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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 486 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥) → 𝑥 ∈ ℝ*) | |
2 | ge0nemnf 12554 | . . . . 5 ⊢ ((𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥) → 𝑥 ≠ -∞) | |
3 | 1, 2 | jca 515 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥) → (𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞)) |
4 | elxrge0 12835 | . . . 4 ⊢ (𝑥 ∈ (0[,]+∞) ↔ (𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥)) | |
5 | eldifsn 4680 | . . . 4 ⊢ (𝑥 ∈ (ℝ* ∖ {-∞}) ↔ (𝑥 ∈ ℝ* ∧ 𝑥 ≠ -∞)) | |
6 | 3, 4, 5 | 3imtr4i 295 | . . 3 ⊢ (𝑥 ∈ (0[,]+∞) → 𝑥 ∈ (ℝ* ∖ {-∞})) |
7 | 6 | ssriv 3919 | . 2 ⊢ (0[,]+∞) ⊆ (ℝ* ∖ {-∞}) |
8 | 0e0iccpnf 12837 | . 2 ⊢ 0 ∈ (0[,]+∞) | |
9 | ge0xaddcl 12840 | . . 3 ⊢ ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞)) → (𝑥 +𝑒 𝑦) ∈ (0[,]+∞)) | |
10 | 9 | rgen2 3168 | . 2 ⊢ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝑥 +𝑒 𝑦) ∈ (0[,]+∞) |
11 | xrs1mnd.1 | . . . 4 ⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) | |
12 | 11 | xrs1mnd 20129 | . . 3 ⊢ 𝑅 ∈ Mnd |
13 | difss 4059 | . . . . 5 ⊢ (ℝ* ∖ {-∞}) ⊆ ℝ* | |
14 | xrsbas 20107 | . . . . . 6 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
15 | 11, 14 | ressbas2 16547 | . . . . 5 ⊢ ((ℝ* ∖ {-∞}) ⊆ ℝ* → (ℝ* ∖ {-∞}) = (Base‘𝑅)) |
16 | 13, 15 | ax-mp 5 | . . . 4 ⊢ (ℝ* ∖ {-∞}) = (Base‘𝑅) |
17 | 11 | xrs10 20130 | . . . 4 ⊢ 0 = (0g‘𝑅) |
18 | xrex 12374 | . . . . . 6 ⊢ ℝ* ∈ V | |
19 | 18 | difexi 5196 | . . . . 5 ⊢ (ℝ* ∖ {-∞}) ∈ V |
20 | xrsadd 20108 | . . . . . 6 ⊢ +𝑒 = (+g‘ℝ*𝑠) | |
21 | 11, 20 | ressplusg 16604 | . . . . 5 ⊢ ((ℝ* ∖ {-∞}) ∈ V → +𝑒 = (+g‘𝑅)) |
22 | 19, 21 | ax-mp 5 | . . . 4 ⊢ +𝑒 = (+g‘𝑅) |
23 | 16, 17, 22 | issubm 17960 | . . 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 1338 | 1 ⊢ (0[,]+∞) ∈ (SubMnd‘𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 {csn 4525 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 0cc0 10526 +∞cpnf 10661 -∞cmnf 10662 ℝ*cxr 10663 ≤ cle 10665 +𝑒 cxad 12493 [,]cicc 12729 Basecbs 16475 ↾s cress 16476 +gcplusg 16557 ℝ*𝑠cxrs 16765 Mndcmnd 17903 SubMndcsubmnd 17947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 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-icc 12733 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-tset 16576 df-ple 16577 df-ds 16579 df-0g 16707 df-xrs 16767 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 |
This theorem is referenced by: xrge0cmn 20133 xrge0gsumle 23438 xrge0tsms 23439 xrge0tsmsd 30742 |
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