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| Mirrors > Home > MPE Home > Th. List > resubdrg | Structured version Visualization version GIF version | ||
| Description: The real numbers form a division subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.) (Revised by Thierry Arnoux, 30-Jun-2019.) |
| Ref | Expression |
|---|---|
| resubdrg | ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | recn 11245 | . . 3 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
| 2 | readdcl 11238 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ) | |
| 3 | renegcl 11572 | . . 3 ⊢ (𝑥 ∈ ℝ → -𝑥 ∈ ℝ) | |
| 4 | 1re 11261 | . . 3 ⊢ 1 ∈ ℝ | |
| 5 | remulcl 11240 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ) | |
| 6 | rereccl 11985 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑥 ≠ 0) → (1 / 𝑥) ∈ ℝ) | |
| 7 | 1, 2, 3, 4, 5, 6 | cnsubdrglem 21436 | . 2 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℝ) ∈ DivRing) |
| 8 | df-refld 21623 | . . . 4 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
| 9 | 8 | eleq1i 2832 | . . 3 ⊢ (ℝfld ∈ DivRing ↔ (ℂfld ↾s ℝ) ∈ DivRing) |
| 10 | 9 | anbi2i 623 | . 2 ⊢ ((ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) ↔ (ℝ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℝ) ∈ DivRing)) |
| 11 | 7, 10 | mpbir 231 | 1 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 ↾s cress 17274 SubRingcsubrg 20569 DivRingcdr 20729 ℂfldccnfld 21364 ℝfldcrefld 21622 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-subrng 20546 df-subrg 20570 df-drng 20731 df-cnfld 21365 df-refld 21623 |
| This theorem is referenced by: resubgval 21627 re1r 21631 redvr 21635 refld 21637 rzgrp 21641 recvs 25179 recvsOLD 25180 taylthlem2 26416 taylthlem2OLD 26417 reefgim 26494 circgrp 26594 circsubm 26595 jensenlem2 27031 amgmlem 27033 nn0archi 33375 rrxdim 33665 ccfldextrr 33699 rezh 33970 rerrext 34010 cnrrext 34011 zrhre 34020 qqhre 34021 bj-rveccmod 37303 amgmwlem 49321 |
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