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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 11209 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
2 | rexr 11209 | . . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
3 | xaddval 13151 | . . 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 11211 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
6 | ifnefalse 4502 | . . . . 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 11212 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
9 | ifnefalse 4502 | . . . . 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 2773 | . . 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 11211 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ +∞) | |
13 | ifnefalse 4502 | . . . . 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 11212 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ -∞) | |
16 | ifnefalse 4502 | . . . . 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 2773 | . . 3 ⊢ (𝐵 ∈ ℝ → if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) = (𝐴 + 𝐵)) |
19 | 11, 18 | sylan9eq 2793 | . 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 2773 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ifcif 4490 (class class class)co 7361 ℝcr 11058 0cc0 11059 + caddc 11062 +∞cpnf 11194 -∞cmnf 11195 ℝ*cxr 11196 +𝑒 cxad 13039 |
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-icn 11118 ax-addcl 11119 ax-mulcl 11121 ax-i2m1 11127 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 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-ov 7364 df-oprab 7365 df-mpo 7366 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-xadd 13042 |
This theorem is referenced by: rexsub 13161 rexaddd 13162 xnn0xaddcl 13163 xaddnemnf 13164 xaddnepnf 13165 xnegid 13166 xaddcom 13168 xaddid1 13169 xnn0xadd0 13175 xnegdi 13176 xaddass 13177 xadddilem 13222 x2times 13227 hashunx 14295 hashunsnggt 14303 isxmet2d 23703 xmeter 23809 vtxdgfival 28466 1loopgrvd2 28500 vdegp1bi 28534 xlt2addrd 31717 xrsmulgzz 31925 xrge0slmod 32194 xrge0iifhom 32582 esumfsupre 32734 esumpfinvallem 32737 omssubadd 32964 probun 33083 heicant 36163 cntotbnd 36305 heiborlem6 36325 supxrgelem 43662 supxrge 43663 infrpge 43676 xrlexaddrp 43677 ovolsplit 44319 sge0tsms 44711 sge0pr 44725 sge0resplit 44737 sge0split 44740 sge0iunmptlemfi 44744 sge0iunmptlemre 44746 sge0xaddlem1 44764 sge0xaddlem2 44765 carageniuncllem1 44852 carageniuncllem2 44853 hoidmv1lelem2 44923 hoidmvlelem2 44927 hspmbllem3 44959 |
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