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 ∉ SGrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 9918 | . . . 4 ⊢ 1 ∈ ℝ | |
2 | 1 | rexri 9976 | . . 3 ⊢ 1 ∈ ℝ* |
3 | mnfxr 9975 | . . 3 ⊢ -∞ ∈ ℝ* | |
4 | pnfxr 9971 | . . 3 ⊢ +∞ ∈ ℝ* | |
5 | 2, 3, 4 | 3pm3.2i 1232 | . 2 ⊢ (1 ∈ ℝ* ∧ -∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) |
6 | xaddcom 11945 | . . . . . . . 8 ⊢ ((1 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (1 +𝑒 -∞) = (-∞ +𝑒 1)) | |
7 | 2, 3, 6 | mp2an 704 | . . . . . . 7 ⊢ (1 +𝑒 -∞) = (-∞ +𝑒 1) |
8 | renepnf 9966 | . . . . . . . . 9 ⊢ (1 ∈ ℝ → 1 ≠ +∞) | |
9 | 1, 8 | ax-mp 5 | . . . . . . . 8 ⊢ 1 ≠ +∞ |
10 | xaddmnf2 11934 | . . . . . . . 8 ⊢ ((1 ∈ ℝ* ∧ 1 ≠ +∞) → (-∞ +𝑒 1) = -∞) | |
11 | 2, 9, 10 | mp2an 704 | . . . . . . 7 ⊢ (-∞ +𝑒 1) = -∞ |
12 | 7, 11 | eqtri 2632 | . . . . . 6 ⊢ (1 +𝑒 -∞) = -∞ |
13 | 12 | oveq1i 6559 | . . . . 5 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) = (-∞ +𝑒 +∞) |
14 | mnfaddpnf 11936 | . . . . 5 ⊢ (-∞ +𝑒 +∞) = 0 | |
15 | 13, 14 | eqtri 2632 | . . . 4 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) = 0 |
16 | 0ne1 10965 | . . . 4 ⊢ 0 ≠ 1 | |
17 | 15, 16 | eqnetri 2852 | . . 3 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) ≠ 1 |
18 | 14 | oveq2i 6560 | . . . 4 ⊢ (1 +𝑒 (-∞ +𝑒 +∞)) = (1 +𝑒 0) |
19 | xaddid1 11946 | . . . . 5 ⊢ (1 ∈ ℝ* → (1 +𝑒 0) = 1) | |
20 | 2, 19 | ax-mp 5 | . . . 4 ⊢ (1 +𝑒 0) = 1 |
21 | 18, 20 | eqtri 2632 | . . 3 ⊢ (1 +𝑒 (-∞ +𝑒 +∞)) = 1 |
22 | 17, 21 | neeqtrri 2855 | . 2 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) ≠ (1 +𝑒 (-∞ +𝑒 +∞)) |
23 | xrsbas 19581 | . . 3 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
24 | xrsadd 19582 | . . 3 ⊢ +𝑒 = (+g‘ℝ*𝑠) | |
25 | 23, 24 | isnsgrp 17111 | . 2 ⊢ ((1 ∈ ℝ* ∧ -∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (((1 +𝑒 -∞) +𝑒 +∞) ≠ (1 +𝑒 (-∞ +𝑒 +∞)) → ℝ*𝑠 ∉ SGrp)) |
26 | 5, 22, 25 | mp2 9 | 1 ⊢ ℝ*𝑠 ∉ SGrp |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∉ wnel 2781 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 +∞cpnf 9950 -∞cmnf 9951 ℝ*cxr 9952 +𝑒 cxad 11820 ℝ*𝑠cxrs 15983 SGrpcsgrp 17106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-xadd 11823 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-tset 15787 df-ple 15788 df-ds 15791 df-xrs 15985 df-sgrp 17107 |
This theorem is referenced by: xrsmgmdifsgrp 19602 |
Copyright terms: Public domain | W3C validator |