5501 . . . . . 6 (𝑗 = 𝐵 → (𝐺 “ {𝑗}) = (𝐺 “ {𝐵}))
85, 7ineqan12d 3849 . . . . 5 ((𝑖 = 𝐴𝑗 = 𝐵) → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) = ((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵})))
98fveq2d 6233 . . . 4 ((𝑖 = 𝐴𝑗 = 𝐵) → (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) = (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵}))))
103, 9ifbieq2d 4144 . . 3 ((𝑖 = 𝐴𝑗 = 𝐵) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))) = if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵})))))
11 itg1add.3 . . 3 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))))
12 c0ex 10072 . . . 4 0 ∈ V
13 fvex 6239 . . . 4 (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵}))) ∈ V
1412, 13ifex 4189 . . 3 if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵})))) ∈ V
1510, 11, 14ovmpt2a 6833 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐼𝐵) = if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵})))))
16 iffalse 4128 . 2 (¬ (𝐴 = 0 ∧ 𝐵 = 0) → if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵})))) = (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵}))))
1715, 16sylan9eq 2705 1 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴𝐼𝐵) = (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵}))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  cin 3606  ifcif 4119  {csn 4210  ccnv 5142  dom cdm 5143  cima 5146  cfv 5926  (class class class)co 6690  cmpt2 6692  cr 9973  0cc0 9974  volcvol 23278  1citg1 23429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-mulcl 10036  ax-i2m1 10042
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695
This theorem is referenced by:  itg1addlem4  23511  itg1addlem5  23512
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