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Theorem xmulval 12277
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 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))))

Proof of Theorem xmulval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 470 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 = 𝐴)
21eqeq1d 2815 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = 0 ↔ 𝐴 = 0))
3 simpr 473 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
43eqeq1d 2815 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 = 0 ↔ 𝐵 = 0))
52, 4orbi12d 933 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥 = 0 ∨ 𝑦 = 0) ↔ (𝐴 = 0 ∨ 𝐵 = 0)))
63breq2d 4863 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (0 < 𝑦 ↔ 0 < 𝐵))
71eqeq1d 2815 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = +∞ ↔ 𝐴 = +∞))
86, 7anbi12d 618 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → ((0 < 𝑦𝑥 = +∞) ↔ (0 < 𝐵𝐴 = +∞)))
93breq1d 4861 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 < 0 ↔ 𝐵 < 0))
101eqeq1d 2815 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = -∞ ↔ 𝐴 = -∞))
119, 10anbi12d 618 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑦 < 0 ∧ 𝑥 = -∞) ↔ (𝐵 < 0 ∧ 𝐴 = -∞)))
128, 11orbi12d 933 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ↔ ((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞))))
131breq2d 4863 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (0 < 𝑥 ↔ 0 < 𝐴))
143eqeq1d 2815 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 = +∞ ↔ 𝐵 = +∞))
1513, 14anbi12d 618 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → ((0 < 𝑥𝑦 = +∞) ↔ (0 < 𝐴𝐵 = +∞)))
161breq1d 4861 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 < 0 ↔ 𝐴 < 0))
173eqeq1d 2815 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 = -∞ ↔ 𝐵 = -∞))
1816, 17anbi12d 618 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥 < 0 ∧ 𝑦 = -∞) ↔ (𝐴 < 0 ∧ 𝐵 = -∞)))
1915, 18orbi12d 933 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞)) ↔ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))))
2012, 19orbi12d 933 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → ((((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞))) ↔ (((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)))))
216, 10anbi12d 618 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → ((0 < 𝑦𝑥 = -∞) ↔ (0 < 𝐵𝐴 = -∞)))
229, 7anbi12d 618 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑦 < 0 ∧ 𝑥 = +∞) ↔ (𝐵 < 0 ∧ 𝐴 = +∞)))
2321, 22orbi12d 933 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ↔ ((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞))))
2413, 17anbi12d 618 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → ((0 < 𝑥𝑦 = -∞) ↔ (0 < 𝐴𝐵 = -∞)))
2516, 14anbi12d 618 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥 < 0 ∧ 𝑦 = +∞) ↔ (𝐴 < 0 ∧ 𝐵 = +∞)))
2624, 25orbi12d 933 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞)) ↔ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))
2723, 26orbi12d 933 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → ((((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞))) ↔ (((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)))))
28 oveq12 6886 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 · 𝑦) = (𝐴 · 𝐵))
2927, 28ifbieq2d 4311 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → if((((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞))), -∞, (𝑥 · 𝑦)) = if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))
3020, 29ifbieq2d 4311 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → if((((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞))), +∞, if((((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞))), -∞, (𝑥 · 𝑦))) = if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))))
315, 30ifbieq2d 4311 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → if((𝑥 = 0 ∨ 𝑦 = 0), 0, if((((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞))), +∞, if((((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞))), -∞, (𝑥 · 𝑦)))) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))))
32 df-xmul 12167 . 2 ·e = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, if((((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞))), +∞, if((((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞))), -∞, (𝑥 · 𝑦)))))
33 c0ex 10322 . . 3 0 ∈ V
34 pnfex 10381 . . . 4 +∞ ∈ V
35 mnfxr 10384 . . . . . 6 -∞ ∈ ℝ*
3635elexi 3414 . . . . 5 -∞ ∈ V
37 ovex 6909 . . . . 5 (𝐴 · 𝐵) ∈ V
3836, 37ifex 4334 . . . 4 if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)) ∈ V
3934, 38ifex 4334 . . 3 if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) ∈ V
4033, 39ifex 4334 . 2 if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))) ∈ V
4131, 32, 40ovmpt2a 7024 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wo 865   = wceq 1637  wcel 2157  ifcif 4286   class class class wbr 4851  (class class class)co 6877  0cc0 10224   · cmul 10229  +∞cpnf 10359  -∞cmnf 10360  *cxr 10361   < clt 10362   ·e cxmu 12164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182  ax-cnex 10280  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-mulcl 10286  ax-i2m1 10292
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3400  df-sbc 3641  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-op 4384  df-uni 4638  df-br 4852  df-opab 4914  df-id 5226  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-iota 6067  df-fun 6106  df-fv 6112  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-pnf 10364  df-mnf 10365  df-xr 10366  df-xmul 12167
This theorem is referenced by:  xmulcom  12317  xmul01  12318  xmulneg1  12320  rexmul  12322  xmulpnf1  12325
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