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Theorem itg1addlem3 23806
Description: Lemma for itg1add 23809. (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 2803 . . . . 5 (𝑖 = 𝐴 → (𝑖 = 0 ↔ 𝐴 = 0))
2 eqeq1 2803 . . . . 5 (𝑗 = 𝐵 → (𝑗 = 0 ↔ 𝐵 = 0))
31, 2bi2anan9 630 . . . 4 ((𝑖 = 𝐴𝑗 = 𝐵) → ((𝑖 = 0 ∧ 𝑗 = 0) ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
4 sneq 4378 . . . . . . 7 (𝑖 = 𝐴 → {𝑖} = {𝐴})
54imaeq2d 5683 . . . . . 6 (𝑖 = 𝐴 → (𝐹 “ {𝑖}) = (𝐹 “ {𝐴}))
6 sneq 4378 . . . . . . 7 (𝑗 = 𝐵 → {𝑗} = {𝐵})
76imaeq2d 5683 . . . . . 6 (𝑗 = 𝐵 → (𝐺 “ {𝑗}) = (𝐺 “ {𝐵}))
85, 7ineqan12d 4014 . . . . 5 ((𝑖 = 𝐴𝑗 = 𝐵) → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) = ((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵})))
98fveq2d 6415 . . . 4 ((𝑖 = 𝐴𝑗 = 𝐵) → (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) = (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵}))))
103, 9ifbieq2d 4302 . . 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 10322 . . . 4 0 ∈ V
13 fvex 6424 . . . 4 (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵}))) ∈ V
1412, 13ifex 4325 . . 3 if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵})))) ∈ V
1510, 11, 14ovmpt2a 7025 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐼𝐵) = if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵})))))
16 iffalse 4286 . 2 (¬ (𝐴 = 0 ∧ 𝐵 = 0) → if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵})))) = (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵}))))
1715, 16sylan9eq 2853 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 385   = wceq 1653  wcel 2157  cin 3768  ifcif 4277  {csn 4368  ccnv 5311  dom cdm 5312  cima 5315  cfv 6101  (class class class)co 6878  cmpt2 6880  cr 10223  0cc0 10224  volcvol 23571  1citg1 23723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-mulcl 10286  ax-i2m1 10292
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883
This theorem is referenced by:  itg1addlem4  23807  itg1addlem5  23808
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