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Mirrors > Home > MPE Home > Th. List > recrng | Structured version Visualization version GIF version |
Description: The real numbers form a star ring. (Contributed by Thierry Arnoux, 19-Apr-2019.) |
Ref | Expression |
---|---|
recrng | ⊢ ℝfld ∈ *-Ring |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rebase 20446 | . . 3 ⊢ ℝ = (Base‘ℝfld) | |
2 | refldcj 20460 | . . 3 ⊢ ∗ = (*𝑟‘ℝfld) | |
3 | refld 20459 | . . . . . 6 ⊢ ℝfld ∈ Field | |
4 | isfld 19228 | . . . . . 6 ⊢ (ℝfld ∈ Field ↔ (ℝfld ∈ DivRing ∧ ℝfld ∈ CRing)) | |
5 | 3, 4 | mpbi 222 | . . . . 5 ⊢ (ℝfld ∈ DivRing ∧ ℝfld ∈ CRing) |
6 | 5 | simpri 478 | . . . 4 ⊢ ℝfld ∈ CRing |
7 | 6 | a1i 11 | . . 3 ⊢ (⊤ → ℝfld ∈ CRing) |
8 | cjre 14353 | . . . 4 ⊢ (𝑥 ∈ ℝ → (∗‘𝑥) = 𝑥) | |
9 | 8 | adantl 474 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (∗‘𝑥) = 𝑥) |
10 | 1, 2, 7, 9 | idsrngd 19349 | . 2 ⊢ (⊤ → ℝfld ∈ *-Ring) |
11 | 10 | mptru 1514 | 1 ⊢ ℝfld ∈ *-Ring |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 387 = wceq 1507 ⊤wtru 1508 ∈ wcel 2050 ‘cfv 6182 ℝcr 10328 ∗ccj 14310 CRingccrg 19015 DivRingcdr 19219 Fieldcfield 19220 *-Ringcsr 19331 ℝfldcrefld 20444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 ax-addf 10408 ax-mulf 10409 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7495 df-2nd 7496 df-tpos 7689 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-oadd 7903 df-er 8083 df-map 8202 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-div 11093 df-nn 11434 df-2 11497 df-3 11498 df-4 11499 df-5 11500 df-6 11501 df-7 11502 df-8 11503 df-9 11504 df-n0 11702 df-z 11788 df-dec 11906 df-uz 12053 df-fz 12703 df-cj 14313 df-re 14314 df-im 14315 df-struct 16335 df-ndx 16336 df-slot 16337 df-base 16339 df-sets 16340 df-ress 16341 df-plusg 16428 df-mulr 16429 df-starv 16430 df-tset 16434 df-ple 16435 df-ds 16437 df-unif 16438 df-0g 16565 df-mgm 17704 df-sgrp 17746 df-mnd 17757 df-mhm 17797 df-grp 17888 df-minusg 17889 df-subg 18054 df-ghm 18121 df-cmn 18662 df-mgp 18957 df-ur 18969 df-ring 19016 df-cring 19017 df-oppr 19090 df-dvdsr 19108 df-unit 19109 df-invr 19139 df-dvr 19150 df-rnghom 19184 df-drng 19221 df-field 19222 df-subrg 19250 df-staf 19332 df-srng 19333 df-cnfld 20242 df-refld 20445 |
This theorem is referenced by: rrxnm 23691 rrxds 23693 rrxplusgvscavalb 23695 |
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