Theorem itg1addlem3 25908

Description: Lemma for itg1add 25911. (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 2769	. . . . 5 ⊢ (𝑖 = 𝐴 → (𝑖 = 0 ↔ 𝐴 = 0))
2		eqeq1 2769	. . . . 5 ⊢ (𝑗 = 𝐵 → (𝑗 = 0 ↔ 𝐵 = 0))
3	1, 2	bi2anan9 650	. . . 4 ⊢ ((𝑖 = 𝐴 ∧ 𝑗 = 𝐵) → ((𝑖 = 0 ∧ 𝑗 = 0) ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
4		sneq 4601	. . . . . . 7 ⊢ (𝑖 = 𝐴 → {𝑖} = {𝐴})
5	4	imaeq2d 6064	. . . . . 6 ⊢ (𝑖 = 𝐴 → (◡𝐹 “ {𝑖}) = (◡𝐹 “ {𝐴}))
6		sneq 4601	. . . . . . 7 ⊢ (𝑗 = 𝐵 → {𝑗} = {𝐵})
7	6	imaeq2d 6064	. . . . . 6 ⊢ (𝑗 = 𝐵 → (◡𝐺 “ {𝑗}) = (◡𝐺 “ {𝐵}))
8	5, 7	ineqan12d 4175	. . . . 5 ⊢ ((𝑖 = 𝐴 ∧ 𝑗 = 𝐵) → ((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) = ((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵})))
9	8	fveq2d 6889	. . . 4 ⊢ ((𝑖 = 𝐴 ∧ 𝑗 = 𝐵) → (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))) = (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵}))))
10	3, 9	ifbieq2d 4516	. . 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 11215	. . . 4 ⊢ 0 ∈ V
13		fvex 6898	. . . 4 ⊢ (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵}))) ∈ V
14	12, 13	ifex 4540	. . 3 ⊢ if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵})))) ∈ V
15	10, 11, 14	ovmpoa 7574	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐼𝐵) = if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵})))))
16		iffalse 4498	. 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 2820	1 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴𝐼𝐵) = (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵}))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ∩ cin 3905  ifcif 4489  {csn 4591  ^◡ccnv 5662  dom cdm 5663   “ cima 5666  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  ℝcr 11114  0cc0 11115  volcvol 25673  ∫₁citg1 25825

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-mulcl 11177  ax-i2m1 11183

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424

This theorem is used by:  itg1addlem4  25909  itg1addlem5  25910