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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 11320 | . . 3 ⊢ 1 ∈ ℝ* | |
| 2 | mnfxr 11318 | . . 3 ⊢ -∞ ∈ ℝ* | |
| 3 | pnfxr 11315 | . . 3 ⊢ +∞ ∈ ℝ* | |
| 4 | 1, 2, 3 | 3pm3.2i 1340 | . 2 ⊢ (1 ∈ ℝ* ∧ -∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) |
| 5 | xaddcom 13282 | . . . . . . . 8 ⊢ ((1 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (1 +𝑒 -∞) = (-∞ +𝑒 1)) | |
| 6 | 1, 2, 5 | mp2an 692 | . . . . . . 7 ⊢ (1 +𝑒 -∞) = (-∞ +𝑒 1) |
| 7 | 1re 11261 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
| 8 | renepnf 11309 | . . . . . . . . 9 ⊢ (1 ∈ ℝ → 1 ≠ +∞) | |
| 9 | 7, 8 | ax-mp 5 | . . . . . . . 8 ⊢ 1 ≠ +∞ |
| 10 | xaddmnf2 13271 | . . . . . . . 8 ⊢ ((1 ∈ ℝ* ∧ 1 ≠ +∞) → (-∞ +𝑒 1) = -∞) | |
| 11 | 1, 9, 10 | mp2an 692 | . . . . . . 7 ⊢ (-∞ +𝑒 1) = -∞ |
| 12 | 6, 11 | eqtri 2765 | . . . . . 6 ⊢ (1 +𝑒 -∞) = -∞ |
| 13 | 12 | oveq1i 7441 | . . . . 5 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) = (-∞ +𝑒 +∞) |
| 14 | mnfaddpnf 13273 | . . . . 5 ⊢ (-∞ +𝑒 +∞) = 0 | |
| 15 | 13, 14 | eqtri 2765 | . . . 4 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) = 0 |
| 16 | 0ne1 12337 | . . . 4 ⊢ 0 ≠ 1 | |
| 17 | 15, 16 | eqnetri 3011 | . . 3 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) ≠ 1 |
| 18 | 14 | oveq2i 7442 | . . . 4 ⊢ (1 +𝑒 (-∞ +𝑒 +∞)) = (1 +𝑒 0) |
| 19 | xaddrid 13283 | . . . . 5 ⊢ (1 ∈ ℝ* → (1 +𝑒 0) = 1) | |
| 20 | 1, 19 | ax-mp 5 | . . . 4 ⊢ (1 +𝑒 0) = 1 |
| 21 | 18, 20 | eqtri 2765 | . . 3 ⊢ (1 +𝑒 (-∞ +𝑒 +∞)) = 1 |
| 22 | 17, 21 | neeqtrri 3014 | . 2 ⊢ ((1 +𝑒 -∞) +𝑒 +∞) ≠ (1 +𝑒 (-∞ +𝑒 +∞)) |
| 23 | xrsbas 21396 | . . 3 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
| 24 | xrsadd 21397 | . . 3 ⊢ +𝑒 = (+g‘ℝ*𝑠) | |
| 25 | 23, 24 | isnsgrp 18736 | . 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 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∉ wnel 3046 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 +∞cpnf 11292 -∞cmnf 11293 ℝ*cxr 11294 +𝑒 cxad 13152 ℝ*𝑠cxrs 17545 Smgrpcsgrp 18731 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-xadd 13155 df-fz 13548 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-tset 17316 df-ple 17317 df-ds 17319 df-xrs 17547 df-sgrp 18732 |
| This theorem is referenced by: xrsmgmdifsgrp 21421 |
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