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 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 12975 | . . . . . . 7 ⊢ -∞ < 0 | |
7 | mnfxr 11146 | . . . . . . . 8 ⊢ -∞ ∈ ℝ* | |
8 | 0xr 11136 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
9 | xrltnle 11156 | . . . . . . . 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 13303 | . . . . 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 13276 | . . . 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 13305 | . 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 12841 | . . . . . 6 ⊢ ℝ* ∈ V | |
26 | difexg 5283 | . . . . . 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 17053 | . . . . 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 20759 | . . 3 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) |
33 | ovex 7383 | . . . . 5 ⊢ (0[,]+∞) ∈ V | |
34 | ressress 17064 | . . . . 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 7361 | . . . 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 18520 | . 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 6492 (class class class)co 7350 0cc0 10985 +∞cpnf 11120 -∞cmnf 11121 ℝ*cxr 11122 < clt 11123 ≤ cle 11124 [,]cicc 13196 Basecbs 17018 ↾s cress 17047 0gc0g 17256 ℝ*𝑠cxrs 17317 Mndcmnd 18491 CMndccmn 19491 |
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 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
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 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-n0 12348 df-z 12434 df-dec 12552 df-uz 12697 df-xadd 12963 df-icc 13200 df-fz 13354 df-struct 16954 df-sets 16971 df-slot 16989 df-ndx 17001 df-base 17019 df-ress 17048 df-plusg 17081 df-mulr 17082 df-tset 17087 df-ple 17088 df-ds 17090 df-0g 17258 df-xrs 17319 df-mgm 18432 df-sgrp 18481 df-mnd 18492 df-submnd 18537 df-cmn 19493 |
This theorem is referenced by: xrge0mulgnn0 31662 xrge0slmod 31921 xrge0iifmhm 32281 esumgsum 32405 esumnul 32408 esum0 32409 gsumesum 32419 esumsnf 32424 esumss 32432 esumpfinval 32435 esumpfinvalf 32436 esumcocn 32440 sitmcl 32712 |
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