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Mirrors > Home > MPE Home > Th. List > xrs1mnd | Structured version Visualization version GIF version |
Description: The extended real numbers, restricted to ℝ* ∖ {-∞}, form an additive monoid - in contrast to the full structure, see xrsmgmdifsgrp 20510. (Contributed by Mario Carneiro, 27-Nov-2014.) |
Ref | Expression |
---|---|
xrs1mnd.1 | ⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) |
Ref | Expression |
---|---|
xrs1mnd | ⊢ 𝑅 ∈ Mnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 4105 | . . . 4 ⊢ (ℝ* ∖ {-∞}) ⊆ ℝ* | |
2 | xrs1mnd.1 | . . . . 5 ⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) | |
3 | xrsbas 20489 | . . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
4 | 2, 3 | ressbas2 16543 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ⊆ ℝ* → (ℝ* ∖ {-∞}) = (Base‘𝑅)) |
5 | 1, 4 | mp1i 13 | . . 3 ⊢ (⊤ → (ℝ* ∖ {-∞}) = (Base‘𝑅)) |
6 | xrex 12374 | . . . . 5 ⊢ ℝ* ∈ V | |
7 | 6 | difexi 5223 | . . . 4 ⊢ (ℝ* ∖ {-∞}) ∈ V |
8 | xrsadd 20490 | . . . . 5 ⊢ +𝑒 = (+g‘ℝ*𝑠) | |
9 | 2, 8 | ressplusg 16600 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ∈ V → +𝑒 = (+g‘𝑅)) |
10 | 7, 9 | mp1i 13 | . . 3 ⊢ (⊤ → +𝑒 = (+g‘𝑅)) |
11 | eldifsn 4711 | . . . . 5 ⊢ (𝑥 ∈ (ℝ* ∖ {-∞}) ↔ (𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞)) | |
12 | eldifsn 4711 | . . . . 5 ⊢ (𝑦 ∈ (ℝ* ∖ {-∞}) ↔ (𝑦 ∈ ℝ* ∧ 𝑦 ≠ -∞)) | |
13 | xaddcl 12620 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 +𝑒 𝑦) ∈ ℝ*) | |
14 | 13 | ad2ant2r 743 | . . . . . 6 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞) ∧ (𝑦 ∈ ℝ* ∧ 𝑦 ≠ -∞)) → (𝑥 +𝑒 𝑦) ∈ ℝ*) |
15 | xaddnemnf 12617 | . . . . . 6 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞) ∧ (𝑦 ∈ ℝ* ∧ 𝑦 ≠ -∞)) → (𝑥 +𝑒 𝑦) ≠ -∞) | |
16 | eldifsn 4711 | . . . . . 6 ⊢ ((𝑥 +𝑒 𝑦) ∈ (ℝ* ∖ {-∞}) ↔ ((𝑥 +𝑒 𝑦) ∈ ℝ* ∧ (𝑥 +𝑒 𝑦) ≠ -∞)) | |
17 | 14, 15, 16 | sylanbrc 583 | . . . . 5 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞) ∧ (𝑦 ∈ ℝ* ∧ 𝑦 ≠ -∞)) → (𝑥 +𝑒 𝑦) ∈ (ℝ* ∖ {-∞})) |
18 | 11, 12, 17 | syl2anb 597 | . . . 4 ⊢ ((𝑥 ∈ (ℝ* ∖ {-∞}) ∧ 𝑦 ∈ (ℝ* ∖ {-∞})) → (𝑥 +𝑒 𝑦) ∈ (ℝ* ∖ {-∞})) |
19 | 18 | 3adant1 1122 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ* ∖ {-∞}) ∧ 𝑦 ∈ (ℝ* ∖ {-∞})) → (𝑥 +𝑒 𝑦) ∈ (ℝ* ∖ {-∞})) |
20 | eldifsn 4711 | . . . . 5 ⊢ (𝑧 ∈ (ℝ* ∖ {-∞}) ↔ (𝑧 ∈ ℝ* ∧ 𝑧 ≠ -∞)) | |
21 | xaddass 12630 | . . . . 5 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞) ∧ (𝑦 ∈ ℝ* ∧ 𝑦 ≠ -∞) ∧ (𝑧 ∈ ℝ* ∧ 𝑧 ≠ -∞)) → ((𝑥 +𝑒 𝑦) +𝑒 𝑧) = (𝑥 +𝑒 (𝑦 +𝑒 𝑧))) | |
22 | 11, 12, 20, 21 | syl3anb 1153 | . . . 4 ⊢ ((𝑥 ∈ (ℝ* ∖ {-∞}) ∧ 𝑦 ∈ (ℝ* ∖ {-∞}) ∧ 𝑧 ∈ (ℝ* ∖ {-∞})) → ((𝑥 +𝑒 𝑦) +𝑒 𝑧) = (𝑥 +𝑒 (𝑦 +𝑒 𝑧))) |
23 | 22 | adantl 482 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (ℝ* ∖ {-∞}) ∧ 𝑦 ∈ (ℝ* ∖ {-∞}) ∧ 𝑧 ∈ (ℝ* ∖ {-∞}))) → ((𝑥 +𝑒 𝑦) +𝑒 𝑧) = (𝑥 +𝑒 (𝑦 +𝑒 𝑧))) |
24 | 0re 10631 | . . . 4 ⊢ 0 ∈ ℝ | |
25 | rexr 10675 | . . . . 5 ⊢ (0 ∈ ℝ → 0 ∈ ℝ*) | |
26 | renemnf 10678 | . . . . 5 ⊢ (0 ∈ ℝ → 0 ≠ -∞) | |
27 | eldifsn 4711 | . . . . 5 ⊢ (0 ∈ (ℝ* ∖ {-∞}) ↔ (0 ∈ ℝ* ∧ 0 ≠ -∞)) | |
28 | 25, 26, 27 | sylanbrc 583 | . . . 4 ⊢ (0 ∈ ℝ → 0 ∈ (ℝ* ∖ {-∞})) |
29 | 24, 28 | mp1i 13 | . . 3 ⊢ (⊤ → 0 ∈ (ℝ* ∖ {-∞})) |
30 | eldifi 4100 | . . . . 5 ⊢ (𝑥 ∈ (ℝ* ∖ {-∞}) → 𝑥 ∈ ℝ*) | |
31 | 30 | adantl 482 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ* ∖ {-∞})) → 𝑥 ∈ ℝ*) |
32 | xaddid2 12623 | . . . 4 ⊢ (𝑥 ∈ ℝ* → (0 +𝑒 𝑥) = 𝑥) | |
33 | 31, 32 | syl 17 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ* ∖ {-∞})) → (0 +𝑒 𝑥) = 𝑥) |
34 | 31 | xaddid1d 12624 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ* ∖ {-∞})) → (𝑥 +𝑒 0) = 𝑥) |
35 | 5, 10, 19, 23, 29, 33, 34 | ismndd 17921 | . 2 ⊢ (⊤ → 𝑅 ∈ Mnd) |
36 | 35 | mptru 1535 | 1 ⊢ 𝑅 ∈ Mnd |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∧ w3a 1079 = wceq 1528 ⊤wtru 1529 ∈ wcel 2105 ≠ wne 3013 Vcvv 3492 ∖ cdif 3930 ⊆ wss 3933 {csn 4557 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 0cc0 10525 -∞cmnf 10661 ℝ*cxr 10662 +𝑒 cxad 12493 Basecbs 16471 ↾s cress 16472 +gcplusg 16553 ℝ*𝑠cxrs 16761 Mndcmnd 17899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 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 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-tset 16572 df-ple 16573 df-ds 16575 df-xrs 16763 df-mgm 17840 df-sgrp 17889 df-mnd 17900 |
This theorem is referenced by: xrs1cmn 20513 xrge0subm 20514 xrge00 30600 |
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