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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 19768 | . . . . . . . 8 ⊢ ℂfld ∈ Ring | |
2 | cnfldbas 19750 | . . . . . . . . 9 ⊢ ℂ = (Base‘ℂfld) | |
3 | cnfld0 19770 | . . . . . . . . . 10 ⊢ 0 = (0g‘ℂfld) | |
4 | cndrng 19775 | . . . . . . . . . 10 ⊢ ℂfld ∈ DivRing | |
5 | 2, 3, 4 | drngui 18753 | . . . . . . . . 9 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
6 | eqid 2622 | . . . . . . . . 9 ⊢ (/r‘ℂfld) = (/r‘ℂfld) | |
7 | cnfldmul 19752 | . . . . . . . . 9 ⊢ · = (.r‘ℂfld) | |
8 | 2, 5, 6, 7 | dvrcan1 18691 | . . . . . . . 8 ⊢ ((ℂfld ∈ Ring ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → ((𝑥(/r‘ℂfld)𝑦) · 𝑦) = 𝑥) |
9 | 1, 8 | mp3an1 1411 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → ((𝑥(/r‘ℂfld)𝑦) · 𝑦) = 𝑥) |
10 | 9 | oveq1d 6665 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (((𝑥(/r‘ℂfld)𝑦) · 𝑦) / 𝑦) = (𝑥 / 𝑦)) |
11 | 2, 5, 6 | dvrcl 18686 | . . . . . . . 8 ⊢ ((ℂfld ∈ Ring ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥(/r‘ℂfld)𝑦) ∈ ℂ) |
12 | 1, 11 | mp3an1 1411 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥(/r‘ℂfld)𝑦) ∈ ℂ) |
13 | simpr 477 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑦 ∈ (ℂ ∖ {0})) | |
14 | eldifsn 4317 | . . . . . . . . 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 475 | . . . . . . 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 10807 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (((𝑥(/r‘ℂfld)𝑦) · 𝑦) / 𝑦) = (𝑥(/r‘ℂfld)𝑦)) |
19 | 10, 18 | eqtr3d 2658 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥 / 𝑦) = (𝑥(/r‘ℂfld)𝑦)) |
20 | simpl 473 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → 𝑥 ∈ ℂ) | |
21 | divval 10687 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (𝑥 / 𝑦) = (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) | |
22 | 20, 16, 17, 21 | syl3anc 1326 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥 / 𝑦) = (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) |
23 | 19, 22 | eqtr3d 2658 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥(/r‘ℂfld)𝑦) = (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) |
24 | eqid 2622 | . . . . 5 ⊢ (invr‘ℂfld) = (invr‘ℂfld) | |
25 | 2, 7, 5, 24, 6 | dvrval 18685 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥(/r‘ℂfld)𝑦) = (𝑥 · ((invr‘ℂfld)‘𝑦))) |
26 | 23, 25 | eqtr3d 2658 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) = (𝑥 · ((invr‘ℂfld)‘𝑦))) |
27 | 26 | mpt2eq3ia 6720 | . 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑥 · ((invr‘ℂfld)‘𝑦))) |
28 | df-div 10685 | . 2 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) | |
29 | 2, 7, 5, 24, 6 | dvrfval 18684 | . 2 ⊢ (/r‘ℂfld) = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑥 · ((invr‘ℂfld)‘𝑦))) |
30 | 27, 28, 29 | 3eqtr4i 2654 | 1 ⊢ / = (/r‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 {csn 4177 ‘cfv 5888 ℩crio 6610 (class class class)co 6650 ↦ cmpt2 6652 ℂcc 9934 0cc0 9936 · cmul 9941 / cdiv 10684 Ringcrg 18547 invrcinvr 18671 /rcdvr 18682 ℂfldccnfld 19746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-cmn 18195 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-cnfld 19747 |
This theorem is referenced by: cnfldinv 19777 cnsubdrglem 19797 qsssubdrg 19805 redvr 19963 cvsdiv 22932 qrngdiv 25313 |
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