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 2737 | . . 3 ⊢ (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) | |
2 | 1 | xrs1mnd 20758 | . 2 ⊢ (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ∈ Mnd |
3 | xrge0cmn 20762 | . . 3 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
4 | cmnmnd 19508 | . . 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 12974 | . . . . . . 7 ⊢ -∞ < 0 | |
7 | mnfxr 11145 | . . . . . . . 8 ⊢ -∞ ∈ ℝ* | |
8 | 0xr 11135 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
9 | xrltnle 11155 | . . . . . . . 8 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ*) → (-∞ < 0 ↔ ¬ 0 ≤ -∞)) | |
10 | 7, 8, 9 | mp2an 690 | . . . . . . 7 ⊢ (-∞ < 0 ↔ ¬ 0 ≤ -∞) |
11 | 6, 10 | mpbi 229 | . . . . . 6 ⊢ ¬ 0 ≤ -∞ |
12 | 11 | intnan 487 | . . . . 5 ⊢ ¬ (-∞ ∈ ℝ* ∧ 0 ≤ -∞) |
13 | elxrge0 13302 | . . . . 5 ⊢ (-∞ ∈ (0[,]+∞) ↔ (-∞ ∈ ℝ* ∧ 0 ≤ -∞)) | |
14 | 12, 13 | mtbir 322 | . . . 4 ⊢ ¬ -∞ ∈ (0[,]+∞) |
15 | difsn 4756 | . . . 4 ⊢ (¬ -∞ ∈ (0[,]+∞) → ((0[,]+∞) ∖ {-∞}) = (0[,]+∞)) | |
16 | 14, 15 | ax-mp 5 | . . 3 ⊢ ((0[,]+∞) ∖ {-∞}) = (0[,]+∞) |
17 | iccssxr 13275 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
18 | ssdif 4097 | . . . 4 ⊢ ((0[,]+∞) ⊆ ℝ* → ((0[,]+∞) ∖ {-∞}) ⊆ (ℝ* ∖ {-∞})) | |
19 | 17, 18 | ax-mp 5 | . . 3 ⊢ ((0[,]+∞) ∖ {-∞}) ⊆ (ℝ* ∖ {-∞}) |
20 | 16, 19 | eqsstrri 3977 | . 2 ⊢ (0[,]+∞) ⊆ (ℝ* ∖ {-∞}) |
21 | 0e0iccpnf 13304 | . 2 ⊢ 0 ∈ (0[,]+∞) | |
22 | difss 4089 | . . . . 5 ⊢ (ℝ* ∖ {-∞}) ⊆ ℝ* | |
23 | df-ss 3925 | . . . . 5 ⊢ ((ℝ* ∖ {-∞}) ⊆ ℝ* ↔ ((ℝ* ∖ {-∞}) ∩ ℝ*) = (ℝ* ∖ {-∞})) | |
24 | 22, 23 | mpbi 229 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ∩ ℝ*) = (ℝ* ∖ {-∞}) |
25 | xrex 12840 | . . . . . 6 ⊢ ℝ* ∈ V | |
26 | difexg 5282 | . . . . . 6 ⊢ (ℝ* ∈ V → (ℝ* ∖ {-∞}) ∈ V) | |
27 | 25, 26 | ax-mp 5 | . . . . 5 ⊢ (ℝ* ∖ {-∞}) ∈ V |
28 | xrsbas 20736 | . . . . . 6 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
29 | 1, 28 | ressbas 17052 | . . . . 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 2767 | . . 3 ⊢ (ℝ* ∖ {-∞}) = (Base‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) |
32 | 1 | xrs10 20759 | . . 3 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) |
33 | ovex 7382 | . . . . 5 ⊢ (0[,]+∞) ∈ V | |
34 | ressress 17063 | . . . . 5 ⊢ (((ℝ* ∖ {-∞}) ∈ V ∧ (0[,]+∞) ∈ V) → ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s ((ℝ* ∖ {-∞}) ∩ (0[,]+∞)))) | |
35 | 27, 33, 34 | mp2an 690 | . . . 4 ⊢ ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s ((ℝ* ∖ {-∞}) ∩ (0[,]+∞))) |
36 | dfss 3926 | . . . . . . 7 ⊢ ((0[,]+∞) ⊆ (ℝ* ∖ {-∞}) ↔ (0[,]+∞) = ((0[,]+∞) ∩ (ℝ* ∖ {-∞}))) | |
37 | 20, 36 | mpbi 229 | . . . . . 6 ⊢ (0[,]+∞) = ((0[,]+∞) ∩ (ℝ* ∖ {-∞})) |
38 | incom 4159 | . . . . . 6 ⊢ ((0[,]+∞) ∩ (ℝ* ∖ {-∞})) = ((ℝ* ∖ {-∞}) ∩ (0[,]+∞)) | |
39 | 37, 38 | eqtr2i 2766 | . . . . 5 ⊢ ((ℝ* ∖ {-∞}) ∩ (0[,]+∞)) = (0[,]+∞) |
40 | 39 | oveq2i 7360 | . . . 4 ⊢ (ℝ*𝑠 ↾s ((ℝ* ∖ {-∞}) ∩ (0[,]+∞))) = (ℝ*𝑠 ↾s (0[,]+∞)) |
41 | 35, 40 | eqtr2i 2766 | . . 3 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) = ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) |
42 | 31, 32, 41 | submnd0 18519 | . 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 691 | 1 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ∖ cdif 3905 ∩ cin 3907 ⊆ wss 3908 {csn 4584 class class class wbr 5103 ‘cfv 6491 (class class class)co 7349 0cc0 10984 +∞cpnf 11119 -∞cmnf 11120 ℝ*cxr 11121 < clt 11122 ≤ cle 11123 [,]cicc 13195 Basecbs 17017 ↾s cress 17046 0gc0g 17255 ℝ*𝑠cxrs 17316 Mndcmnd 18490 CMndccmn 19491 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 ax-pre-mulgt0 11061 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7793 df-1st 7911 df-2nd 7912 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-1o 8379 df-er 8581 df-en 8817 df-dom 8818 df-sdom 8819 df-fin 8820 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-nn 12087 df-2 12149 df-3 12150 df-4 12151 df-5 12152 df-6 12153 df-7 12154 df-8 12155 df-9 12156 df-n0 12347 df-z 12433 df-dec 12551 df-uz 12696 df-xadd 12962 df-icc 13199 df-fz 13353 df-struct 16953 df-sets 16970 df-slot 16988 df-ndx 17000 df-base 17018 df-ress 17047 df-plusg 17080 df-mulr 17081 df-tset 17086 df-ple 17087 df-ds 17089 df-0g 17257 df-xrs 17318 df-mgm 18431 df-sgrp 18480 df-mnd 18491 df-submnd 18536 df-cmn 19493 |
This theorem is referenced by: xrge0mulgnn0 31674 xrge0slmod 31933 xrge0iifmhm 32293 esumgsum 32417 esumnul 32420 esum0 32421 gsumesum 32431 esumsnf 32436 esumss 32444 esumpfinval 32447 esumpfinvalf 32448 esumcocn 32452 sitmcl 32724 |
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