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Mirrors > Home > MPE Home > Th. List > xrs10 | Structured version Visualization version GIF version |
Description: The zero of the extended real number monoid. (Contributed by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
xrs1mnd.1 | ⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) |
Ref | Expression |
---|---|
xrs10 | ⊢ 0 = (0g‘𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 4046 | . . . 4 ⊢ (ℝ* ∖ {-∞}) ⊆ ℝ* | |
2 | xrs1mnd.1 | . . . . 5 ⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) | |
3 | xrsbas 20379 | . . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
4 | 2, 3 | ressbas2 16791 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ⊆ ℝ* → (ℝ* ∖ {-∞}) = (Base‘𝑅)) |
5 | 1, 4 | ax-mp 5 | . . 3 ⊢ (ℝ* ∖ {-∞}) = (Base‘𝑅) |
6 | eqid 2737 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | xrex 12583 | . . . . 5 ⊢ ℝ* ∈ V | |
8 | 7 | difexi 5221 | . . . 4 ⊢ (ℝ* ∖ {-∞}) ∈ V |
9 | xrsadd 20380 | . . . . 5 ⊢ +𝑒 = (+g‘ℝ*𝑠) | |
10 | 2, 9 | ressplusg 16834 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ∈ V → +𝑒 = (+g‘𝑅)) |
11 | 8, 10 | ax-mp 5 | . . 3 ⊢ +𝑒 = (+g‘𝑅) |
12 | 0re 10835 | . . . 4 ⊢ 0 ∈ ℝ | |
13 | rexr 10879 | . . . . 5 ⊢ (0 ∈ ℝ → 0 ∈ ℝ*) | |
14 | renemnf 10882 | . . . . 5 ⊢ (0 ∈ ℝ → 0 ≠ -∞) | |
15 | eldifsn 4700 | . . . . 5 ⊢ (0 ∈ (ℝ* ∖ {-∞}) ↔ (0 ∈ ℝ* ∧ 0 ≠ -∞)) | |
16 | 13, 14, 15 | sylanbrc 586 | . . . 4 ⊢ (0 ∈ ℝ → 0 ∈ (ℝ* ∖ {-∞})) |
17 | 12, 16 | mp1i 13 | . . 3 ⊢ (⊤ → 0 ∈ (ℝ* ∖ {-∞})) |
18 | eldifi 4041 | . . . . 5 ⊢ (𝑥 ∈ (ℝ* ∖ {-∞}) → 𝑥 ∈ ℝ*) | |
19 | 18 | adantl 485 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ* ∖ {-∞})) → 𝑥 ∈ ℝ*) |
20 | xaddid2 12832 | . . . 4 ⊢ (𝑥 ∈ ℝ* → (0 +𝑒 𝑥) = 𝑥) | |
21 | 19, 20 | syl 17 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ* ∖ {-∞})) → (0 +𝑒 𝑥) = 𝑥) |
22 | 19 | xaddid1d 12833 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ* ∖ {-∞})) → (𝑥 +𝑒 0) = 𝑥) |
23 | 5, 6, 11, 17, 21, 22 | ismgmid2 18140 | . 2 ⊢ (⊤ → 0 = (0g‘𝑅)) |
24 | 23 | mptru 1550 | 1 ⊢ 0 = (0g‘𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ⊤wtru 1544 ∈ wcel 2110 ≠ wne 2940 Vcvv 3408 ∖ cdif 3863 ⊆ wss 3866 {csn 4541 ‘cfv 6380 (class class class)co 7213 ℝcr 10728 0cc0 10729 -∞cmnf 10865 ℝ*cxr 10866 +𝑒 cxad 12702 Basecbs 16760 ↾s cress 16784 +gcplusg 16802 0gc0g 16944 ℝ*𝑠cxrs 17005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-xadd 12705 df-fz 13096 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-tset 16821 df-ple 16822 df-ds 16824 df-0g 16946 df-xrs 17007 |
This theorem is referenced by: xrge0subm 20404 imasdsf1olem 23271 xrge0gsumle 23730 xrge0tsms 23731 xrge00 31014 xrge0tsmsd 31036 gsumge0cl 43584 |
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