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Theorem itg1addlem3 24214
Description: Lemma for itg1add 24217. (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 2828 . . . . 5 (𝑖 = 𝐴 → (𝑖 = 0 ↔ 𝐴 = 0))
2 eqeq1 2828 . . . . 5 (𝑗 = 𝐵 → (𝑗 = 0 ↔ 𝐵 = 0))
31, 2bi2anan9 635 . . . 4 ((𝑖 = 𝐴𝑗 = 𝐵) → ((𝑖 = 0 ∧ 𝑗 = 0) ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
4 sneq 4573 . . . . . . 7 (𝑖 = 𝐴 → {𝑖} = {𝐴})
54imaeq2d 5926 . . . . . 6 (𝑖 = 𝐴 → (𝐹 “ {𝑖}) = (𝐹 “ {𝐴}))
6 sneq 4573 . . . . . . 7 (𝑗 = 𝐵 → {𝑗} = {𝐵})
76imaeq2d 5926 . . . . . 6 (𝑗 = 𝐵 → (𝐺 “ {𝑗}) = (𝐺 “ {𝐵}))
85, 7ineqan12d 4194 . . . . 5 ((𝑖 = 𝐴𝑗 = 𝐵) → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) = ((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵})))
98fveq2d 6670 . . . 4 ((𝑖 = 𝐴𝑗 = 𝐵) → (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) = (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵}))))
103, 9ifbieq2d 4494 . . 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 10627 . . . 4 0 ∈ V
13 fvex 6679 . . . 4 (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵}))) ∈ V
1412, 13ifex 4517 . . 3 if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵})))) ∈ V
1510, 11, 14ovmpoa 7298 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐼𝐵) = if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵})))))
16 iffalse 4478 . 2 (¬ (𝐴 = 0 ∧ 𝐵 = 0) → if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵})))) = (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵}))))
1715, 16sylan9eq 2880 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 396   = wceq 1530  wcel 2106  cin 3938  ifcif 4469  {csn 4563  ccnv 5552  dom cdm 5553  cima 5556  cfv 6351  (class class class)co 7151  cmpo 7153  cr 10528  0cc0 10529  volcvol 23979  1citg1 24131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pr 5325  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-mulcl 10591  ax-i2m1 10597
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156
This theorem is referenced by:  itg1addlem4  24215  itg1addlem5  24216
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