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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 10687 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
2 | rexr 10687 | . . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
3 | xaddval 12617 | . . 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 10689 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
6 | ifnefalse 4479 | . . . . 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 10690 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
9 | ifnefalse 4479 | . . . . 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 2856 | . . 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 10689 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ +∞) | |
13 | ifnefalse 4479 | . . . . 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 10690 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ -∞) | |
16 | ifnefalse 4479 | . . . . 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 2856 | . . 3 ⊢ (𝐵 ∈ ℝ → if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) = (𝐴 + 𝐵)) |
19 | 11, 18 | sylan9eq 2876 | . 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 2856 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ifcif 4467 (class class class)co 7156 ℝcr 10536 0cc0 10537 + caddc 10540 +∞cpnf 10672 -∞cmnf 10673 ℝ*cxr 10674 +𝑒 cxad 12506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-mulcl 10599 ax-i2m1 10605 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-xadd 12509 |
This theorem is referenced by: rexsub 12627 rexaddd 12628 xnn0xaddcl 12629 xaddnemnf 12630 xaddnepnf 12631 xnegid 12632 xaddcom 12634 xaddid1 12635 xnn0xadd0 12641 xnegdi 12642 xaddass 12643 xadddilem 12688 x2times 12693 hashunx 13748 hashunsnggt 13756 isxmet2d 22937 xmeter 23043 vtxdgfival 27251 1loopgrvd2 27285 vdegp1bi 27319 xlt2addrd 30482 xrsmulgzz 30665 xrge0slmod 30917 xrge0iifhom 31180 esumfsupre 31330 esumpfinvallem 31333 omssubadd 31558 probun 31677 heicant 34942 cntotbnd 35089 heiborlem6 35109 supxrgelem 41654 supxrge 41655 infrpge 41668 xrlexaddrp 41669 ovolsplit 42322 sge0tsms 42711 sge0pr 42725 sge0resplit 42737 sge0split 42740 sge0iunmptlemfi 42744 sge0iunmptlemre 42746 sge0xaddlem1 42764 sge0xaddlem2 42765 carageniuncllem1 42852 carageniuncllem2 42853 hoidmv1lelem2 42923 hoidmvlelem2 42927 hspmbllem3 42959 |
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