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Theorem xaddf 13203
Description: The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
xaddf +𝑒 :(ℝ* Γ— ℝ*)βŸΆβ„*

Proof of Theorem xaddf
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0xr 11261 . . . . . 6 0 ∈ ℝ*
2 pnfxr 11268 . . . . . 6 +∞ ∈ ℝ*
31, 2ifcli 4576 . . . . 5 if(𝑦 = -∞, 0, +∞) ∈ ℝ*
43a1i 11 . . . 4 (((π‘₯ ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ π‘₯ = +∞) β†’ if(𝑦 = -∞, 0, +∞) ∈ ℝ*)
5 mnfxr 11271 . . . . . . 7 -∞ ∈ ℝ*
61, 5ifcli 4576 . . . . . 6 if(𝑦 = +∞, 0, -∞) ∈ ℝ*
76a1i 11 . . . . 5 ((((π‘₯ ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ Β¬ π‘₯ = +∞) ∧ π‘₯ = -∞) β†’ if(𝑦 = +∞, 0, -∞) ∈ ℝ*)
82a1i 11 . . . . . . . 8 ((((π‘₯ ∈ ℝ* ∧ (Β¬ π‘₯ = +∞ ∧ Β¬ π‘₯ = -∞)) ∧ 𝑦 ∈ ℝ*) ∧ 𝑦 = +∞) β†’ +∞ ∈ ℝ*)
95a1i 11 . . . . . . . . 9 (((((π‘₯ ∈ ℝ* ∧ (Β¬ π‘₯ = +∞ ∧ Β¬ π‘₯ = -∞)) ∧ 𝑦 ∈ ℝ*) ∧ Β¬ 𝑦 = +∞) ∧ 𝑦 = -∞) β†’ -∞ ∈ ℝ*)
10 ioran 983 . . . . . . . . . . . . . 14 (Β¬ (π‘₯ = +∞ ∨ π‘₯ = -∞) ↔ (Β¬ π‘₯ = +∞ ∧ Β¬ π‘₯ = -∞))
11 elxr 13096 . . . . . . . . . . . . . . . . . 18 (π‘₯ ∈ ℝ* ↔ (π‘₯ ∈ ℝ ∨ π‘₯ = +∞ ∨ π‘₯ = -∞))
12 3orass 1091 . . . . . . . . . . . . . . . . . 18 ((π‘₯ ∈ ℝ ∨ π‘₯ = +∞ ∨ π‘₯ = -∞) ↔ (π‘₯ ∈ ℝ ∨ (π‘₯ = +∞ ∨ π‘₯ = -∞)))
1311, 12sylbb 218 . . . . . . . . . . . . . . . . 17 (π‘₯ ∈ ℝ* β†’ (π‘₯ ∈ ℝ ∨ (π‘₯ = +∞ ∨ π‘₯ = -∞)))
1413ord 863 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ ℝ* β†’ (Β¬ π‘₯ ∈ ℝ β†’ (π‘₯ = +∞ ∨ π‘₯ = -∞)))
1514con1d 145 . . . . . . . . . . . . . . 15 (π‘₯ ∈ ℝ* β†’ (Β¬ (π‘₯ = +∞ ∨ π‘₯ = -∞) β†’ π‘₯ ∈ ℝ))
1615imp 408 . . . . . . . . . . . . . 14 ((π‘₯ ∈ ℝ* ∧ Β¬ (π‘₯ = +∞ ∨ π‘₯ = -∞)) β†’ π‘₯ ∈ ℝ)
1710, 16sylan2br 596 . . . . . . . . . . . . 13 ((π‘₯ ∈ ℝ* ∧ (Β¬ π‘₯ = +∞ ∧ Β¬ π‘₯ = -∞)) β†’ π‘₯ ∈ ℝ)
18 ioran 983 . . . . . . . . . . . . . 14 (Β¬ (𝑦 = +∞ ∨ 𝑦 = -∞) ↔ (Β¬ 𝑦 = +∞ ∧ Β¬ 𝑦 = -∞))
19 elxr 13096 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℝ* ↔ (𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞))
20 3orass 1091 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞) ↔ (𝑦 ∈ ℝ ∨ (𝑦 = +∞ ∨ 𝑦 = -∞)))
2119, 20sylbb 218 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℝ* β†’ (𝑦 ∈ ℝ ∨ (𝑦 = +∞ ∨ 𝑦 = -∞)))
2221ord 863 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℝ* β†’ (Β¬ 𝑦 ∈ ℝ β†’ (𝑦 = +∞ ∨ 𝑦 = -∞)))
2322con1d 145 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℝ* β†’ (Β¬ (𝑦 = +∞ ∨ 𝑦 = -∞) β†’ 𝑦 ∈ ℝ))
2423imp 408 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ* ∧ Β¬ (𝑦 = +∞ ∨ 𝑦 = -∞)) β†’ 𝑦 ∈ ℝ)
2518, 24sylan2br 596 . . . . . . . . . . . . 13 ((𝑦 ∈ ℝ* ∧ (Β¬ 𝑦 = +∞ ∧ Β¬ 𝑦 = -∞)) β†’ 𝑦 ∈ ℝ)
26 readdcl 11193 . . . . . . . . . . . . 13 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ (π‘₯ + 𝑦) ∈ ℝ)
2717, 25, 26syl2an 597 . . . . . . . . . . . 12 (((π‘₯ ∈ ℝ* ∧ (Β¬ π‘₯ = +∞ ∧ Β¬ π‘₯ = -∞)) ∧ (𝑦 ∈ ℝ* ∧ (Β¬ 𝑦 = +∞ ∧ Β¬ 𝑦 = -∞))) β†’ (π‘₯ + 𝑦) ∈ ℝ)
2827rexrd 11264 . . . . . . . . . . 11 (((π‘₯ ∈ ℝ* ∧ (Β¬ π‘₯ = +∞ ∧ Β¬ π‘₯ = -∞)) ∧ (𝑦 ∈ ℝ* ∧ (Β¬ 𝑦 = +∞ ∧ Β¬ 𝑦 = -∞))) β†’ (π‘₯ + 𝑦) ∈ ℝ*)
2928anassrs 469 . . . . . . . . . 10 ((((π‘₯ ∈ ℝ* ∧ (Β¬ π‘₯ = +∞ ∧ Β¬ π‘₯ = -∞)) ∧ 𝑦 ∈ ℝ*) ∧ (Β¬ 𝑦 = +∞ ∧ Β¬ 𝑦 = -∞)) β†’ (π‘₯ + 𝑦) ∈ ℝ*)
3029anassrs 469 . . . . . . . . 9 (((((π‘₯ ∈ ℝ* ∧ (Β¬ π‘₯ = +∞ ∧ Β¬ π‘₯ = -∞)) ∧ 𝑦 ∈ ℝ*) ∧ Β¬ 𝑦 = +∞) ∧ Β¬ 𝑦 = -∞) β†’ (π‘₯ + 𝑦) ∈ ℝ*)
319, 30ifclda 4564 . . . . . . . 8 ((((π‘₯ ∈ ℝ* ∧ (Β¬ π‘₯ = +∞ ∧ Β¬ π‘₯ = -∞)) ∧ 𝑦 ∈ ℝ*) ∧ Β¬ 𝑦 = +∞) β†’ if(𝑦 = -∞, -∞, (π‘₯ + 𝑦)) ∈ ℝ*)
328, 31ifclda 4564 . . . . . . 7 (((π‘₯ ∈ ℝ* ∧ (Β¬ π‘₯ = +∞ ∧ Β¬ π‘₯ = -∞)) ∧ 𝑦 ∈ ℝ*) β†’ if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (π‘₯ + 𝑦))) ∈ ℝ*)
3332an32s 651 . . . . . 6 (((π‘₯ ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ (Β¬ π‘₯ = +∞ ∧ Β¬ π‘₯ = -∞)) β†’ if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (π‘₯ + 𝑦))) ∈ ℝ*)
3433anassrs 469 . . . . 5 ((((π‘₯ ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ Β¬ π‘₯ = +∞) ∧ Β¬ π‘₯ = -∞) β†’ if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (π‘₯ + 𝑦))) ∈ ℝ*)
357, 34ifclda 4564 . . . 4 (((π‘₯ ∈ ℝ* ∧ 𝑦 ∈ ℝ*) ∧ Β¬ π‘₯ = +∞) β†’ if(π‘₯ = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (π‘₯ + 𝑦)))) ∈ ℝ*)
364, 35ifclda 4564 . . 3 ((π‘₯ ∈ ℝ* ∧ 𝑦 ∈ ℝ*) β†’ if(π‘₯ = +∞, if(𝑦 = -∞, 0, +∞), if(π‘₯ = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (π‘₯ + 𝑦))))) ∈ ℝ*)
3736rgen2 3198 . 2 βˆ€π‘₯ ∈ ℝ* βˆ€π‘¦ ∈ ℝ* if(π‘₯ = +∞, if(𝑦 = -∞, 0, +∞), if(π‘₯ = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (π‘₯ + 𝑦))))) ∈ ℝ*
38 df-xadd 13093 . . 3 +𝑒 = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(π‘₯ = +∞, if(𝑦 = -∞, 0, +∞), if(π‘₯ = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (π‘₯ + 𝑦))))))
3938fmpo 8054 . 2 (βˆ€π‘₯ ∈ ℝ* βˆ€π‘¦ ∈ ℝ* if(π‘₯ = +∞, if(𝑦 = -∞, 0, +∞), if(π‘₯ = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (π‘₯ + 𝑦))))) ∈ ℝ* ↔ +𝑒 :(ℝ* Γ— ℝ*)βŸΆβ„*)
4037, 39mpbi 229 1 +𝑒 :(ℝ* Γ— ℝ*)βŸΆβ„*
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∧ wa 397   ∨ wo 846   ∨ w3o 1087   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  ifcif 4529   Γ— cxp 5675  βŸΆwf 6540  (class class class)co 7409  β„cr 11109  0cc0 11110   + caddc 11113  +∞cpnf 11245  -∞cmnf 11246  β„*cxr 11247   +𝑒 cxad 13090
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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-1cn 11168  ax-addrcl 11171  ax-rnegex 11181  ax-cnre 11183
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-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-pnf 11250  df-mnf 11251  df-xr 11252  df-xadd 13093
This theorem is referenced by:  xaddcl  13218  xrsadd  20962  xrofsup  31980  xrge0pluscn  32920  xrge0tmdALT  32926
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