Theorem xaddval 13244

Description: Value of the extended real addition operation. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xaddval	⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))))

Proof of Theorem xaddval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴)
2	1	eqeq1d 2738	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = +∞ ↔ 𝐴 = +∞))
3		simpr 484	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
4	3	eqeq1d 2738	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 = -∞ ↔ 𝐵 = -∞))
5	4	ifbid 4529	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = -∞, 0, +∞) = if(𝐵 = -∞, 0, +∞))
6	1	eqeq1d 2738	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = -∞ ↔ 𝐴 = -∞))
7	3	eqeq1d 2738	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 = +∞ ↔ 𝐵 = +∞))
8	7	ifbid 4529	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = +∞, 0, -∞) = if(𝐵 = +∞, 0, -∞))
9		oveq12 7419	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 + 𝑦) = (𝐴 + 𝐵))
10	4, 9	ifbieq2d 4532	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = -∞, -∞, (𝑥 + 𝑦)) = if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))
11	7, 10	ifbieq2d 4532	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))
12	6, 8, 11	ifbieq12d 4534	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦)))) = if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))))
13	2, 5, 12	ifbieq12d 4534	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))))
14		df-xadd 13134	. 2 ⊢ +𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
15		c0ex 11234	. . . 4 ⊢ 0 ∈ V
16		pnfex 11293	. . . 4 ⊢ +∞ ∈ V
17	15, 16	ifex 4556	. . 3 ⊢ if(𝐵 = -∞, 0, +∞) ∈ V
18		mnfxr 11297	. . . . . 6 ⊢ -∞ ∈ ℝ*
19	18	elexi 3487	. . . . 5 ⊢ -∞ ∈ V
20	15, 19	ifex 4556	. . . 4 ⊢ if(𝐵 = +∞, 0, -∞) ∈ V
21		ovex 7443	. . . . . 6 ⊢ (𝐴 + 𝐵) ∈ V
22	19, 21	ifex 4556	. . . . 5 ⊢ if(𝐵 = -∞, -∞, (𝐴 + 𝐵)) ∈ V
23	16, 22	ifex 4556	. . . 4 ⊢ if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) ∈ V
24	20, 23	ifex 4556	. . 3 ⊢ if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) ∈ V
25	17, 24	ifex 4556	. 2 ⊢ if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) ∈ V
26	13, 14, 25	ovmpoa 7567	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ifcif 4505  (class class class)co 7410  0cc0 11134   + caddc 11137  +∞cpnf 11271  -∞cmnf 11272  ℝ^*cxr 11273   +_𝑒 cxad 13131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-mulcl 11196  ax-i2m1 11202

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-pnf 11276  df-mnf 11277  df-xr 11278  df-xadd 13134

This theorem is referenced by:  xaddpnf1  13247  xaddpnf2  13248  xaddmnf1  13249  xaddmnf2  13250  pnfaddmnf  13251  mnfaddpnf  13252  rexadd  13253