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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 8023 | . . 3 ⊢ +∞ ∈ ℝ* | |
2 | xaddval 9858 | . . 3 ⊢ ((+∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (+∞ +𝑒 𝐴) = if(+∞ = +∞, if(𝐴 = -∞, 0, +∞), if(+∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (+∞ + 𝐴)))))) | |
3 | 1, 2 | mpan 424 | . 2 ⊢ (𝐴 ∈ ℝ* → (+∞ +𝑒 𝐴) = if(+∞ = +∞, if(𝐴 = -∞, 0, +∞), if(+∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (+∞ + 𝐴)))))) |
4 | eqid 2187 | . . . 4 ⊢ +∞ = +∞ | |
5 | 4 | iftruei 3552 | . . 3 ⊢ if(+∞ = +∞, if(𝐴 = -∞, 0, +∞), if(+∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (+∞ + 𝐴))))) = if(𝐴 = -∞, 0, +∞) |
6 | ifnefalse 3557 | . . 3 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, 0, +∞) = +∞) | |
7 | 5, 6 | eqtrid 2232 | . 2 ⊢ (𝐴 ≠ -∞ → if(+∞ = +∞, if(𝐴 = -∞, 0, +∞), if(+∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (+∞ + 𝐴))))) = +∞) |
8 | 3, 7 | sylan9eq 2240 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (+∞ +𝑒 𝐴) = +∞) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1363 ∈ wcel 2158 ≠ wne 2357 ifcif 3546 (class class class)co 5888 0cc0 7824 + caddc 7827 +∞cpnf 8002 -∞cmnf 8003 ℝ*cxr 8004 +𝑒 cxad 9783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1re 7918 ax-addrcl 7921 ax-rnegex 7933 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8007 df-mnf 8008 df-xr 8009 df-xadd 9786 |
This theorem is referenced by: xaddnemnf 9870 xaddcom 9874 xaddid1 9875 xnn0xadd0 9880 xnegdi 9881 xaddass 9882 xleadd1a 9886 xltadd1 9889 xposdif 9895 xleaddadd 9900 xrmaxadd 11282 xrbdtri 11297 isxmet2d 14088 |
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