Theorem xmulval 13140

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

Assertion

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

Proof of Theorem xmulval

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

Step	Hyp	Ref	Expression
1		simpl 482	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴)
2	1	eqeq1d 2738	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 0 ↔ 𝐴 = 0))
3		simpr 484	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
4	3	eqeq1d 2738	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 = 0 ↔ 𝐵 = 0))
5	2, 4	orbi12d 918	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 = 0 ∨ 𝑦 = 0) ↔ (𝐴 = 0 ∨ 𝐵 = 0)))
6	3	breq2d 5110	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (0 < 𝑦 ↔ 0 < 𝐵))
7	1	eqeq1d 2738	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = +∞ ↔ 𝐴 = +∞))
8	6, 7	anbi12d 632	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((0 < 𝑦 ∧ 𝑥 = +∞) ↔ (0 < 𝐵 ∧ 𝐴 = +∞)))
9	3	breq1d 5108	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 < 0 ↔ 𝐵 < 0))
10	1	eqeq1d 2738	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = -∞ ↔ 𝐴 = -∞))
11	9, 10	anbi12d 632	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑦 < 0 ∧ 𝑥 = -∞) ↔ (𝐵 < 0 ∧ 𝐴 = -∞)))
12	8, 11	orbi12d 918	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (((0 < 𝑦 ∧ 𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ↔ ((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞))))
13	1	breq2d 5110	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (0 < 𝑥 ↔ 0 < 𝐴))
14	3	eqeq1d 2738	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 = +∞ ↔ 𝐵 = +∞))
15	13, 14	anbi12d 632	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((0 < 𝑥 ∧ 𝑦 = +∞) ↔ (0 < 𝐴 ∧ 𝐵 = +∞)))
16	1	breq1d 5108	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 < 0 ↔ 𝐴 < 0))
17	3	eqeq1d 2738	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 = -∞ ↔ 𝐵 = -∞))
18	16, 17	anbi12d 632	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 < 0 ∧ 𝑦 = -∞) ↔ (𝐴 < 0 ∧ 𝐵 = -∞)))
19	15, 18	orbi12d 918	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (((0 < 𝑥 ∧ 𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞)) ↔ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))))
20	12, 19	orbi12d 918	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((((0 < 𝑦 ∧ 𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥 ∧ 𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞))) ↔ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))))
21	6, 10	anbi12d 632	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((0 < 𝑦 ∧ 𝑥 = -∞) ↔ (0 < 𝐵 ∧ 𝐴 = -∞)))
22	9, 7	anbi12d 632	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑦 < 0 ∧ 𝑥 = +∞) ↔ (𝐵 < 0 ∧ 𝐴 = +∞)))
23	21, 22	orbi12d 918	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (((0 < 𝑦 ∧ 𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ↔ ((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞))))
24	13, 17	anbi12d 632	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((0 < 𝑥 ∧ 𝑦 = -∞) ↔ (0 < 𝐴 ∧ 𝐵 = -∞)))
25	16, 14	anbi12d 632	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 < 0 ∧ 𝑦 = +∞) ↔ (𝐴 < 0 ∧ 𝐵 = +∞)))
26	24, 25	orbi12d 918	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (((0 < 𝑥 ∧ 𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞)) ↔ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))
27	23, 26	orbi12d 918	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((((0 < 𝑦 ∧ 𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥 ∧ 𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞))) ↔ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))))
28		oveq12 7367	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 · 𝑦) = (𝐴 · 𝐵))
29	27, 28	ifbieq2d 4506	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if((((0 < 𝑦 ∧ 𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥 ∧ 𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞))), -∞, (𝑥 · 𝑦)) = if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))
30	20, 29	ifbieq2d 4506	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if((((0 < 𝑦 ∧ 𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥 ∧ 𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞))), +∞, if((((0 < 𝑦 ∧ 𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥 ∧ 𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞))), -∞, (𝑥 · 𝑦))) = if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))))
31	5, 30	ifbieq2d 4506	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if((𝑥 = 0 ∨ 𝑦 = 0), 0, if((((0 < 𝑦 ∧ 𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥 ∧ 𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞))), +∞, if((((0 < 𝑦 ∧ 𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥 ∧ 𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞))), -∞, (𝑥 · 𝑦)))) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))))
32		df-xmul 13028	. 2 ⊢ ·e = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, if((((0 < 𝑦 ∧ 𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥 ∧ 𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞))), +∞, if((((0 < 𝑦 ∧ 𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥 ∧ 𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞))), -∞, (𝑥 · 𝑦)))))
33		c0ex 11126	. . 3 ⊢ 0 ∈ V
34		pnfex 11185	. . . 4 ⊢ +∞ ∈ V
35		mnfxr 11189	. . . . . 6 ⊢ -∞ ∈ ℝ*
36	35	elexi 3463	. . . . 5 ⊢ -∞ ∈ V
37		ovex 7391	. . . . 5 ⊢ (𝐴 · 𝐵) ∈ V
38	36, 37	ifex 4530	. . . 4 ⊢ if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)) ∈ V
39	34, 38	ifex 4530	. . 3 ⊢ if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) ∈ V
40	33, 39	ifex 4530	. 2 ⊢ if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))) ∈ V
41	31, 32, 40	ovmpoa 7513	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113  ifcif 4479   class class class wbr 5098  (class class class)co 7358  0cc0 11026   · cmul 11031  +∞cpnf 11163  -∞cmnf 11164  ℝ^*cxr 11165   < clt 11166   ·_e cxmu 13025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-mulcl 11088  ax-i2m1 11094

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-pnf 11168  df-mnf 11169  df-xr 11170  df-xmul 13028

This theorem is referenced by:  xmulcom  13181  xmul01  13182  xmulneg1  13184  rexmul  13186  xmulpnf1  13189