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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 21292. (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 4126 | . . . 4 ⊢ (ℝ* ∖ {-∞}) ⊆ ℝ* | |
2 | xrs1mnd.1 | . . . . 5 ⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) | |
3 | xrsbas 21267 | . . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
4 | 2, 3 | ressbas2 17188 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ⊆ ℝ* → (ℝ* ∖ {-∞}) = (Base‘𝑅)) |
5 | 1, 4 | mp1i 13 | . . 3 ⊢ (⊤ → (ℝ* ∖ {-∞}) = (Base‘𝑅)) |
6 | xrex 12972 | . . . . 5 ⊢ ℝ* ∈ V | |
7 | 6 | difexi 5321 | . . . 4 ⊢ (ℝ* ∖ {-∞}) ∈ V |
8 | xrsadd 21268 | . . . . 5 ⊢ +𝑒 = (+g‘ℝ*𝑠) | |
9 | 2, 8 | ressplusg 17241 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ∈ V → +𝑒 = (+g‘𝑅)) |
10 | 7, 9 | mp1i 13 | . . 3 ⊢ (⊤ → +𝑒 = (+g‘𝑅)) |
11 | eldifsn 4785 | . . . . 5 ⊢ (𝑥 ∈ (ℝ* ∖ {-∞}) ↔ (𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞)) | |
12 | eldifsn 4785 | . . . . 5 ⊢ (𝑦 ∈ (ℝ* ∖ {-∞}) ↔ (𝑦 ∈ ℝ* ∧ 𝑦 ≠ -∞)) | |
13 | xaddcl 13221 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 +𝑒 𝑦) ∈ ℝ*) | |
14 | 13 | ad2ant2r 744 | . . . . . 6 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞) ∧ (𝑦 ∈ ℝ* ∧ 𝑦 ≠ -∞)) → (𝑥 +𝑒 𝑦) ∈ ℝ*) |
15 | xaddnemnf 13218 | . . . . . 6 ⊢ (((𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞) ∧ (𝑦 ∈ ℝ* ∧ 𝑦 ≠ -∞)) → (𝑥 +𝑒 𝑦) ≠ -∞) | |
16 | eldifsn 4785 | . . . . . 6 ⊢ ((𝑥 +𝑒 𝑦) ∈ (ℝ* ∖ {-∞}) ↔ ((𝑥 +𝑒 𝑦) ∈ ℝ* ∧ (𝑥 +𝑒 𝑦) ≠ -∞)) | |
17 | 14, 15, 16 | sylanbrc 582 | . . . . 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 1127 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ* ∖ {-∞}) ∧ 𝑦 ∈ (ℝ* ∖ {-∞})) → (𝑥 +𝑒 𝑦) ∈ (ℝ* ∖ {-∞})) |
20 | eldifsn 4785 | . . . . 5 ⊢ (𝑧 ∈ (ℝ* ∖ {-∞}) ↔ (𝑧 ∈ ℝ* ∧ 𝑧 ≠ -∞)) | |
21 | xaddass 13231 | . . . . 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 1158 | . . . 4 ⊢ ((𝑥 ∈ (ℝ* ∖ {-∞}) ∧ 𝑦 ∈ (ℝ* ∖ {-∞}) ∧ 𝑧 ∈ (ℝ* ∖ {-∞})) → ((𝑥 +𝑒 𝑦) +𝑒 𝑧) = (𝑥 +𝑒 (𝑦 +𝑒 𝑧))) |
23 | 22 | adantl 481 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (ℝ* ∖ {-∞}) ∧ 𝑦 ∈ (ℝ* ∖ {-∞}) ∧ 𝑧 ∈ (ℝ* ∖ {-∞}))) → ((𝑥 +𝑒 𝑦) +𝑒 𝑧) = (𝑥 +𝑒 (𝑦 +𝑒 𝑧))) |
24 | 0re 11217 | . . . 4 ⊢ 0 ∈ ℝ | |
25 | rexr 11261 | . . . . 5 ⊢ (0 ∈ ℝ → 0 ∈ ℝ*) | |
26 | renemnf 11264 | . . . . 5 ⊢ (0 ∈ ℝ → 0 ≠ -∞) | |
27 | eldifsn 4785 | . . . . 5 ⊢ (0 ∈ (ℝ* ∖ {-∞}) ↔ (0 ∈ ℝ* ∧ 0 ≠ -∞)) | |
28 | 25, 26, 27 | sylanbrc 582 | . . . 4 ⊢ (0 ∈ ℝ → 0 ∈ (ℝ* ∖ {-∞})) |
29 | 24, 28 | mp1i 13 | . . 3 ⊢ (⊤ → 0 ∈ (ℝ* ∖ {-∞})) |
30 | eldifi 4121 | . . . . 5 ⊢ (𝑥 ∈ (ℝ* ∖ {-∞}) → 𝑥 ∈ ℝ*) | |
31 | 30 | adantl 481 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ* ∖ {-∞})) → 𝑥 ∈ ℝ*) |
32 | xaddlid 13224 | . . . 4 ⊢ (𝑥 ∈ ℝ* → (0 +𝑒 𝑥) = 𝑥) | |
33 | 31, 32 | syl 17 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ* ∖ {-∞})) → (0 +𝑒 𝑥) = 𝑥) |
34 | 31 | xaddridd 13225 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ* ∖ {-∞})) → (𝑥 +𝑒 0) = 𝑥) |
35 | 5, 10, 19, 23, 29, 33, 34 | ismndd 18686 | . 2 ⊢ (⊤ → 𝑅 ∈ Mnd) |
36 | 35 | mptru 1540 | 1 ⊢ 𝑅 ∈ Mnd |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∧ w3a 1084 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ≠ wne 2934 Vcvv 3468 ∖ cdif 3940 ⊆ wss 3943 {csn 4623 ‘cfv 6536 (class class class)co 7404 ℝcr 11108 0cc0 11109 -∞cmnf 11247 ℝ*cxr 11248 +𝑒 cxad 13093 Basecbs 17150 ↾s cress 17179 +gcplusg 17203 ℝ*𝑠cxrs 17452 Mndcmnd 18664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-xadd 13096 df-fz 13488 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-tset 17222 df-ple 17223 df-ds 17225 df-xrs 17454 df-mgm 18570 df-sgrp 18649 df-mnd 18665 |
This theorem is referenced by: xrs1cmn 21295 xrge0subm 21296 xrge00 32687 |
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