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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 11317 | . . 3 ⊢ 1 ∈ ℝ* | |
2 | mnfxr 11315 | . . 3 ⊢ -∞ ∈ ℝ* | |
3 | pnfxr 11312 | . . 3 ⊢ +∞ ∈ ℝ* | |
4 | 1, 2, 3 | 3pm3.2i 1338 | . 2 ⊢ (1 ∈ ℝ* ∧ -∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) |
5 | xaddcom 13278 | . . . . . . . 8 ⊢ ((1 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (1 +𝑒 -∞) = (-∞ +𝑒 1)) | |
6 | 1, 2, 5 | mp2an 692 | . . . . . . 7 ⊢ (1 +𝑒 -∞) = (-∞ +𝑒 1) |
7 | 1re 11258 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
8 | renepnf 11306 | . . . . . . . . 9 ⊢ (1 ∈ ℝ → 1 ≠ +∞) | |
9 | 7, 8 | ax-mp 5 | . . . . . . . 8 ⊢ 1 ≠ +∞ |
10 | xaddmnf2 13267 | . . . . . . . 8 ⊢ ((1 ∈ ℝ* ∧ 1 ≠ +∞) → (-∞ +𝑒 1) = -∞) | |
11 | 1, 9, 10 | mp2an 692 | . . . . . . 7 ⊢ (-∞ +𝑒 1) = -∞ |
12 | 6, 11 | eqtri 2762 | . . . . . 6 ⊢ (1 +𝑒 -∞) = -∞ |
13 | 12 | oveq1i 7440 | . . . . 5 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) = (-∞ +𝑒 +∞) |
14 | mnfaddpnf 13269 | . . . . 5 ⊢ (-∞ +𝑒 +∞) = 0 | |
15 | 13, 14 | eqtri 2762 | . . . 4 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) = 0 |
16 | 0ne1 12334 | . . . 4 ⊢ 0 ≠ 1 | |
17 | 15, 16 | eqnetri 3008 | . . 3 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) ≠ 1 |
18 | 14 | oveq2i 7441 | . . . 4 ⊢ (1 +𝑒 (-∞ +𝑒 +∞)) = (1 +𝑒 0) |
19 | xaddrid 13279 | . . . . 5 ⊢ (1 ∈ ℝ* → (1 +𝑒 0) = 1) | |
20 | 1, 19 | ax-mp 5 | . . . 4 ⊢ (1 +𝑒 0) = 1 |
21 | 18, 20 | eqtri 2762 | . . 3 ⊢ (1 +𝑒 (-∞ +𝑒 +∞)) = 1 |
22 | 17, 21 | neeqtrri 3011 | . 2 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) ≠ (1 +𝑒 (-∞ +𝑒 +∞)) |
23 | xrsbas 21413 | . . 3 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
24 | xrsadd 21414 | . . 3 ⊢ +𝑒 = (+g‘ℝ*𝑠) | |
25 | 23, 24 | isnsgrp 18748 | . 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 1536 ∈ wcel 2105 ≠ wne 2937 ∉ wnel 3043 (class class class)co 7430 ℝcr 11151 0cc0 11152 1c1 11153 +∞cpnf 11289 -∞cmnf 11290 ℝ*cxr 11291 +𝑒 cxad 13149 ℝ*𝑠cxrs 17546 Smgrpcsgrp 18743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-xadd 13152 df-fz 13544 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17245 df-plusg 17310 df-mulr 17311 df-tset 17316 df-ple 17317 df-ds 17319 df-xrs 17548 df-sgrp 18744 |
This theorem is referenced by: xrsmgmdifsgrp 21438 |
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