Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > refld | Structured version Visualization version GIF version |
Description: The real numbers form a field. (Contributed by Thierry Arnoux, 1-Nov-2017.) |
Ref | Expression |
---|---|
refld | ⊢ ℝfld ∈ Field |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resubdrg 20801 | . . 3 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
2 | 1 | simpri 486 | . 2 ⊢ ℝfld ∈ DivRing |
3 | df-refld 20798 | . . 3 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
4 | cncrng 20607 | . . . 4 ⊢ ℂfld ∈ CRing | |
5 | 1 | simpli 484 | . . . 4 ⊢ ℝ ∈ (SubRing‘ℂfld) |
6 | eqid 2738 | . . . . 5 ⊢ (ℂfld ↾s ℝ) = (ℂfld ↾s ℝ) | |
7 | 6 | subrgcrng 20016 | . . . 4 ⊢ ((ℂfld ∈ CRing ∧ ℝ ∈ (SubRing‘ℂfld)) → (ℂfld ↾s ℝ) ∈ CRing) |
8 | 4, 5, 7 | mp2an 689 | . . 3 ⊢ (ℂfld ↾s ℝ) ∈ CRing |
9 | 3, 8 | eqeltri 2835 | . 2 ⊢ ℝfld ∈ CRing |
10 | isfld 19988 | . 2 ⊢ (ℝfld ∈ Field ↔ (ℝfld ∈ DivRing ∧ ℝfld ∈ CRing)) | |
11 | 2, 9, 10 | mpbir2an 708 | 1 ⊢ ℝfld ∈ Field |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 ‘cfv 6427 (class class class)co 7268 ℝcr 10858 ↾s cress 16929 CRingccrg 19772 DivRingcdr 19979 Fieldcfield 19980 SubRingcsubrg 20008 ℂfldccnfld 20585 ℝfldcrefld 20797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 ax-addf 10938 ax-mulf 10939 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4841 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7704 df-1st 7821 df-2nd 7822 df-tpos 8030 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-er 8486 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-div 11621 df-nn 11962 df-2 12024 df-3 12025 df-4 12026 df-5 12027 df-6 12028 df-7 12029 df-8 12030 df-9 12031 df-n0 12222 df-z 12308 df-dec 12426 df-uz 12571 df-fz 13228 df-struct 16836 df-sets 16853 df-slot 16871 df-ndx 16883 df-base 16901 df-ress 16930 df-plusg 16963 df-mulr 16964 df-starv 16965 df-tset 16969 df-ple 16970 df-ds 16972 df-unif 16973 df-0g 17140 df-mgm 18314 df-sgrp 18363 df-mnd 18374 df-grp 18568 df-minusg 18569 df-subg 18740 df-cmn 19376 df-mgp 19709 df-ur 19726 df-ring 19773 df-cring 19774 df-oppr 19850 df-dvdsr 19871 df-unit 19872 df-invr 19902 df-dvr 19913 df-drng 19981 df-field 19982 df-subrg 20010 df-cnfld 20586 df-refld 20798 |
This theorem is referenced by: recrng 20814 recvs 24297 recvsOLD 24298 rrxbase 24540 rrxprds 24541 rrxip 24542 rrxcph 24544 rrxsca 24548 rrx0 24549 reofld 31530 rearchi 31532 ccfldextrr 31709 ccfldsrarelvec 31727 bj-rrdrg 35447 |
Copyright terms: Public domain | W3C validator |