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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 473 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥) → 𝑥 ∈ ℝ*) | |
2 | ge0nemnf 12004 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥) → 𝑥 ≠ -∞) | |
3 | 1, 2 | jca 554 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥) → (𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞)) |
4 | elxrge0 12281 | . . . 4 ⊢ (𝑥 ∈ (0[,]+∞) ↔ (𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥)) | |
5 | eldifsn 4317 | . . . 4 ⊢ (𝑥 ∈ (ℝ* ∖ {-∞}) ↔ (𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞)) | |
6 | 3, 4, 5 | 3imtr4i 281 | . . 3 ⊢ (𝑥 ∈ (0[,]+∞) → 𝑥 ∈ (ℝ* ∖ {-∞})) |
7 | 6 | ssriv 3607 | . 2 ⊢ (0[,]+∞) ⊆ (ℝ* ∖ {-∞}) |
8 | 0e0iccpnf 12283 | . 2 ⊢ 0 ∈ (0[,]+∞) | |
9 | ge0xaddcl 12286 | . . 3 ⊢ ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞)) → (𝑥 +𝑒 𝑦) ∈ (0[,]+∞)) | |
10 | 9 | rgen2a 2977 | . 2 ⊢ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 +𝑒 𝑦) ∈ (0[,]+∞) |
11 | xrs1mnd.1 | . . . 4 ⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) | |
12 | 11 | xrs1mnd 19784 | . . 3 ⊢ 𝑅 ∈ Mnd |
13 | difss 3737 | . . . . 5 ⊢ (ℝ* ∖ {-∞}) ⊆ ℝ* | |
14 | xrsbas 19762 | . . . . . 6 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
15 | 11, 14 | ressbas2 15931 | . . . . 5 ⊢ ((ℝ* ∖ {-∞}) ⊆ ℝ* → (ℝ* ∖ {-∞}) = (Base‘𝑅)) |
16 | 13, 15 | ax-mp 5 | . . . 4 ⊢ (ℝ* ∖ {-∞}) = (Base‘𝑅) |
17 | 11 | xrs10 19785 | . . . 4 ⊢ 0 = (0g‘𝑅) |
18 | xrex 11829 | . . . . . 6 ⊢ ℝ* ∈ V | |
19 | difexg 4808 | . . . . . 6 ⊢ (ℝ* ∈ V → (ℝ* ∖ {-∞}) ∈ V) | |
20 | 18, 19 | ax-mp 5 | . . . . 5 ⊢ (ℝ* ∖ {-∞}) ∈ V |
21 | xrsadd 19763 | . . . . . 6 ⊢ +𝑒 = (+g‘ℝ*𝑠) | |
22 | 11, 21 | ressplusg 15993 | . . . . 5 ⊢ ((ℝ* ∖ {-∞}) ∈ V → +𝑒 = (+g‘𝑅)) |
23 | 20, 22 | ax-mp 5 | . . . 4 ⊢ +𝑒 = (+g‘𝑅) |
24 | 16, 17, 23 | issubm 17347 | . . 3 ⊢ (𝑅 ∈ Mnd → ((0[,]+∞) ∈ (SubMnd‘𝑅) ↔ ((0[,]+∞) ⊆ (ℝ* ∖ {-∞}) ∧ 0 ∈ (0[,]+∞) ∧ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 +𝑒 𝑦) ∈ (0[,]+∞)))) |
25 | 12, 24 | ax-mp 5 | . 2 ⊢ ((0[,]+∞) ∈ (SubMnd‘𝑅) ↔ ((0[,]+∞) ⊆ (ℝ* ∖ {-∞}) ∧ 0 ∈ (0[,]+∞) ∧ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 +𝑒 𝑦) ∈ (0[,]+∞))) |
26 | 7, 8, 10, 25 | mpbir3an 1244 | 1 ⊢ (0[,]+∞) ∈ (SubMnd‘𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 Vcvv 3200 ∖ cdif 3571 ⊆ wss 3574 {csn 4177 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 0cc0 9936 +∞cpnf 10071 -∞cmnf 10072 ℝ*cxr 10073 ≤ cle 10075 +𝑒 cxad 11944 [,]cicc 12178 Basecbs 15857 ↾s cress 15858 +gcplusg 15941 ℝ*𝑠cxrs 16160 Mndcmnd 17294 SubMndcsubmnd 17334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-xadd 11947 df-icc 12182 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-tset 15960 df-ple 15961 df-ds 15964 df-0g 16102 df-xrs 16162 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 |
This theorem is referenced by: xrge0cmn 19788 xrge0gsumle 22636 xrge0tsms 22637 xrge0tsmsd 29785 |
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