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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 4024 | . . . 4 ⊢ (ℝ* ∖ {-∞}) ⊆ ℝ* | |
2 | xrs1mnd.1 | . . . . 5 ⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) | |
3 | xrsbas 20231 | . . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
4 | 2, 3 | ressbas2 16372 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ⊆ ℝ* → (ℝ* ∖ {-∞}) = (Base‘𝑅)) |
5 | 1, 4 | ax-mp 5 | . . 3 ⊢ (ℝ* ∖ {-∞}) = (Base‘𝑅) |
6 | eqid 2793 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | xrex 12225 | . . . . 5 ⊢ ℝ* ∈ V | |
8 | 7 | difexi 5116 | . . . 4 ⊢ (ℝ* ∖ {-∞}) ∈ V |
9 | xrsadd 20232 | . . . . 5 ⊢ +𝑒 = (+g‘ℝ*𝑠) | |
10 | 2, 9 | ressplusg 16429 | . . . 4 ⊢ ((ℝ* ∖ {-∞}) ∈ V → +𝑒 = (+g‘𝑅)) |
11 | 8, 10 | ax-mp 5 | . . 3 ⊢ +𝑒 = (+g‘𝑅) |
12 | 0re 10478 | . . . 4 ⊢ 0 ∈ ℝ | |
13 | rexr 10522 | . . . . 5 ⊢ (0 ∈ ℝ → 0 ∈ ℝ*) | |
14 | renemnf 10525 | . . . . 5 ⊢ (0 ∈ ℝ → 0 ≠ -∞) | |
15 | eldifsn 4620 | . . . . 5 ⊢ (0 ∈ (ℝ* ∖ {-∞}) ↔ (0 ∈ ℝ* ∧ 0 ≠ -∞)) | |
16 | 13, 14, 15 | sylanbrc 583 | . . . 4 ⊢ (0 ∈ ℝ → 0 ∈ (ℝ* ∖ {-∞})) |
17 | 12, 16 | mp1i 13 | . . 3 ⊢ (⊤ → 0 ∈ (ℝ* ∖ {-∞})) |
18 | eldifi 4019 | . . . . 5 ⊢ (𝑥 ∈ (ℝ* ∖ {-∞}) → 𝑥 ∈ ℝ*) | |
19 | 18 | adantl 482 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ* ∖ {-∞})) → 𝑥 ∈ ℝ*) |
20 | xaddid2 12474 | . . . 4 ⊢ (𝑥 ∈ ℝ* → (0 +𝑒 𝑥) = 𝑥) | |
21 | 19, 20 | syl 17 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ* ∖ {-∞})) → (0 +𝑒 𝑥) = 𝑥) |
22 | 19 | xaddid1d 12475 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ* ∖ {-∞})) → (𝑥 +𝑒 0) = 𝑥) |
23 | 5, 6, 11, 17, 21, 22 | ismgmid2 17694 | . 2 ⊢ (⊤ → 0 = (0g‘𝑅)) |
24 | 23 | mptru 1527 | 1 ⊢ 0 = (0g‘𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1520 ⊤wtru 1521 ∈ wcel 2079 ≠ wne 2982 Vcvv 3432 ∖ cdif 3851 ⊆ wss 3854 {csn 4466 ‘cfv 6217 (class class class)co 7007 ℝcr 10371 0cc0 10372 -∞cmnf 10508 ℝ*cxr 10509 +𝑒 cxad 12344 Basecbs 16300 ↾s cress 16301 +gcplusg 16382 0gc0g 16530 ℝ*𝑠cxrs 16590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-oadd 7948 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-7 11542 df-8 11543 df-9 11544 df-n0 11735 df-z 11819 df-dec 11937 df-uz 12083 df-xadd 12347 df-fz 12732 df-struct 16302 df-ndx 16303 df-slot 16304 df-base 16306 df-sets 16307 df-ress 16308 df-plusg 16395 df-mulr 16396 df-tset 16401 df-ple 16402 df-ds 16404 df-0g 16532 df-xrs 16592 |
This theorem is referenced by: xrge0subm 20256 imasdsf1olem 22654 xrge0gsumle 23112 xrge0tsms 23113 xrge00 30317 xrge0tsmsd 30460 gsumge0cl 42149 |
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