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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 4107 | . . . 4 ⊢ (ℝ* ∖ {-∞}) ⊆ ℝ* | |
2 | xrs1mnd.1 | . . . . 5 ⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) | |
3 | xrsbas 20560 | . . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
4 | 2, 3 | ressbas2 16554 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ⊆ ℝ* → (ℝ* ∖ {-∞}) = (Base‘𝑅)) |
5 | 1, 4 | ax-mp 5 | . . 3 ⊢ (ℝ* ∖ {-∞}) = (Base‘𝑅) |
6 | eqid 2821 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | xrex 12385 | . . . . 5 ⊢ ℝ* ∈ V | |
8 | 7 | difexi 5231 | . . . 4 ⊢ (ℝ* ∖ {-∞}) ∈ V |
9 | xrsadd 20561 | . . . . 5 ⊢ +𝑒 = (+g‘ℝ*𝑠) | |
10 | 2, 9 | ressplusg 16611 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ∈ V → +𝑒 = (+g‘𝑅)) |
11 | 8, 10 | ax-mp 5 | . . 3 ⊢ +𝑒 = (+g‘𝑅) |
12 | 0re 10642 | . . . 4 ⊢ 0 ∈ ℝ | |
13 | rexr 10686 | . . . . 5 ⊢ (0 ∈ ℝ → 0 ∈ ℝ*) | |
14 | renemnf 10689 | . . . . 5 ⊢ (0 ∈ ℝ → 0 ≠ -∞) | |
15 | eldifsn 4718 | . . . . 5 ⊢ (0 ∈ (ℝ* ∖ {-∞}) ↔ (0 ∈ ℝ* ∧ 0 ≠ -∞)) | |
16 | 13, 14, 15 | sylanbrc 585 | . . . 4 ⊢ (0 ∈ ℝ → 0 ∈ (ℝ* ∖ {-∞})) |
17 | 12, 16 | mp1i 13 | . . 3 ⊢ (⊤ → 0 ∈ (ℝ* ∖ {-∞})) |
18 | eldifi 4102 | . . . . 5 ⊢ (𝑥 ∈ (ℝ* ∖ {-∞}) → 𝑥 ∈ ℝ*) | |
19 | 18 | adantl 484 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ* ∖ {-∞})) → 𝑥 ∈ ℝ*) |
20 | xaddid2 12634 | . . . 4 ⊢ (𝑥 ∈ ℝ* → (0 +𝑒 𝑥) = 𝑥) | |
21 | 19, 20 | syl 17 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ* ∖ {-∞})) → (0 +𝑒 𝑥) = 𝑥) |
22 | 19 | xaddid1d 12635 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ* ∖ {-∞})) → (𝑥 +𝑒 0) = 𝑥) |
23 | 5, 6, 11, 17, 21, 22 | ismgmid2 17877 | . 2 ⊢ (⊤ → 0 = (0g‘𝑅)) |
24 | 23 | mptru 1540 | 1 ⊢ 0 = (0g‘𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ⊤wtru 1534 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 ∖ cdif 3932 ⊆ wss 3935 {csn 4566 ‘cfv 6354 (class class class)co 7155 ℝcr 10535 0cc0 10536 -∞cmnf 10672 ℝ*cxr 10673 +𝑒 cxad 12504 Basecbs 16482 ↾s cress 16483 +gcplusg 16564 0gc0g 16712 ℝ*𝑠cxrs 16772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-xadd 12507 df-fz 12892 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-tset 16583 df-ple 16584 df-ds 16586 df-0g 16714 df-xrs 16774 |
This theorem is referenced by: xrge0subm 20585 imasdsf1olem 22982 xrge0gsumle 23440 xrge0tsms 23441 xrge00 30673 xrge0tsmsd 30692 gsumge0cl 42652 |
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