Theorem itg1addlem3 25688

Description: Lemma for itg1add 25691. (Contributed by Mario Carneiro, 26-Jun-2014.)

Hypotheses

Ref	Expression
i1fadd.1	⊢ (𝜑 → 𝐹 ∈ dom ∫1)
i1fadd.2	⊢ (𝜑 → 𝐺 ∈ 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

Step	Hyp	Ref	Expression
1		eqeq1 2738	. . . . 5 ⊢ (𝑖 = 𝐴 → (𝑖 = 0 ↔ 𝐴 = 0))
2		eqeq1 2738	. . . . 5 ⊢ (𝑗 = 𝐵 → (𝑗 = 0 ↔ 𝐵 = 0))
3	1, 2	bi2anan9 638	. . . 4 ⊢ ((𝑖 = 𝐴 ∧ 𝑗 = 𝐵) → ((𝑖 = 0 ∧ 𝑗 = 0) ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
4		sneq 4618	. . . . . . 7 ⊢ (𝑖 = 𝐴 → {𝑖} = {𝐴})
5	4	imaeq2d 6060	. . . . . 6 ⊢ (𝑖 = 𝐴 → (◡𝐹 “ {𝑖}) = (◡𝐹 “ {𝐴}))
6		sneq 4618	. . . . . . 7 ⊢ (𝑗 = 𝐵 → {𝑗} = {𝐵})
7	6	imaeq2d 6060	. . . . . 6 ⊢ (𝑗 = 𝐵 → (◡𝐺 “ {𝑗}) = (◡𝐺 “ {𝐵}))
8	5, 7	ineqan12d 4204	. . . . 5 ⊢ ((𝑖 = 𝐴 ∧ 𝑗 = 𝐵) → ((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) = ((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵})))
9	8	fveq2d 6891	. . . 4 ⊢ ((𝑖 = 𝐴 ∧ 𝑗 = 𝐵) → (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))) = (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵}))))
10	3, 9	ifbieq2d 4534	. . 3 ⊢ ((𝑖 = 𝐴 ∧ 𝑗 = 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 11238	. . . 4 ⊢ 0 ∈ V
13		fvex 6900	. . . 4 ⊢ (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵}))) ∈ V
14	12, 13	ifex 4558	. . 3 ⊢ if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵})))) ∈ V
15	10, 11, 14	ovmpoa 7571	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐼𝐵) = if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵})))))
16		iffalse 4516	. 2 ⊢ (¬ (𝐴 = 0 ∧ 𝐵 = 0) → if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵})))) = (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵}))))
17	15, 16	sylan9eq 2789	1 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴𝐼𝐵) = (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵}))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ∩ cin 3932  ifcif 4507  {csn 4608  ^◡ccnv 5666  dom cdm 5667   “ cima 5670  ‘cfv 6542  (class class class)co 7414   ∈ cmpo 7416  ℝcr 11137  0cc0 11138  volcvol 25453  ∫₁citg1 25605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-mulcl 11200  ax-i2m1 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-sbc 3773  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-br 5126  df-opab 5188  df-id 5560  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6495  df-fun 6544  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419

This theorem is referenced by:  itg1addlem4  25689  itg1addlem5  25690