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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge00 | Structured version Visualization version GIF version |
Description: The zero of the extended nonnegative real numbers monoid. (Contributed by Thierry Arnoux, 30-Jan-2017.) |
Ref | Expression |
---|---|
xrge00 | ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) | |
2 | 1 | xrs1mnd 20788 | . 2 ⊢ (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ∈ Mnd |
3 | xrge0cmn 20792 | . . 3 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
4 | cmnmnd 19538 | . . 3 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd |
6 | mnflt0 13001 | . . . . . . 7 ⊢ -∞ < 0 | |
7 | mnfxr 11171 | . . . . . . . 8 ⊢ -∞ ∈ ℝ* | |
8 | 0xr 11161 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
9 | xrltnle 11181 | . . . . . . . 8 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ*) → (-∞ < 0 ↔ ¬ 0 ≤ -∞)) | |
10 | 7, 8, 9 | mp2an 691 | . . . . . . 7 ⊢ (-∞ < 0 ↔ ¬ 0 ≤ -∞) |
11 | 6, 10 | mpbi 229 | . . . . . 6 ⊢ ¬ 0 ≤ -∞ |
12 | 11 | intnan 488 | . . . . 5 ⊢ ¬ (-∞ ∈ ℝ* ∧ 0 ≤ -∞) |
13 | elxrge0 13329 | . . . . 5 ⊢ (-∞ ∈ (0[,]+∞) ↔ (-∞ ∈ ℝ* ∧ 0 ≤ -∞)) | |
14 | 12, 13 | mtbir 323 | . . . 4 ⊢ ¬ -∞ ∈ (0[,]+∞) |
15 | difsn 4757 | . . . 4 ⊢ (¬ -∞ ∈ (0[,]+∞) → ((0[,]+∞) ∖ {-∞}) = (0[,]+∞)) | |
16 | 14, 15 | ax-mp 5 | . . 3 ⊢ ((0[,]+∞) ∖ {-∞}) = (0[,]+∞) |
17 | iccssxr 13302 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
18 | ssdif 4098 | . . . 4 ⊢ ((0[,]+∞) ⊆ ℝ* → ((0[,]+∞) ∖ {-∞}) ⊆ (ℝ* ∖ {-∞})) | |
19 | 17, 18 | ax-mp 5 | . . 3 ⊢ ((0[,]+∞) ∖ {-∞}) ⊆ (ℝ* ∖ {-∞}) |
20 | 16, 19 | eqsstrri 3978 | . 2 ⊢ (0[,]+∞) ⊆ (ℝ* ∖ {-∞}) |
21 | 0e0iccpnf 13331 | . 2 ⊢ 0 ∈ (0[,]+∞) | |
22 | difss 4090 | . . . . 5 ⊢ (ℝ* ∖ {-∞}) ⊆ ℝ* | |
23 | df-ss 3926 | . . . . 5 ⊢ ((ℝ* ∖ {-∞}) ⊆ ℝ* ↔ ((ℝ* ∖ {-∞}) ∩ ℝ*) = (ℝ* ∖ {-∞})) | |
24 | 22, 23 | mpbi 229 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ∩ ℝ*) = (ℝ* ∖ {-∞}) |
25 | xrex 12867 | . . . . . 6 ⊢ ℝ* ∈ V | |
26 | difexg 5283 | . . . . . 6 ⊢ (ℝ* ∈ V → (ℝ* ∖ {-∞}) ∈ V) | |
27 | 25, 26 | ax-mp 5 | . . . . 5 ⊢ (ℝ* ∖ {-∞}) ∈ V |
28 | xrsbas 20766 | . . . . . 6 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
29 | 1, 28 | ressbas 17078 | . . . . 5 ⊢ ((ℝ* ∖ {-∞}) ∈ V → ((ℝ* ∖ {-∞}) ∩ ℝ*) = (Base‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞})))) |
30 | 27, 29 | ax-mp 5 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ∩ ℝ*) = (Base‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) |
31 | 24, 30 | eqtr3i 2768 | . . 3 ⊢ (ℝ* ∖ {-∞}) = (Base‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) |
32 | 1 | xrs10 20789 | . . 3 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) |
33 | ovex 7385 | . . . . 5 ⊢ (0[,]+∞) ∈ V | |
34 | ressress 17089 | . . . . 5 ⊢ (((ℝ* ∖ {-∞}) ∈ V ∧ (0[,]+∞) ∈ V) → ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s ((ℝ* ∖ {-∞}) ∩ (0[,]+∞)))) | |
35 | 27, 33, 34 | mp2an 691 | . . . 4 ⊢ ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s ((ℝ* ∖ {-∞}) ∩ (0[,]+∞))) |
36 | dfss 3927 | . . . . . . 7 ⊢ ((0[,]+∞) ⊆ (ℝ* ∖ {-∞}) ↔ (0[,]+∞) = ((0[,]+∞) ∩ (ℝ* ∖ {-∞}))) | |
37 | 20, 36 | mpbi 229 | . . . . . 6 ⊢ (0[,]+∞) = ((0[,]+∞) ∩ (ℝ* ∖ {-∞})) |
38 | incom 4160 | . . . . . 6 ⊢ ((0[,]+∞) ∩ (ℝ* ∖ {-∞})) = ((ℝ* ∖ {-∞}) ∩ (0[,]+∞)) | |
39 | 37, 38 | eqtr2i 2767 | . . . . 5 ⊢ ((ℝ* ∖ {-∞}) ∩ (0[,]+∞)) = (0[,]+∞) |
40 | 39 | oveq2i 7363 | . . . 4 ⊢ (ℝ*𝑠 ↾s ((ℝ* ∖ {-∞}) ∩ (0[,]+∞))) = (ℝ*𝑠 ↾s (0[,]+∞)) |
41 | 35, 40 | eqtr2i 2767 | . . 3 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) = ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) |
42 | 31, 32, 41 | submnd0 18545 | . 2 ⊢ ((((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ∈ Mnd ∧ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd) ∧ ((0[,]+∞) ⊆ (ℝ* ∖ {-∞}) ∧ 0 ∈ (0[,]+∞))) → 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞)))) |
43 | 2, 5, 20, 21, 42 | mp4an 692 | 1 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ∖ cdif 3906 ∩ cin 3908 ⊆ wss 3909 {csn 4585 class class class wbr 5104 ‘cfv 6494 (class class class)co 7352 0cc0 11010 +∞cpnf 11145 -∞cmnf 11146 ℝ*cxr 11147 < clt 11148 ≤ cle 11149 [,]cicc 13222 Basecbs 17043 ↾s cress 17072 0gc0g 17281 ℝ*𝑠cxrs 17342 Mndcmnd 18516 CMndccmn 19521 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-1st 7914 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-1o 8405 df-er 8607 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-nn 12113 df-2 12175 df-3 12176 df-4 12177 df-5 12178 df-6 12179 df-7 12180 df-8 12181 df-9 12182 df-n0 12373 df-z 12459 df-dec 12578 df-uz 12723 df-xadd 12989 df-icc 13226 df-fz 13380 df-struct 16979 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-mulr 17107 df-tset 17112 df-ple 17113 df-ds 17115 df-0g 17283 df-xrs 17344 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-submnd 18562 df-cmn 19523 |
This theorem is referenced by: xrge0mulgnn0 31705 xrge0slmod 31964 xrge0iifmhm 32324 esumgsum 32448 esumnul 32451 esum0 32452 gsumesum 32462 esumsnf 32467 esumss 32475 esumpfinval 32478 esumpfinvalf 32479 esumcocn 32483 sitmcl 32755 |
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