Theorem itg1addlem3 25667

Description: Lemma for itg1add 25670. (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 2741	. . . . 5 ⊢ (𝑖 = 𝐴 → (𝑖 = 0 ↔ 𝐴 = 0))
2		eqeq1 2741	. . . . 5 ⊢ (𝑗 = 𝐵 → (𝑗 = 0 ↔ 𝐵 = 0))
3	1, 2	bi2anan9 639	. . . 4 ⊢ ((𝑖 = 𝐴 ∧ 𝑗 = 𝐵) → ((𝑖 = 0 ∧ 𝑗 = 0) ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
4		sneq 4592	. . . . . . 7 ⊢ (𝑖 = 𝐴 → {𝑖} = {𝐴})
5	4	imaeq2d 6027	. . . . . 6 ⊢ (𝑖 = 𝐴 → (◡𝐹 “ {𝑖}) = (◡𝐹 “ {𝐴}))
6		sneq 4592	. . . . . . 7 ⊢ (𝑗 = 𝐵 → {𝑗} = {𝐵})
7	6	imaeq2d 6027	. . . . . 6 ⊢ (𝑗 = 𝐵 → (◡𝐺 “ {𝑗}) = (◡𝐺 “ {𝐵}))
8	5, 7	ineqan12d 4176	. . . . 5 ⊢ ((𝑖 = 𝐴 ∧ 𝑗 = 𝐵) → ((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) = ((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵})))
9	8	fveq2d 6846	. . . 4 ⊢ ((𝑖 = 𝐴 ∧ 𝑗 = 𝐵) → (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))) = (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵}))))
10	3, 9	ifbieq2d 4508	. . 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 11138	. . . 4 ⊢ 0 ∈ V
13		fvex 6855	. . . 4 ⊢ (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵}))) ∈ V
14	12, 13	ifex 4532	. . 3 ⊢ if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵})))) ∈ V
15	10, 11, 14	ovmpoa 7523	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐼𝐵) = if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵})))))
16		iffalse 4490	. 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 2792	1 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴𝐼𝐵) = (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵}))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∩ cin 3902  ifcif 4481  {csn 4582  ^◡ccnv 5631  dom cdm 5632   “ cima 5635  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  ℝcr 11037  0cc0 11038  volcvol 25432  ∫₁citg1 25584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-mulcl 11100  ax-i2m1 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373

This theorem is referenced by:  itg1addlem4  25668  itg1addlem5  25669