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Mirrors > Home > MPE Home > Th. List > xrge0subm | Structured version Visualization version GIF version |
Description: The nonnegative extended real numbers are a submonoid of the nonnegative-infinite extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
xrs1mnd.1 | ⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) |
Ref | Expression |
---|---|
xrge0subm | ⊢ (0[,]+∞) ∈ (SubMnd‘𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 483 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥) → 𝑥 ∈ ℝ*) | |
2 | ge0nemnf 12905 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥) → 𝑥 ≠ -∞) | |
3 | 1, 2 | jca 512 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥) → (𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞)) |
4 | elxrge0 13187 | . . . 4 ⊢ (𝑥 ∈ (0[,]+∞) ↔ (𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥)) | |
5 | eldifsn 4722 | . . . 4 ⊢ (𝑥 ∈ (ℝ* ∖ {-∞}) ↔ (𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞)) | |
6 | 3, 4, 5 | 3imtr4i 292 | . . 3 ⊢ (𝑥 ∈ (0[,]+∞) → 𝑥 ∈ (ℝ* ∖ {-∞})) |
7 | 6 | ssriv 3926 | . 2 ⊢ (0[,]+∞) ⊆ (ℝ* ∖ {-∞}) |
8 | 0e0iccpnf 13189 | . 2 ⊢ 0 ∈ (0[,]+∞) | |
9 | ge0xaddcl 13192 | . . 3 ⊢ ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞)) → (𝑥 +𝑒 𝑦) ∈ (0[,]+∞)) | |
10 | 9 | rgen2 3128 | . 2 ⊢ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 +𝑒 𝑦) ∈ (0[,]+∞) |
11 | xrs1mnd.1 | . . . 4 ⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) | |
12 | 11 | xrs1mnd 20634 | . . 3 ⊢ 𝑅 ∈ Mnd |
13 | difss 4067 | . . . . 5 ⊢ (ℝ* ∖ {-∞}) ⊆ ℝ* | |
14 | xrsbas 20612 | . . . . . 6 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
15 | 11, 14 | ressbas2 16947 | . . . . 5 ⊢ ((ℝ* ∖ {-∞}) ⊆ ℝ* → (ℝ* ∖ {-∞}) = (Base‘𝑅)) |
16 | 13, 15 | ax-mp 5 | . . . 4 ⊢ (ℝ* ∖ {-∞}) = (Base‘𝑅) |
17 | 11 | xrs10 20635 | . . . 4 ⊢ 0 = (0g‘𝑅) |
18 | xrex 12725 | . . . . . 6 ⊢ ℝ* ∈ V | |
19 | 18 | difexi 5254 | . . . . 5 ⊢ (ℝ* ∖ {-∞}) ∈ V |
20 | xrsadd 20613 | . . . . . 6 ⊢ +𝑒 = (+g‘ℝ*𝑠) | |
21 | 11, 20 | ressplusg 16998 | . . . . 5 ⊢ ((ℝ* ∖ {-∞}) ∈ V → +𝑒 = (+g‘𝑅)) |
22 | 19, 21 | ax-mp 5 | . . . 4 ⊢ +𝑒 = (+g‘𝑅) |
23 | 16, 17, 22 | issubm 18440 | . . 3 ⊢ (𝑅 ∈ Mnd → ((0[,]+∞) ∈ (SubMnd‘𝑅) ↔ ((0[,]+∞) ⊆ (ℝ* ∖ {-∞}) ∧ 0 ∈ (0[,]+∞) ∧ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 +𝑒 𝑦) ∈ (0[,]+∞)))) |
24 | 12, 23 | ax-mp 5 | . 2 ⊢ ((0[,]+∞) ∈ (SubMnd‘𝑅) ↔ ((0[,]+∞) ⊆ (ℝ* ∖ {-∞}) ∧ 0 ∈ (0[,]+∞) ∧ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 +𝑒 𝑦) ∈ (0[,]+∞))) |
25 | 7, 8, 10, 24 | mpbir3an 1340 | 1 ⊢ (0[,]+∞) ∈ (SubMnd‘𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 Vcvv 3431 ∖ cdif 3885 ⊆ wss 3888 {csn 4563 class class class wbr 5076 ‘cfv 6435 (class class class)co 7277 0cc0 10869 +∞cpnf 11004 -∞cmnf 11005 ℝ*cxr 11006 ≤ cle 11008 +𝑒 cxad 12844 [,]cicc 13080 Basecbs 16910 ↾s cress 16939 +gcplusg 16960 ℝ*𝑠cxrs 17209 Mndcmnd 18383 SubMndcsubmnd 18427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-1o 8295 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-fin 8735 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12436 df-uz 12581 df-xadd 12847 df-icc 13084 df-fz 13238 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-tset 16979 df-ple 16980 df-ds 16982 df-0g 17150 df-xrs 17211 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 |
This theorem is referenced by: xrge0cmn 20638 xrge0gsumle 23994 xrge0tsms 23995 xrge0tsmsd 31314 |
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