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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 10689 | . . 3 ⊢ 1 ∈ ℝ* | |
2 | mnfxr 10687 | . . 3 ⊢ -∞ ∈ ℝ* | |
3 | pnfxr 10684 | . . 3 ⊢ +∞ ∈ ℝ* | |
4 | 1, 2, 3 | 3pm3.2i 1336 | . 2 ⊢ (1 ∈ ℝ* ∧ -∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) |
5 | xaddcom 12621 | . . . . . . . 8 ⊢ ((1 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (1 +𝑒 -∞) = (-∞ +𝑒 1)) | |
6 | 1, 2, 5 | mp2an 691 | . . . . . . 7 ⊢ (1 +𝑒 -∞) = (-∞ +𝑒 1) |
7 | 1re 10630 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
8 | renepnf 10678 | . . . . . . . . 9 ⊢ (1 ∈ ℝ → 1 ≠ +∞) | |
9 | 7, 8 | ax-mp 5 | . . . . . . . 8 ⊢ 1 ≠ +∞ |
10 | xaddmnf2 12610 | . . . . . . . 8 ⊢ ((1 ∈ ℝ* ∧ 1 ≠ +∞) → (-∞ +𝑒 1) = -∞) | |
11 | 1, 9, 10 | mp2an 691 | . . . . . . 7 ⊢ (-∞ +𝑒 1) = -∞ |
12 | 6, 11 | eqtri 2821 | . . . . . 6 ⊢ (1 +𝑒 -∞) = -∞ |
13 | 12 | oveq1i 7145 | . . . . 5 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) = (-∞ +𝑒 +∞) |
14 | mnfaddpnf 12612 | . . . . 5 ⊢ (-∞ +𝑒 +∞) = 0 | |
15 | 13, 14 | eqtri 2821 | . . . 4 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) = 0 |
16 | 0ne1 11696 | . . . 4 ⊢ 0 ≠ 1 | |
17 | 15, 16 | eqnetri 3057 | . . 3 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) ≠ 1 |
18 | 14 | oveq2i 7146 | . . . 4 ⊢ (1 +𝑒 (-∞ +𝑒 +∞)) = (1 +𝑒 0) |
19 | xaddid1 12622 | . . . . 5 ⊢ (1 ∈ ℝ* → (1 +𝑒 0) = 1) | |
20 | 1, 19 | ax-mp 5 | . . . 4 ⊢ (1 +𝑒 0) = 1 |
21 | 18, 20 | eqtri 2821 | . . 3 ⊢ (1 +𝑒 (-∞ +𝑒 +∞)) = 1 |
22 | 17, 21 | neeqtrri 3060 | . 2 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) ≠ (1 +𝑒 (-∞ +𝑒 +∞)) |
23 | xrsbas 20107 | . . 3 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
24 | xrsadd 20108 | . . 3 ⊢ +𝑒 = (+g‘ℝ*𝑠) | |
25 | 23, 24 | isnsgrp 17897 | . 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 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∉ wnel 3091 (class class class)co 7135 ℝcr 10525 0cc0 10526 1c1 10527 +∞cpnf 10661 -∞cmnf 10662 ℝ*cxr 10663 +𝑒 cxad 12493 ℝ*𝑠cxrs 16765 Smgrpcsgrp 17892 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-xadd 12496 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 df-mulr 16571 df-tset 16576 df-ple 16577 df-ds 16579 df-xrs 16767 df-sgrp 17893 |
This theorem is referenced by: xrsmgmdifsgrp 20128 |
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