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 2610 | . . 3 ⊢ (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) | |
2 | 1 | xrs1mnd 19603 | . 2 ⊢ (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ∈ Mnd |
3 | xrge0cmn 19607 | . . 3 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
4 | cmnmnd 18031 | . . 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 11835 | . . . . . . 7 ⊢ -∞ < 0 | |
7 | mnfxr 9975 | . . . . . . . 8 ⊢ -∞ ∈ ℝ* | |
8 | 0xr 9965 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
9 | xrltnle 9984 | . . . . . . . 8 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ*) → (-∞ < 0 ↔ ¬ 0 ≤ -∞)) | |
10 | 7, 8, 9 | mp2an 704 | . . . . . . 7 ⊢ (-∞ < 0 ↔ ¬ 0 ≤ -∞) |
11 | 6, 10 | mpbi 219 | . . . . . 6 ⊢ ¬ 0 ≤ -∞ |
12 | 11 | intnan 951 | . . . . 5 ⊢ ¬ (-∞ ∈ ℝ* ∧ 0 ≤ -∞) |
13 | elxrge0 12152 | . . . . 5 ⊢ (-∞ ∈ (0[,]+∞) ↔ (-∞ ∈ ℝ* ∧ 0 ≤ -∞)) | |
14 | 12, 13 | mtbir 312 | . . . 4 ⊢ ¬ -∞ ∈ (0[,]+∞) |
15 | difsn 4269 | . . . 4 ⊢ (¬ -∞ ∈ (0[,]+∞) → ((0[,]+∞) ∖ {-∞}) = (0[,]+∞)) | |
16 | 14, 15 | ax-mp 5 | . . 3 ⊢ ((0[,]+∞) ∖ {-∞}) = (0[,]+∞) |
17 | iccssxr 12127 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
18 | ssdif 3707 | . . . 4 ⊢ ((0[,]+∞) ⊆ ℝ* → ((0[,]+∞) ∖ {-∞}) ⊆ (ℝ* ∖ {-∞})) | |
19 | 17, 18 | ax-mp 5 | . . 3 ⊢ ((0[,]+∞) ∖ {-∞}) ⊆ (ℝ* ∖ {-∞}) |
20 | 16, 19 | eqsstr3i 3599 | . 2 ⊢ (0[,]+∞) ⊆ (ℝ* ∖ {-∞}) |
21 | 0e0iccpnf 12154 | . 2 ⊢ 0 ∈ (0[,]+∞) | |
22 | difss 3699 | . . . . 5 ⊢ (ℝ* ∖ {-∞}) ⊆ ℝ* | |
23 | df-ss 3554 | . . . . 5 ⊢ ((ℝ* ∖ {-∞}) ⊆ ℝ* ↔ ((ℝ* ∖ {-∞}) ∩ ℝ*) = (ℝ* ∖ {-∞})) | |
24 | 22, 23 | mpbi 219 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ∩ ℝ*) = (ℝ* ∖ {-∞}) |
25 | xrex 11705 | . . . . . 6 ⊢ ℝ* ∈ V | |
26 | difexg 4735 | . . . . . 6 ⊢ (ℝ* ∈ V → (ℝ* ∖ {-∞}) ∈ V) | |
27 | 25, 26 | ax-mp 5 | . . . . 5 ⊢ (ℝ* ∖ {-∞}) ∈ V |
28 | xrsbas 19581 | . . . . . 6 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
29 | 1, 28 | ressbas 15757 | . . . . 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 2634 | . . 3 ⊢ (ℝ* ∖ {-∞}) = (Base‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) |
32 | 1 | xrs10 19604 | . . 3 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (ℝ* ∖ {-∞}))) |
33 | ovex 6577 | . . . . 5 ⊢ (0[,]+∞) ∈ V | |
34 | ressress 15765 | . . . . 5 ⊢ (((ℝ* ∖ {-∞}) ∈ V ∧ (0[,]+∞) ∈ V) → ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s ((ℝ* ∖ {-∞}) ∩ (0[,]+∞)))) | |
35 | 27, 33, 34 | mp2an 704 | . . . 4 ⊢ ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s ((ℝ* ∖ {-∞}) ∩ (0[,]+∞))) |
36 | dfss 3555 | . . . . . . 7 ⊢ ((0[,]+∞) ⊆ (ℝ* ∖ {-∞}) ↔ (0[,]+∞) = ((0[,]+∞) ∩ (ℝ* ∖ {-∞}))) | |
37 | 20, 36 | mpbi 219 | . . . . . 6 ⊢ (0[,]+∞) = ((0[,]+∞) ∩ (ℝ* ∖ {-∞})) |
38 | incom 3767 | . . . . . 6 ⊢ ((0[,]+∞) ∩ (ℝ* ∖ {-∞})) = ((ℝ* ∖ {-∞}) ∩ (0[,]+∞)) | |
39 | 37, 38 | eqtr2i 2633 | . . . . 5 ⊢ ((ℝ* ∖ {-∞}) ∩ (0[,]+∞)) = (0[,]+∞) |
40 | 39 | oveq2i 6560 | . . . 4 ⊢ (ℝ*𝑠 ↾s ((ℝ* ∖ {-∞}) ∩ (0[,]+∞))) = (ℝ*𝑠 ↾s (0[,]+∞)) |
41 | 35, 40 | eqtr2i 2633 | . . 3 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) = ((ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ↾s (0[,]+∞)) |
42 | 31, 32, 41 | submnd0 17143 | . 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 705 | 1 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 {csn 4125 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 0cc0 9815 +∞cpnf 9950 -∞cmnf 9951 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 [,]cicc 12049 Basecbs 15695 ↾s cress 15696 0gc0g 15923 ℝ*𝑠cxrs 15983 Mndcmnd 17117 CMndccmn 18016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-xadd 11823 df-icc 12053 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-tset 15787 df-ple 15788 df-ds 15791 df-0g 15925 df-xrs 15985 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-cmn 18018 |
This theorem is referenced by: xrge0mulgnn0 29020 xrge0slmod 29175 xrge0iifmhm 29313 esumgsum 29434 esumnul 29437 esum0 29438 gsumesum 29448 esumsnf 29453 esumss 29461 esumpfinval 29464 esumpfinvalf 29465 esumcocn 29469 sitmcl 29740 |
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