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Theorem xmulf 13200
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 11210 . . . . 5 0 ∈ ℝ*
21a1i 11 . . . 4 (((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ (𝑥 = 0 ∨ 𝑦 = 0)) → 0 ∈ ℝ*)
3 pnfxr 11217 . . . . . 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 11220 . . . . . . 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 13192 . . . . . . . 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 462 . . . . . . . . . . 11 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ↔ (𝑦 ∈ ℝ*𝑥 ∈ ℝ*))
9 orcom 869 . . . . . . . . . . . 12 ((𝑥 = 0 ∨ 𝑦 = 0) ↔ (𝑦 = 0 ∨ 𝑥 = 0))
109notbii 320 . . . . . . . . . . 11 (¬ (𝑥 = 0 ∨ 𝑦 = 0) ↔ ¬ (𝑦 = 0 ∨ 𝑥 = 0))
118, 10anbi12i 628 . . . . . . . . . 10 (((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) ∧ ¬ (𝑥 = 0 ∨ 𝑦 = 0)) ↔ ((𝑦 ∈ ℝ*𝑥 ∈ ℝ*) ∧ ¬ (𝑦 = 0 ∨ 𝑥 = 0)))
12 orcom 869 . . . . . . . . . . 11 ((((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞))) ↔ (((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞)) ∨ ((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞))))
1312notbii 320 . . . . . . . . . 10 (¬ (((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞))) ↔ ¬ (((0 < 𝑥𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞)) ∨ ((0 < 𝑦𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞))))
1411, 13anbi12i 628 . . . . . . . . 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 869 . . . . . . . . . 10 ((((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞))) ↔ (((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞)) ∨ ((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞))))
1615notbii 320 . . . . . . . . 9 (¬ (((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞))) ↔ ¬ (((0 < 𝑥𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞)) ∨ ((0 < 𝑦𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞))))
17 xmullem 13192 . . . . . . . . 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 599 . . . . . . . 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 11193 . . . . . . 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 11213 . . . . . 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 4525 . . . . 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 4525 . . . 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 4525 . . 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*)
2423rgen2 3191 . 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 13043 . . 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)))))
2625fmpo 8004 . 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 229 1 ·e :(ℝ* × ℝ*)⟶ℝ*
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397  wo 846   = wceq 1542  wcel 2107  wral 3061  ifcif 4490   class class class wbr 5109   × cxp 5635  wf 6496  (class class class)co 7361  cr 11058  0cc0 11059   · cmul 11064  +∞cpnf 11194  -∞cmnf 11195  *cxr 11196   < clt 11197   ·e cxmu 13040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-addrcl 11120  ax-mulrcl 11122  ax-rnegex 11130  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-po 5549  df-so 5550  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-xmul 13043
This theorem is referenced by:  xmulcl  13201  xrsmul  20838
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