Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 2823 | . . 3 ⊢ (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) | |
2 | 1 | xrs1mnd 20585 | . 2 ⊢ (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ∈ Mnd |
3 | xrge0cmn 20589 | . . 3 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
4 | cmnmnd 18924 | . . 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 12523 | . . . . . . 7 ⊢ -∞ < 0 | |
7 | mnfxr 10700 | . . . . . . . 8 ⊢ -∞ ∈ ℝ* | |
8 | 0xr 10690 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
9 | xrltnle 10710 | . . . . . . . 8 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ*) → (-∞ < 0 ↔ ¬ 0 ≤ -∞)) | |
10 | 7, 8, 9 | mp2an 690 | . . . . . . 7 ⊢ (-∞ < 0 ↔ ¬ 0 ≤ -∞) |
11 | 6, 10 | mpbi 232 | . . . . . 6 ⊢ ¬ 0 ≤ -∞ |
12 | 11 | intnan 489 | . . . . 5 ⊢ ¬ (-∞ ∈ ℝ* ∧ 0 ≤ -∞) |
13 | elxrge0 12848 | . . . . 5 ⊢ (-∞ ∈ (0[,]+∞) ↔ (-∞ ∈ ℝ* ∧ 0 ≤ -∞)) | |
14 | 12, 13 | mtbir 325 | . . . 4 ⊢ ¬ -∞ ∈ (0[,]+∞) |
15 | difsn 4733 | . . . 4 ⊢ (¬ -∞ ∈ (0[,]+∞) → ((0[,]+∞) ∖ {-∞}) = (0[,]+∞)) | |
16 | 14, 15 | ax-mp 5 | . . 3 ⊢ ((0[,]+∞) ∖ {-∞}) = (0[,]+∞) |
17 | iccssxr 12822 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
18 | ssdif 4118 | . . . 4 ⊢ ((0[,]+∞) ⊆ ℝ* → ((0[,]+∞) ∖ {-∞}) ⊆ (ℝ* ∖ {-∞})) | |
19 | 17, 18 | ax-mp 5 | . . 3 ⊢ ((0[,]+∞) ∖ {-∞}) ⊆ (ℝ* ∖ {-∞}) |
20 | 16, 19 | eqsstrri 4004 | . 2 ⊢ (0[,]+∞) ⊆ (ℝ* ∖ {-∞}) |
21 | 0e0iccpnf 12850 | . 2 ⊢ 0 ∈ (0[,]+∞) | |
22 | difss 4110 | . . . . 5 ⊢ (ℝ* ∖ {-∞}) ⊆ ℝ* | |
23 | df-ss 3954 | . . . . 5 ⊢ ((ℝ* ∖ {-∞}) ⊆ ℝ* ↔ ((ℝ* ∖ {-∞}) ∩ ℝ*) = (ℝ* ∖ {-∞})) | |
24 | 22, 23 | mpbi 232 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ∩ ℝ*) = (ℝ* ∖ {-∞}) |
25 | xrex 12389 | . . . . . 6 ⊢ ℝ* ∈ V | |
26 | difexg 5233 | . . . . . 6 ⊢ (ℝ* ∈ V → (ℝ* ∖ {-∞}) ∈ V) | |
27 | 25, 26 | ax-mp 5 | . . . . 5 ⊢ (ℝ* ∖ {-∞}) ∈ V |
28 | xrsbas 20563 | . . . . . 6 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
29 | 1, 28 | ressbas 16556 | . . . . 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 2848 | . . 3 ⊢ (ℝ* ∖ {-∞}) = (Base‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) |
32 | 1 | xrs10 20586 | . . 3 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) |
33 | ovex 7191 | . . . . 5 ⊢ (0[,]+∞) ∈ V | |
34 | ressress 16564 | . . . . 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 3955 | . . . . . . 7 ⊢ ((0[,]+∞) ⊆ (ℝ* ∖ {-∞}) ↔ (0[,]+∞) = ((0[,]+∞) ∩ (ℝ* ∖ {-∞}))) | |
37 | 20, 36 | mpbi 232 | . . . . . 6 ⊢ (0[,]+∞) = ((0[,]+∞) ∩ (ℝ* ∖ {-∞})) |
38 | incom 4180 | . . . . . 6 ⊢ ((0[,]+∞) ∩ (ℝ* ∖ {-∞})) = ((ℝ* ∖ {-∞}) ∩ (0[,]+∞)) | |
39 | 37, 38 | eqtr2i 2847 | . . . . 5 ⊢ ((ℝ* ∖ {-∞}) ∩ (0[,]+∞)) = (0[,]+∞) |
40 | 39 | oveq2i 7169 | . . . 4 ⊢ (ℝ*𝑠 ↾s ((ℝ* ∖ {-∞}) ∩ (0[,]+∞))) = (ℝ*𝑠 ↾s (0[,]+∞)) |
41 | 35, 40 | eqtr2i 2847 | . . 3 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) = ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) |
42 | 31, 32, 41 | submnd0 17942 | . 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∖ cdif 3935 ∩ cin 3937 ⊆ wss 3938 {csn 4569 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 0cc0 10539 +∞cpnf 10674 -∞cmnf 10675 ℝ*cxr 10676 < clt 10677 ≤ cle 10678 [,]cicc 12744 Basecbs 16485 ↾s cress 16486 0gc0g 16715 ℝ*𝑠cxrs 16775 Mndcmnd 17913 CMndccmn 18908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-xadd 12511 df-icc 12748 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-tset 16586 df-ple 16587 df-ds 16589 df-0g 16717 df-xrs 16777 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-cmn 18910 |
This theorem is referenced by: xrge0mulgnn0 30678 xrge0slmod 30919 xrge0iifmhm 31184 esumgsum 31306 esumnul 31309 esum0 31310 gsumesum 31320 esumsnf 31325 esumss 31333 esumpfinval 31336 esumpfinvalf 31337 esumcocn 31341 sitmcl 31611 |
Copyright terms: Public domain | W3C validator |