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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 11032 | . . 3 ⊢ 1 ∈ ℝ* | |
2 | mnfxr 11030 | . . 3 ⊢ -∞ ∈ ℝ* | |
3 | pnfxr 11027 | . . 3 ⊢ +∞ ∈ ℝ* | |
4 | 1, 2, 3 | 3pm3.2i 1338 | . 2 ⊢ (1 ∈ ℝ* ∧ -∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) |
5 | xaddcom 12972 | . . . . . . . 8 ⊢ ((1 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (1 +𝑒 -∞) = (-∞ +𝑒 1)) | |
6 | 1, 2, 5 | mp2an 689 | . . . . . . 7 ⊢ (1 +𝑒 -∞) = (-∞ +𝑒 1) |
7 | 1re 10973 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
8 | renepnf 11021 | . . . . . . . . 9 ⊢ (1 ∈ ℝ → 1 ≠ +∞) | |
9 | 7, 8 | ax-mp 5 | . . . . . . . 8 ⊢ 1 ≠ +∞ |
10 | xaddmnf2 12961 | . . . . . . . 8 ⊢ ((1 ∈ ℝ* ∧ 1 ≠ +∞) → (-∞ +𝑒 1) = -∞) | |
11 | 1, 9, 10 | mp2an 689 | . . . . . . 7 ⊢ (-∞ +𝑒 1) = -∞ |
12 | 6, 11 | eqtri 2766 | . . . . . 6 ⊢ (1 +𝑒 -∞) = -∞ |
13 | 12 | oveq1i 7287 | . . . . 5 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) = (-∞ +𝑒 +∞) |
14 | mnfaddpnf 12963 | . . . . 5 ⊢ (-∞ +𝑒 +∞) = 0 | |
15 | 13, 14 | eqtri 2766 | . . . 4 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) = 0 |
16 | 0ne1 12042 | . . . 4 ⊢ 0 ≠ 1 | |
17 | 15, 16 | eqnetri 3014 | . . 3 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) ≠ 1 |
18 | 14 | oveq2i 7288 | . . . 4 ⊢ (1 +𝑒 (-∞ +𝑒 +∞)) = (1 +𝑒 0) |
19 | xaddid1 12973 | . . . . 5 ⊢ (1 ∈ ℝ* → (1 +𝑒 0) = 1) | |
20 | 1, 19 | ax-mp 5 | . . . 4 ⊢ (1 +𝑒 0) = 1 |
21 | 18, 20 | eqtri 2766 | . . 3 ⊢ (1 +𝑒 (-∞ +𝑒 +∞)) = 1 |
22 | 17, 21 | neeqtrri 3017 | . 2 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) ≠ (1 +𝑒 (-∞ +𝑒 +∞)) |
23 | xrsbas 20612 | . . 3 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
24 | xrsadd 20613 | . . 3 ⊢ +𝑒 = (+g‘ℝ*𝑠) | |
25 | 23, 24 | isnsgrp 18377 | . 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 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∉ wnel 3049 (class class class)co 7277 ℝcr 10868 0cc0 10869 1c1 10870 +∞cpnf 11004 -∞cmnf 11005 ℝ*cxr 11006 +𝑒 cxad 12844 ℝ*𝑠cxrs 17209 Smgrpcsgrp 18372 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-1o 8295 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-fin 8735 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12436 df-uz 12581 df-xadd 12847 df-fz 13238 df-struct 16846 df-slot 16881 df-ndx 16893 df-base 16911 df-plusg 16973 df-mulr 16974 df-tset 16979 df-ple 16980 df-ds 16982 df-xrs 17211 df-sgrp 18373 |
This theorem is referenced by: xrsmgmdifsgrp 20633 |
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