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Mirrors > Home > ILE Home > Th. List > xaddpnf2 | GIF version |
Description: Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddpnf2 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (+∞ +𝑒 𝐴) = +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfxr 7739 | . . 3 ⊢ +∞ ∈ ℝ* | |
2 | xaddval 9518 | . . 3 ⊢ ((+∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (+∞ +𝑒 𝐴) = if(+∞ = +∞, if(𝐴 = -∞, 0, +∞), if(+∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (+∞ + 𝐴)))))) | |
3 | 1, 2 | mpan 418 | . 2 ⊢ (𝐴 ∈ ℝ* → (+∞ +𝑒 𝐴) = if(+∞ = +∞, if(𝐴 = -∞, 0, +∞), if(+∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (+∞ + 𝐴)))))) |
4 | eqid 2115 | . . . 4 ⊢ +∞ = +∞ | |
5 | 4 | iftruei 3446 | . . 3 ⊢ if(+∞ = +∞, if(𝐴 = -∞, 0, +∞), if(+∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (+∞ + 𝐴))))) = if(𝐴 = -∞, 0, +∞) |
6 | ifnefalse 3451 | . . 3 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, 0, +∞) = +∞) | |
7 | 5, 6 | syl5eq 2159 | . 2 ⊢ (𝐴 ≠ -∞ → if(+∞ = +∞, if(𝐴 = -∞, 0, +∞), if(+∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (+∞ + 𝐴))))) = +∞) |
8 | 3, 7 | sylan9eq 2167 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (+∞ +𝑒 𝐴) = +∞) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1314 ∈ wcel 1463 ≠ wne 2282 ifcif 3440 (class class class)co 5728 0cc0 7544 + caddc 7547 +∞cpnf 7718 -∞cmnf 7719 ℝ*cxr 7720 +𝑒 cxad 9447 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-cnex 7633 ax-resscn 7634 ax-1re 7636 ax-addrcl 7639 ax-rnegex 7651 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 946 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-rab 2399 df-v 2659 df-sbc 2879 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-if 3441 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-iota 5046 df-fun 5083 df-fv 5089 df-ov 5731 df-oprab 5732 df-mpo 5733 df-pnf 7723 df-mnf 7724 df-xr 7725 df-xadd 9450 |
This theorem is referenced by: xaddnemnf 9530 xaddcom 9534 xaddid1 9535 xnn0xadd0 9540 xnegdi 9541 xaddass 9542 xleadd1a 9546 xltadd1 9549 xposdif 9555 xleaddadd 9560 xrmaxadd 10919 xrbdtri 10934 isxmet2d 12334 |
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