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Theorem xmulf 12140
Description: The extended real multiplication operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
xmulf ·e :(ℝ* × ℝ*)⟶ℝ*

Proof of Theorem xmulf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0xr 10124 . . . . 5 0 ∈ ℝ*
21a1i 11 . . . 4 (((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ (𝑥 = 0 ∨ 𝑦 = 0)) → 0 ∈ ℝ*)
3 pnfxr 10130 . . . . . 6 +∞ ∈ ℝ*
43a1i 11 . . . . 5 ((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ (𝑥 = 0 ∨ 𝑦 = 0)) ∧ (((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞)))) → +∞ ∈ ℝ*)
5 mnfxr 10134 . . . . . . 7 -∞ ∈ ℝ*
65a1i 11 . . . . . 6 (((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ (𝑥 = 0 ∨ 𝑦 = 0)) ∧ ¬ (((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞)))) ∧ (((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞)))) → -∞ ∈ ℝ*)
7 xmullem 12132 . . . . . . . 8 (((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ (𝑥 = 0 ∨ 𝑦 = 0)) ∧ ¬ (((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞)))) ∧ ¬ (((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞)))) → 𝑥 ∈ ℝ)
8 ancom 465 . . . . . . . . . . 11 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ↔ (𝑦 ∈ ℝ*𝑥 ∈ ℝ*))
9 orcom 401 . . . . . . . . . . . 12 ((𝑥 = 0 ∨ 𝑦 = 0) ↔ (𝑦 = 0 ∨ 𝑥 = 0))
109notbii 309 . . . . . . . . . . 11 (¬ (𝑥 = 0 ∨ 𝑦 = 0) ↔ ¬ (𝑦 = 0 ∨ 𝑥 = 0))
118, 10anbi12i 733 . . . . . . . . . 10 (((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ (𝑥 = 0 ∨ 𝑦 = 0)) ↔ ((𝑦 ∈ ℝ*𝑥 ∈ ℝ*) ∧ ¬ (𝑦 = 0 ∨ 𝑥 = 0)))
12 orcom 401 . . . . . . . . . . 11 ((((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞))) ↔ (((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞)) ∨ ((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞))))
1312notbii 309 . . . . . . . . . 10 (¬ (((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞))) ↔ ¬ (((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞)) ∨ ((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞))))
1411, 13anbi12i 733 . . . . . . . . 9 ((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ (𝑥 = 0 ∨ 𝑦 = 0)) ∧ ¬ (((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞)))) ↔ (((𝑦 ∈ ℝ*𝑥 ∈ ℝ*) ∧ ¬ (𝑦 = 0 ∨ 𝑥 = 0)) ∧ ¬ (((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞)) ∨ ((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)))))
15 orcom 401 . . . . . . . . . 10 ((((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞))) ↔ (((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞)) ∨ ((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞))))
1615notbii 309 . . . . . . . . 9 (¬ (((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞))) ↔ ¬ (((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞)) ∨ ((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞))))
17 xmullem 12132 . . . . . . . . 9 (((((𝑦 ∈ ℝ*𝑥 ∈ ℝ*) ∧ ¬ (𝑦 = 0 ∨ 𝑥 = 0)) ∧ ¬ (((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞)) ∨ ((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)))) ∧ ¬ (((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞)) ∨ ((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)))) → 𝑦 ∈ ℝ)
1814, 16, 17syl2anb 495 . . . . . . . 8 (((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ (𝑥 = 0 ∨ 𝑦 = 0)) ∧ ¬ (((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞)))) ∧ ¬ (((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞)))) → 𝑦 ∈ ℝ)
197, 18remulcld 10108 . . . . . . 7 (((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ (𝑥 = 0 ∨ 𝑦 = 0)) ∧ ¬ (((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞)))) ∧ ¬ (((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞)))) → (𝑥 · 𝑦) ∈ ℝ)
2019rexrd 10127 . . . . . 6 (((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ (𝑥 = 0 ∨ 𝑦 = 0)) ∧ ¬ (((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞)))) ∧ ¬ (((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞)))) → (𝑥 · 𝑦) ∈ ℝ*)
216, 20ifclda 4153 . . . . 5 ((((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ (𝑥 = 0 ∨ 𝑦 = 0)) ∧ ¬ (((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞)))) → if((((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞))), -∞, (𝑥 · 𝑦)) ∈ ℝ*)
224, 21ifclda 4153 . . . 4 (((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ (𝑥 = 0 ∨ 𝑦 = 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))) ∈ ℝ*)
232, 22ifclda 4153 . . 3 ((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)))) ∈ ℝ*)
2423rgen2a 3006 . 2 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)))) ∈ ℝ*
25 df-xmul 11986 . . 3 ·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)))))
2625fmpt2 7282 . 2 (∀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)))) ∈ ℝ* ↔ ·e :(ℝ* × ℝ*)⟶ℝ*)
2724, 26mpbi 220 1 ·e :(ℝ* × ℝ*)⟶ℝ*
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 382  wa 383   = wceq 1523  wcel 2030  wral 2941  ifcif 4119   class class class wbr 4685   × cxp 5141  wf 5922  (class class class)co 6690  cr 9973  0cc0 9974   · cmul 9979  +∞cpnf 10109  -∞cmnf 10110  *cxr 10111   < clt 10112   ·e cxmu 11983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-i2m1 10042  ax-1ne0 10043  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-xmul 11986
This theorem is referenced by:  xmulcl  12141  xrsmul  19812
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