![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > xrsnsgrp | Structured version Visualization version GIF version |
Description: The "additive group" of the extended reals is not a semigroup. (Contributed by AV, 30-Jan-2020.) |
Ref | Expression |
---|---|
xrsnsgrp | ⊢ ℝ*𝑠 ∉ Smgrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1xr 11224 | . . 3 ⊢ 1 ∈ ℝ* | |
2 | mnfxr 11222 | . . 3 ⊢ -∞ ∈ ℝ* | |
3 | pnfxr 11219 | . . 3 ⊢ +∞ ∈ ℝ* | |
4 | 1, 2, 3 | 3pm3.2i 1340 | . 2 ⊢ (1 ∈ ℝ* ∧ -∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) |
5 | xaddcom 13170 | . . . . . . . 8 ⊢ ((1 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (1 +𝑒 -∞) = (-∞ +𝑒 1)) | |
6 | 1, 2, 5 | mp2an 691 | . . . . . . 7 ⊢ (1 +𝑒 -∞) = (-∞ +𝑒 1) |
7 | 1re 11165 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
8 | renepnf 11213 | . . . . . . . . 9 ⊢ (1 ∈ ℝ → 1 ≠ +∞) | |
9 | 7, 8 | ax-mp 5 | . . . . . . . 8 ⊢ 1 ≠ +∞ |
10 | xaddmnf2 13159 | . . . . . . . 8 ⊢ ((1 ∈ ℝ* ∧ 1 ≠ +∞) → (-∞ +𝑒 1) = -∞) | |
11 | 1, 9, 10 | mp2an 691 | . . . . . . 7 ⊢ (-∞ +𝑒 1) = -∞ |
12 | 6, 11 | eqtri 2760 | . . . . . 6 ⊢ (1 +𝑒 -∞) = -∞ |
13 | 12 | oveq1i 7373 | . . . . 5 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) = (-∞ +𝑒 +∞) |
14 | mnfaddpnf 13161 | . . . . 5 ⊢ (-∞ +𝑒 +∞) = 0 | |
15 | 13, 14 | eqtri 2760 | . . . 4 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) = 0 |
16 | 0ne1 12234 | . . . 4 ⊢ 0 ≠ 1 | |
17 | 15, 16 | eqnetri 3011 | . . 3 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) ≠ 1 |
18 | 14 | oveq2i 7374 | . . . 4 ⊢ (1 +𝑒 (-∞ +𝑒 +∞)) = (1 +𝑒 0) |
19 | xaddrid 13171 | . . . . 5 ⊢ (1 ∈ ℝ* → (1 +𝑒 0) = 1) | |
20 | 1, 19 | ax-mp 5 | . . . 4 ⊢ (1 +𝑒 0) = 1 |
21 | 18, 20 | eqtri 2760 | . . 3 ⊢ (1 +𝑒 (-∞ +𝑒 +∞)) = 1 |
22 | 17, 21 | neeqtrri 3014 | . 2 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) ≠ (1 +𝑒 (-∞ +𝑒 +∞)) |
23 | xrsbas 20851 | . . 3 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
24 | xrsadd 20852 | . . 3 ⊢ +𝑒 = (+g‘ℝ*𝑠) | |
25 | 23, 24 | isnsgrp 18565 | . 2 ⊢ ((1 ∈ ℝ* ∧ -∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (((1 +𝑒 -∞) +𝑒 +∞) ≠ (1 +𝑒 (-∞ +𝑒 +∞)) → ℝ*𝑠 ∉ Smgrp)) |
26 | 4, 22, 25 | mp2 9 | 1 ⊢ ℝ*𝑠 ∉ Smgrp |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∉ wnel 3046 (class class class)co 7363 ℝcr 11060 0cc0 11061 1c1 11062 +∞cpnf 11196 -∞cmnf 11197 ℝ*cxr 11198 +𝑒 cxad 13041 ℝ*𝑠cxrs 17397 Smgrpcsgrp 18560 |
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 2703 ax-sep 5262 ax-nul 5269 ax-pow 5326 ax-pr 5390 ax-un 7678 ax-cnex 11117 ax-resscn 11118 ax-1cn 11119 ax-icn 11120 ax-addcl 11121 ax-addrcl 11122 ax-mulcl 11123 ax-mulrcl 11124 ax-mulcom 11125 ax-addass 11126 ax-mulass 11127 ax-distr 11128 ax-i2m1 11129 ax-1ne0 11130 ax-1rid 11131 ax-rnegex 11132 ax-rrecex 11133 ax-cnre 11134 ax-pre-lttri 11135 ax-pre-lttrn 11136 ax-pre-ltadd 11137 ax-pre-mulgt0 11138 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 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-pss 3933 df-nul 4289 df-if 4493 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4872 df-iun 4962 df-br 5112 df-opab 5174 df-mpt 5195 df-tr 5229 df-id 5537 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5594 df-we 5596 df-xp 5645 df-rel 5646 df-cnv 5647 df-co 5648 df-dm 5649 df-rn 5650 df-res 5651 df-ima 5652 df-pred 6259 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-riota 7319 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7809 df-1st 7927 df-2nd 7928 df-frecs 8218 df-wrecs 8249 df-recs 8323 df-rdg 8362 df-1o 8418 df-er 8656 df-en 8892 df-dom 8893 df-sdom 8894 df-fin 8895 df-pnf 11201 df-mnf 11202 df-xr 11203 df-ltxr 11204 df-le 11205 df-sub 11397 df-neg 11398 df-nn 12164 df-2 12226 df-3 12227 df-4 12228 df-5 12229 df-6 12230 df-7 12231 df-8 12232 df-9 12233 df-n0 12424 df-z 12510 df-dec 12629 df-uz 12774 df-xadd 13044 df-fz 13436 df-struct 17031 df-slot 17066 df-ndx 17078 df-base 17096 df-plusg 17161 df-mulr 17162 df-tset 17167 df-ple 17168 df-ds 17170 df-xrs 17399 df-sgrp 18561 |
This theorem is referenced by: xrsmgmdifsgrp 20872 |
Copyright terms: Public domain | W3C validator |