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Mirrors > Home > MPE Home > Th. List > resrng | Structured version Visualization version GIF version |
Description: The real numbers form a star ring. (Contributed by Thierry Arnoux, 19-Apr-2019.) (Proof shortened by Thierry Arnoux, 11-Jan-2025.) |
Ref | Expression |
---|---|
resrng | ⊢ ℝfld ∈ *-Ring |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rebase 21469 | . . 3 ⊢ ℝ = (Base‘ℝfld) | |
2 | refldcj 21483 | . . 3 ⊢ ∗ = (*𝑟‘ℝfld) | |
3 | refld 21482 | . . . . 5 ⊢ ℝfld ∈ Field | |
4 | 3 | a1i 11 | . . . 4 ⊢ (⊤ → ℝfld ∈ Field) |
5 | 4 | fldcrngd 20592 | . . 3 ⊢ (⊤ → ℝfld ∈ CRing) |
6 | cjre 15084 | . . . 4 ⊢ (𝑥 ∈ ℝ → (∗‘𝑥) = 𝑥) | |
7 | 6 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (∗‘𝑥) = 𝑥) |
8 | 1, 2, 5, 7 | idsrngd 20697 | . 2 ⊢ (⊤ → ℝfld ∈ *-Ring) |
9 | 8 | mptru 1540 | 1 ⊢ ℝfld ∈ *-Ring |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ‘cfv 6534 ℝcr 11106 ∗ccj 15041 Fieldcfield 20580 *-Ringcsr 20679 ℝfldcrefld 21467 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-addf 11186 ax-mulf 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12471 df-z 12557 df-dec 12676 df-uz 12821 df-fz 13483 df-cj 15044 df-re 15045 df-im 15046 df-struct 17081 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-mulr 17212 df-starv 17213 df-tset 17217 df-ple 17218 df-ds 17220 df-unif 17221 df-0g 17388 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-grp 18858 df-minusg 18859 df-subg 19042 df-ghm 19131 df-cmn 19694 df-abl 19695 df-mgp 20032 df-rng 20050 df-ur 20079 df-ring 20132 df-cring 20133 df-oppr 20228 df-dvdsr 20251 df-unit 20252 df-invr 20282 df-dvr 20295 df-rhm 20366 df-subrng 20438 df-subrg 20463 df-drng 20581 df-field 20582 df-staf 20680 df-srng 20681 df-cnfld 21231 df-refld 21468 |
This theorem is referenced by: rrxnm 25243 rrxds 25245 rrxplusgvscavalb 25247 |
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