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Mirrors > Home > MPE Home > Th. List > rexadd | Structured version Visualization version GIF version |
Description: The extended real addition operation when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
rexadd | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 11131 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
2 | rexr 11131 | . . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
3 | xaddval 13067 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))))) | |
4 | 1, 2, 3 | syl2an 597 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))))) |
5 | renepnf 11133 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
6 | ifnefalse 4493 | . . . . 5 ⊢ (𝐴 ≠ +∞ → if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) = if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ → if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) = if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) |
8 | renemnf 11134 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
9 | ifnefalse 4493 | . . . . 5 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ → if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) |
11 | 7, 10 | eqtrd 2777 | . . 3 ⊢ (𝐴 ∈ ℝ → if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) |
12 | renepnf 11133 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ +∞) | |
13 | ifnefalse 4493 | . . . . 5 ⊢ (𝐵 ≠ +∞ → if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) = if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝐵 ∈ ℝ → if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) = if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) |
15 | renemnf 11134 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ -∞) | |
16 | ifnefalse 4493 | . . . . 5 ⊢ (𝐵 ≠ -∞ → if(𝐵 = -∞, -∞, (𝐴 + 𝐵)) = (𝐴 + 𝐵)) | |
17 | 15, 16 | syl 17 | . . . 4 ⊢ (𝐵 ∈ ℝ → if(𝐵 = -∞, -∞, (𝐴 + 𝐵)) = (𝐴 + 𝐵)) |
18 | 14, 17 | eqtrd 2777 | . . 3 ⊢ (𝐵 ∈ ℝ → if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) = (𝐴 + 𝐵)) |
19 | 11, 18 | sylan9eq 2797 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) = (𝐴 + 𝐵)) |
20 | 4, 19 | eqtrd 2777 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ifcif 4481 (class class class)co 7346 ℝcr 10980 0cc0 10981 + caddc 10984 +∞cpnf 11116 -∞cmnf 11117 ℝ*cxr 11118 +𝑒 cxad 12956 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5251 ax-nul 5258 ax-pow 5315 ax-pr 5379 ax-un 7659 ax-cnex 11037 ax-resscn 11038 ax-1cn 11039 ax-icn 11040 ax-addcl 11041 ax-mulcl 11043 ax-i2m1 11049 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3735 df-csb 3851 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4278 df-if 4482 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4861 df-br 5101 df-opab 5163 df-mpt 5184 df-id 5525 df-xp 5633 df-rel 5634 df-cnv 5635 df-co 5636 df-dm 5637 df-rn 5638 df-res 5639 df-ima 5640 df-iota 6440 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-ov 7349 df-oprab 7350 df-mpo 7351 df-er 8578 df-en 8814 df-dom 8815 df-sdom 8816 df-pnf 11121 df-mnf 11122 df-xr 11123 df-xadd 12959 |
This theorem is referenced by: rexsub 13077 rexaddd 13078 xnn0xaddcl 13079 xaddnemnf 13080 xaddnepnf 13081 xnegid 13082 xaddcom 13084 xaddid1 13085 xnn0xadd0 13091 xnegdi 13092 xaddass 13093 xadddilem 13138 x2times 13143 hashunx 14210 hashunsnggt 14218 isxmet2d 23590 xmeter 23696 vtxdgfival 28191 1loopgrvd2 28225 vdegp1bi 28259 xlt2addrd 31432 xrsmulgzz 31638 xrge0slmod 31908 xrge0iifhom 32249 esumfsupre 32401 esumpfinvallem 32404 omssubadd 32631 probun 32750 heicant 35968 cntotbnd 36110 heiborlem6 36130 supxrgelem 43263 supxrge 43264 infrpge 43277 xrlexaddrp 43278 ovolsplit 43917 sge0tsms 44307 sge0pr 44321 sge0resplit 44333 sge0split 44336 sge0iunmptlemfi 44340 sge0iunmptlemre 44342 sge0xaddlem1 44360 sge0xaddlem2 44361 carageniuncllem1 44448 carageniuncllem2 44449 hoidmv1lelem2 44519 hoidmvlelem2 44523 hspmbllem3 44555 |
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