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 Description: Lemma for itg1add 23374. (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
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)

StepHypRef Expression
1 eqeq1 2625 . . . . 5 (𝑖 = 𝐴 → (𝑖 = 0 ↔ 𝐴 = 0))
2 eqeq1 2625 . . . . 5 (𝑗 = 𝐵 → (𝑗 = 0 ↔ 𝐵 = 0))
31, 2bi2anan9 916 . . . 4 ((𝑖 = 𝐴𝑗 = 𝐵) → ((𝑖 = 0 ∧ 𝑗 = 0) ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
4 sneq 4158 . . . . . . 7 (𝑖 = 𝐴 → {𝑖} = {𝐴})
54imaeq2d 5425 . . . . . 6 (𝑖 = 𝐴 → (𝐹 “ {𝑖}) = (𝐹 “ {𝐴}))
6 sneq 4158 . . . . . . 7 (𝑗 = 𝐵 → {𝑗} = {𝐵})
76imaeq2d 5425 . . . . . 6 (𝑗 = 𝐵 → (𝐺 “ {𝑗}) = (𝐺 “ {𝐵}))
85, 7ineqan12d 3794 . . . . 5 ((𝑖 = 𝐴𝑗 = 𝐵) → ((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})) = ((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵})))
98fveq2d 6152 . . . 4 ((𝑖 = 𝐴𝑗 = 𝐵) → (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗}))) = (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵}))))
103, 9ifbieq2d 4083 . . 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 9978 . . . 4 0 ∈ V
13 fvex 6158 . . . 4 (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵}))) ∈ V
1412, 13ifex 4128 . . 3 if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵})))) ∈ V
1510, 11, 14ovmpt2a 6744 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐼𝐵) = if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵})))))
16 iffalse 4067 . 2 (¬ (𝐴 = 0 ∧ 𝐵 = 0) → if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵})))) = (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵}))))
1715, 16sylan9eq 2675 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 384   = wceq 1480   ∈ wcel 1987   ∩ cin 3554  ifcif 4058  {csn 4148  ◡ccnv 5073  dom cdm 5074   “ cima 5077  ‘cfv 5847  (class class class)co 6604   ↦ cmpt2 6606  ℝcr 9879  0cc0 9880  volcvol 23139  ∫1citg1 23290 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-mulcl 9942  ax-i2m1 9948 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609 This theorem is referenced by:  itg1addlem4  23372  itg1addlem5  23373
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