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Mirrors > Home > MPE Home > Th. List > cnflddiv | Structured version Visualization version GIF version |
Description: The division operation in the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
cnflddiv | ⊢ / = (/r‘ℂfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnring 19984 | . . . . . . . 8 ⊢ ℂfld ∈ Ring | |
2 | cnfldbas 19966 | . . . . . . . . 9 ⊢ ℂ = (Base‘ℂfld) | |
3 | cnfld0 19986 | . . . . . . . . . 10 ⊢ 0 = (0g‘ℂfld) | |
4 | cndrng 19991 | . . . . . . . . . 10 ⊢ ℂfld ∈ DivRing | |
5 | 2, 3, 4 | drngui 18964 | . . . . . . . . 9 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
6 | eqid 2771 | . . . . . . . . 9 ⊢ (/r‘ℂfld) = (/r‘ℂfld) | |
7 | cnfldmul 19968 | . . . . . . . . 9 ⊢ · = (.r‘ℂfld) | |
8 | 2, 5, 6, 7 | dvrcan1 18900 | . . . . . . . 8 ⊢ ((ℂfld ∈ Ring ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → ((𝑥(/r‘ℂfld)𝑦) · 𝑦) = 𝑥) |
9 | 1, 8 | mp3an1 1559 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → ((𝑥(/r‘ℂfld)𝑦) · 𝑦) = 𝑥) |
10 | 9 | oveq1d 6809 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (((𝑥(/r‘ℂfld)𝑦) · 𝑦) / 𝑦) = (𝑥 / 𝑦)) |
11 | 2, 5, 6 | dvrcl 18895 | . . . . . . . 8 ⊢ ((ℂfld ∈ Ring ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥(/r‘ℂfld)𝑦) ∈ ℂ) |
12 | 1, 11 | mp3an1 1559 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥(/r‘ℂfld)𝑦) ∈ ℂ) |
13 | simpr 471 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ∈ (ℂ ∖ {0})) | |
14 | eldifsn 4454 | . . . . . . . . 9 ⊢ (𝑦 ∈ (ℂ ∖ {0}) ↔ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0)) | |
15 | 13, 14 | sylib 208 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0)) |
16 | 15 | simpld 478 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ∈ ℂ) |
17 | 15 | simprd 479 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ≠ 0) |
18 | 12, 16, 17 | divcan4d 11010 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (((𝑥(/r‘ℂfld)𝑦) · 𝑦) / 𝑦) = (𝑥(/r‘ℂfld)𝑦)) |
19 | 10, 18 | eqtr3d 2807 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥 / 𝑦) = (𝑥(/r‘ℂfld)𝑦)) |
20 | simpl 468 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑥 ∈ ℂ) | |
21 | divval 10890 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (𝑥 / 𝑦) = (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) | |
22 | 20, 16, 17, 21 | syl3anc 1476 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥 / 𝑦) = (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) |
23 | 19, 22 | eqtr3d 2807 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥(/r‘ℂfld)𝑦) = (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) |
24 | eqid 2771 | . . . . 5 ⊢ (invr‘ℂfld) = (invr‘ℂfld) | |
25 | 2, 7, 5, 24, 6 | dvrval 18894 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥(/r‘ℂfld)𝑦) = (𝑥 · ((invr‘ℂfld)‘𝑦))) |
26 | 23, 25 | eqtr3d 2807 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) = (𝑥 · ((invr‘ℂfld)‘𝑦))) |
27 | 26 | mpt2eq3ia 6868 | . 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑥 · ((invr‘ℂfld)‘𝑦))) |
28 | df-div 10888 | . 2 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) | |
29 | 2, 7, 5, 24, 6 | dvrfval 18893 | . 2 ⊢ (/r‘ℂfld) = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑥 · ((invr‘ℂfld)‘𝑦))) |
30 | 27, 28, 29 | 3eqtr4i 2803 | 1 ⊢ / = (/r‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∖ cdif 3721 {csn 4317 ‘cfv 6032 ℩crio 6754 (class class class)co 6794 ↦ cmpt2 6796 ℂcc 10137 0cc0 10139 · cmul 10144 / cdiv 10887 Ringcrg 18756 invrcinvr 18880 /rcdvr 18891 ℂfldccnfld 19962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 ax-cnex 10195 ax-resscn 10196 ax-1cn 10197 ax-icn 10198 ax-addcl 10199 ax-addrcl 10200 ax-mulcl 10201 ax-mulrcl 10202 ax-mulcom 10203 ax-addass 10204 ax-mulass 10205 ax-distr 10206 ax-i2m1 10207 ax-1ne0 10208 ax-1rid 10209 ax-rnegex 10210 ax-rrecex 10211 ax-cnre 10212 ax-pre-lttri 10213 ax-pre-lttrn 10214 ax-pre-ltadd 10215 ax-pre-mulgt0 10216 ax-addf 10218 ax-mulf 10219 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-pss 3740 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5824 df-ord 5870 df-on 5871 df-lim 5872 df-suc 5873 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-f1 6037 df-fo 6038 df-f1o 6039 df-fv 6040 df-riota 6755 df-ov 6797 df-oprab 6798 df-mpt2 6799 df-om 7214 df-1st 7316 df-2nd 7317 df-tpos 7505 df-wrecs 7560 df-recs 7622 df-rdg 7660 df-1o 7714 df-oadd 7718 df-er 7897 df-en 8111 df-dom 8112 df-sdom 8113 df-fin 8114 df-pnf 10279 df-mnf 10280 df-xr 10281 df-ltxr 10282 df-le 10283 df-sub 10471 df-neg 10472 df-div 10888 df-nn 11224 df-2 11282 df-3 11283 df-4 11284 df-5 11285 df-6 11286 df-7 11287 df-8 11288 df-9 11289 df-n0 11496 df-z 11581 df-dec 11697 df-uz 11890 df-fz 12535 df-struct 16067 df-ndx 16068 df-slot 16069 df-base 16071 df-sets 16072 df-ress 16073 df-plusg 16163 df-mulr 16164 df-starv 16165 df-tset 16169 df-ple 16170 df-ds 16173 df-unif 16174 df-0g 16311 df-mgm 17451 df-sgrp 17493 df-mnd 17504 df-grp 17634 df-minusg 17635 df-cmn 18403 df-mgp 18699 df-ur 18711 df-ring 18758 df-cring 18759 df-oppr 18832 df-dvdsr 18850 df-unit 18851 df-invr 18881 df-dvr 18892 df-drng 18960 df-cnfld 19963 |
This theorem is referenced by: cnfldinv 19993 cnsubdrglem 20013 qsssubdrg 20021 redvr 20181 cvsdiv 23152 qrngdiv 25535 |
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