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Mirrors > Home > MPE Home > Th. List > cnfldsub | Structured version Visualization version GIF version |
Description: The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.) |
Ref | Expression |
---|---|
cnfldsub | ⊢ − = (-g‘ℂfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnfldbas 19750 | . . . . 5 ⊢ ℂ = (Base‘ℂfld) | |
2 | cnfldadd 19751 | . . . . 5 ⊢ + = (+g‘ℂfld) | |
3 | eqid 2622 | . . . . 5 ⊢ (invg‘ℂfld) = (invg‘ℂfld) | |
4 | eqid 2622 | . . . . 5 ⊢ (-g‘ℂfld) = (-g‘ℂfld) | |
5 | 1, 2, 3, 4 | grpsubval 17465 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥(-g‘ℂfld)𝑦) = (𝑥 + ((invg‘ℂfld)‘𝑦))) |
6 | cnfldneg 19772 | . . . . . 6 ⊢ (𝑦 ∈ ℂ → ((invg‘ℂfld)‘𝑦) = -𝑦) | |
7 | 6 | adantl 482 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((invg‘ℂfld)‘𝑦) = -𝑦) |
8 | 7 | oveq2d 6666 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + ((invg‘ℂfld)‘𝑦)) = (𝑥 + -𝑦)) |
9 | negsub 10329 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + -𝑦) = (𝑥 − 𝑦)) | |
10 | 5, 8, 9 | 3eqtrrd 2661 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 − 𝑦) = (𝑥(-g‘ℂfld)𝑦)) |
11 | 10 | mpt2eq3ia 6720 | . 2 ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 − 𝑦)) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥(-g‘ℂfld)𝑦)) |
12 | subf 10283 | . . . 4 ⊢ − :(ℂ × ℂ)⟶ℂ | |
13 | ffn 6045 | . . . 4 ⊢ ( − :(ℂ × ℂ)⟶ℂ → − Fn (ℂ × ℂ)) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ − Fn (ℂ × ℂ) |
15 | fnov 6768 | . . 3 ⊢ ( − Fn (ℂ × ℂ) ↔ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 − 𝑦))) | |
16 | 14, 15 | mpbi 220 | . 2 ⊢ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 − 𝑦)) |
17 | cnring 19768 | . . . . 5 ⊢ ℂfld ∈ Ring | |
18 | ringgrp 18552 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Grp) | |
19 | 17, 18 | ax-mp 5 | . . . 4 ⊢ ℂfld ∈ Grp |
20 | 1, 4 | grpsubf 17494 | . . . 4 ⊢ (ℂfld ∈ Grp → (-g‘ℂfld):(ℂ × ℂ)⟶ℂ) |
21 | ffn 6045 | . . . 4 ⊢ ((-g‘ℂfld):(ℂ × ℂ)⟶ℂ → (-g‘ℂfld) Fn (ℂ × ℂ)) | |
22 | 19, 20, 21 | mp2b 10 | . . 3 ⊢ (-g‘ℂfld) Fn (ℂ × ℂ) |
23 | fnov 6768 | . . 3 ⊢ ((-g‘ℂfld) Fn (ℂ × ℂ) ↔ (-g‘ℂfld) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥(-g‘ℂfld)𝑦))) | |
24 | 22, 23 | mpbi 220 | . 2 ⊢ (-g‘ℂfld) = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥(-g‘ℂfld)𝑦)) |
25 | 11, 16, 24 | 3eqtr4i 2654 | 1 ⊢ − = (-g‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∈ wcel 1990 × cxp 5112 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ℂcc 9934 + caddc 9939 − cmin 10266 -cneg 10267 Grpcgrp 17422 invgcminusg 17423 -gcsg 17424 Ringcrg 18547 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-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-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-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-sbg 17427 df-cmn 18195 df-mgp 18490 df-ring 18549 df-cring 18550 df-cnfld 19747 |
This theorem is referenced by: zndvds 19898 resubgval 19955 cnngp 22583 cnfldtgp 22672 clmsub 22880 clmsubcl 22886 cnindmet 22962 qqhucn 30036 zringsubgval 42183 |
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